Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry

Abstract

Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. The key parameter in helical symmetry is the step or pitch, the magnitude of the translation after rotating one full turn around the symmetry axis. In this article we study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular pipe, with respectively no-slip and no-penetration boundary conditions, as the step approaches infinity. We show that, as the step becomes large, the three-dimensional helical flow approaches a planar flow, which is governed by the so-called two-and-half Navier–Stokes and Euler equations, respectively.

1 Introduction

The helical groups are a family of one-dimensional subgroups of the rigid motions of three-dimensional Euclidean space consisting of simultaneous rotation around an axis and translation along the same axis, for which the ratio of angular rotation to translation is kept fixed. Each helical group is characterized by a parameter $\sigma \in \mathbb {R}\setminus \{0\}$, which we call the step or pitch, defined as the translation displacement along the symmetry axis after one full clockwise turn around the axis. The incompressible Navier–Stokes and Euler equations are covariant under the action of the helical group. Helically-symmetric or, simply, “helical” flows represent a physically interesting class of fluid motions, which interpolate between two-dimensional flows and axisymmetric flows, see for instance [2]. Indeed, the helical groups lie between rigid translations in one direction, associated with 2D flows, and rotation around a fixed axis, associated with axisymmetric flows. These regimes correspond to formally taking the limits $\sigma \rightarrow \infty$ and $\sigma \rightarrow 0$, respectively. The main goal of this work is to examine the precise nature of the limit $\sigma \rightarrow \infty$ for helical flows, in the case of viscous and inviscid incompressible flows in a circular pipe satisfying, respectively, no-slip and no-penetration boundary conditions. The limit $\sigma \rightarrow 0$ is more technical and, in some sense, less interesting, as we expect that helical flows will converge in the limit to axisymmetric, planar flows, a trivial special case of axisymmetric flows. In fact, periodicity in this case implies asymptotically high-frequency oscillations, with weak averaging in the vertical direction. The analysis of the limit $\sigma \rightarrow 0$ is closely related to that in some of the thin domain literature, particularly the special case referred to as PD, or periodic Dirichlet (see [8] for more details.) We reserve to study the limit $\sigma \rightarrow 0$ in future work.

We begin by recalling the known mathematical results concerning helical flows. As it is the case of two-dimensional flows and axisymmetric flows in cylindrical domains bounded away from the axis of symmetry viscous incompressible helical flows are globally well posed. This result was proved by Mahalov et al. in [12]. In fact, for the case of a circular pipe they established both global existence of a weak helical solution with initial data in $L^2$, and global existence and uniqueness of a strong solution with initial data in the Sobolev space $H^1$. (For a discussion about uniqueness of weak solutions, within the class of all Leray–Hopf weak solutions of the three-dimensional Navier–Stokes with helical initial data, see [1].) The situation is different, and rather interesting, in the case of ideal fluid governed by the Euler equations, see [5, 6]. As a matter of fact, an additional geometric condition is imposed on inviscid flows, akin to assuming no swirl in the axisymmetric setting [13, 15], which we call no helical swirl or no helical stretching. Under this condition, Ettinger and Titi [6] showed global existence and uniqueness of weak solutions in an appropriate vorticity-stream function formulation. This formulation can be used, because, even for finite $\sigma$, the flow is essentially two-dimensional, in the sense that it is completely determined by the dynamics of the first two components of the velocity field restricted to any cross section of the pipe.

The main result of this work is a convergence result of helical flows to certain flows, the dynamics of which is two dimensional. For this reason, we will call such limits planar flows, even though the velocity field can still have three non-zero components. More precisely, we show that, in the limit $\sigma \rightarrow \infty$, helical flows converge, respectively, to so-called $2$ and $1/2$ dimensional flows in the viscous case, and to 2D Euler flows in the inviscid case. These results are established by first obtaining a set of symmetry-reduced equations equivalent to the original fluid equations, at least for regular flows. The unknowns in these equations are fields on a cross section of the pipe and, hence, depend on two spatial variables only. Convergence is then investigated via energy methods and compactness arguments. For the Navier–Stokes equations, energy estimates are sufficient to pass to the limit and give us a rate of convergence of order $1/\sqrt{\sigma }$ in the energy norm.

One special difficulty in the viscous case is the way in which the divergence-free condition and the symmetry reduction interact when we vary $\sigma$. To be more precise, the symmetry reduction amounts to the fact that a helical vector field is entirely determined by its trace on a horizontal slice, say $D = \{x_1^2 + x_2^2 < 1, x_3 = 0\}$, the trace being a three-component vector field in the plane. For a given $\sigma >0$ all three-component fields in $D$ may be extended in a unique way to helical vector fields in $D \times (0,\sigma )$. However, the resulting extension will not be divergence-free unless the original field in the slice satisfies a certain $\sigma$-dependent condition. In other words, after symmetry reduction, problems with different $\sigma$ reside in different function spaces, even if their physical domain $D$ is the same. This difficulty is bypassed in the inviscid case with the use of a stream function, under the “no helical swirl” condition.

The remainder of this article is divided into four sections. In Sect. 2, we fix notation and derive an equivalent formulation of helical symmetry for functions and vector fields. In Sect. 3, we perform the symmetry reduction on the Navier–Stokes equations. In Sect. 4 we study the limit $\sigma \rightarrow \infty$ for the viscous case, while in Sect. 5 we discuss the case of the Euler equations.

2 Preliminaries and Symmetry Reduction

We begin by recalling some standard notation for function spaces that will appear throughout the paper. If $\Omega$ is a domain in $\mathbb {R}^d$, we denote by $H^k(\Omega ), k\in \mathbb {N}$, the standard $L^2$-based Sobolev spaces:

$$\begin{aligned} H^k(\Omega )= \left\{ f:\Omega \rightarrow \mathbb {R}\ ; \ f, \partial ^\alpha f \in L^2(\Omega ), |\alpha |\le k \right\} , \end{aligned}$$

where we employed the usual multi-index notation for derivatives, which are interpreted in the weak sense, while $W^{k,p}(\Omega )$ denotes $L^p$-based Sobolev spaces. By abuse of notation, if $\mathbf {u}:\Omega \rightarrow \mathbb {R}^d$ is a vector field, we will often write $\mathbf {u}\in H^k(\Omega )$ for $u\in (H^k(\Omega ))^d$, and we will drop the explicit dependence on the domain $\Omega$ when no confusion can arise. $H^1_0(\Omega )$ will denote the subspace of $H^1(\Omega )$ of functions with zero trace at the boundary $\partial \Omega$. If $\Omega$ is an unbounded domain, $L^p_{\text {loc}}(\Omega )$ is the space of functions with $p$-th integrable power on each bounded open subset of $\Omega$. Lastly, we denote Hölder spaces by $C^\alpha (\Omega ), \alpha \in \mathbb {R}_+$. Later in the paper, we will introduce other spaces adapted to the symmetry and geometry of the problem. Throughout, $(,)$ will denote the standard $L^2$ inner product.

One tool that will be used repeatedly in the analysis is the following interpolation inequality in two space dimensions, the so-called Ladyzhenskaya inequality. If $D$ is a smooth domain in $\mathbb {R}^2$ and $f \in H^1_0(D)$, then

$$\begin{aligned} \Vert f\Vert _{L^4(D)}^4 \le 2 \Vert f\Vert _{L^2(D)}^2\Vert \nabla f\Vert _{L^2(D)}^2. \end{aligned}$$

(2.1)

This inequality follows immediately from Lemma 1 on page 8 of [10].

Let $\Omega = \{x=(x_1,x_2,x_3)\in \mathbb {R}^3 \;|\; x_1^2+x_2^2\le 1\}= D\times \mathbb {R}$ be the infinite pipe parallel to the $x_3$ axis with circular cross-section D.

We consider the initial-boundary-value problem for the incompressible Navier–Stokes (NSE) and Euler equations (EE) in $\Omega$. We recall the notion of helically symmetric solutions of these equations, studied in [6, 12].

We first give the definition of a helical vector field and a helical (scalar) function. We denote a point in $\mathbb {R}^3$ by $x=(x_1,x_2,x_3)$ in Cartesian coordinates. Given a non-zero number $\sigma \in \mathbb {R}$, we define the action of the helical group of transformations $G_\sigma$ on $\mathbb {R}^3$ by:

$$\begin{aligned} S(\rho ) (x) = \left( \begin{matrix} x_1\, \cos \rho + x_2 \sin \rho \\ -x_1 \, \sin \rho + x_2 \, \cos \rho \\ x_3 + \frac{\sigma }{2\pi } \rho \end{matrix} \right) , \qquad \rho \in \mathbb {R}, \end{aligned}$$

that is, a rotation around the $x_3$ axis with simultaneous translation along the $x_3$ axis. $G_\sigma$ is uniquely determined by $\sigma$, which we will call the step (or pitch). Invariant curves for the action of the helical group $G_\sigma$ are helices having the $x_3$ axis as axis of symmetry. The cylinder $\Omega$ is an invariant set for the action of $G_\sigma$ for all $\sigma$. A change of sign in $\sigma$ corresponds to switching the orientation of the helices preserved by the group action from right-handed to left-handed. Without loss of generality, we will restrict our attention to the case of $\sigma >0$.

We will say that the smooth function $f(x)$ is helically symmetric, or simply helical, if $f$ is invariant under the action of $G_\sigma$, i.e., $f(S(\rho ) x) = f(x), \forall \rho \in \mathbb {R}$. Similarly, we say that the smooth vector field $\mathbf {u}(x)$ is helically symmetric, or simply helical, if it is covariant with respect to the action of $G_\sigma$, i.e., $M(\rho ) \mathbf {u}(x) = \mathbf {u}(S(\rho ) x)$ for all $\rho \in \mathbb {R}$, where

$$\begin{aligned} M(\rho ) : = \left[ \begin{array}{ccc} \cos \rho &{} \sin \rho &{} 0 \\ \\ -\sin \rho &{} \cos \rho &{} 0 \\ \\ 0 &{} 0 &{} 1 \end{array}\right] . \end{aligned}$$

(2.2)

We find it convenient to give an alternative definition of helical symmetry as follows. We re-write a vector field $\mathbf {u}(x)=(u^1,u^2,u^3)(x_1,x_2,x_3)$ with respect to the moving orthonormal frame associated to standard cylindrical coordinates $(r,\theta ,z)$,

$$\begin{aligned} \mathbf{e}_\mathbf{r}=(\cos \theta ,\sin \theta ,0), \;\;\; \mathbf{e}_{\varvec{\theta }}=(-\sin \theta ,\cos \theta ,0), \;\;\; \mathbf{e}_\mathbf{z}=(0,0,1), \end{aligned}$$

as:

$$\begin{aligned} \mathbf {u} = u_r \mathbf{e}_\mathbf{r} + u_{\theta } \mathbf{e}_{\varvec{\theta }} + u_z \mathbf{e}_\mathbf{z}, \end{aligned}$$

where $u_r, u_{\theta }, u_z$ are functions of $(r,\theta ,z)$. We introduce two new independent variables in place of $\theta$ and $z$:

$$\begin{aligned} \eta : =\frac{\sigma }{2\pi } \theta + z,\ \xi : = \frac{\sigma }{2\pi } \theta - z. \end{aligned}$$

(2.3)

As shown in [6] for instance, a (smooth) function $p=p(r,\theta ,z)$ is a helical function if and only if, when expressed in the $(r,\xi ,\eta )$ variables, it is independent of $\xi : \, p=q(r,\frac{\sigma }{2\pi }\theta + z)$, for some $q=q(r,\eta )$. Indeed, by definition $f$ is helical if and only if $f(S(\rho ) x)$ is actually independent of $\rho$, being equal to $f(x)$. Then, by the Chain Rule $d f(S(\rho ) x)/d \rho =0$ is equivalent to $d f(r,\xi , \eta )/d\xi =0$.

Similarly, a (smooth) vector field $\mathbf {u}$ is helical if and only if there exist $v_r, v_{\theta }, v_z$, functions of $(r,\eta )$ such that $u_r = v_r(r,\frac{\sigma }{2\pi } \theta + z), u_{\theta } = v_{\theta }(r,\frac{\sigma }{2\pi }\theta + z), u_z = v_z(r, \frac{\sigma }{2\pi } \theta + z)$.

We note that a vector field $\mathbf {u}$ is invariant under the action of $G_\sigma$ for all $\sigma \ne 0$ if and only if $v_r, v_{\theta }, v_z$ are functions of $r$ only. In particular, planar, circularly symmetric flows, that is flows for which $v_r=v_z\equiv 0$ and $v_\theta$ is a radial function, are a (very) special case of helical flows.

The change of variables $(x_1,x_2,x_3) \mapsto (r,\xi ,\eta )$ introduced above has often been used to characterize helical symmetry, and, in fact, it does provide a simple, geometrically elegant description of invariance for both scalar functions and vector fields. However, to obtain estimates on solutions of the fluid equations, we find that an alternative characterization actually simplifies calculations, by avoiding moving frames. As a matter of fact, we show in the following proposition that sufficiently smooth functions and fields with helical symmetry are essentially two dimensional, in the sense that they are uniquely determined by their trace on any “slice” $\Omega \cap \{z=\text {constant}\}$, which can be canonically identified with the unit disk $D\subset \mathbb {R}^2$.

Below we will make use of the following notation, where we employ Cartesian coordinates and frames. Given $y=(y_1,y_2)$ we let $y^{\perp } = (-y_2,y_1)$ and we set

$$\begin{aligned} E \equiv y^{\perp }\cdot \nabla _y. \end{aligned}$$

(2.4)

The operator $E$ is scalar and will be applied component wise to vectors as well in what follows.

We also use the notation $\mathbf {V}_H=(V^1,V^2,0)$ for the horizontal component of the vector $\mathbf {V}=(V^1,V^2,V^3)$, and we denote the vector $(- V^2,V^1,0)$ by $\mathbf {V}_H^{\perp }$.

Proposition 2.1

Let $\mathbf {u} = \mathbf {u}(x)$ be a smooth helical vector field and let $p=p(x)$ be a smooth helical function, where $x=(x_1,x_2,x_3)$. Then there exist unique $\mathbf {w}=(w^1,w^2,w^3) =(w^1,w^2,w^3)(y_1,y_2)$ and $q=q(y_1,y_2)$ such that

$$\begin{aligned} \mathbf {u}(x) = M(2\pi x_3/\sigma )\mathbf {w}(y(x)), \;\;\; p=p(x)=q(y(x)), \end{aligned}$$

(2.5)

with $M(\rho )$ given in (2.2), and

$$\begin{aligned} y(x) = \left[ \begin{array}{l} y_1 \\ \\ y_2 \end{array}\right] = \left[ \begin{array}{cc} \cos (2\pi x_3/\sigma ) &{} -\sin (2\pi x_3/\sigma ) \\ \\ \sin (2\pi x_3/\sigma ) &{} \cos (2\pi x_3/\sigma ) \end{array}\right] \left[ \begin{array}{l} x_1\\ \\ x_2 \end{array}\right] . \end{aligned}$$

(2.6)

Conversely, if $\mathbf {u}$ and $p$ are defined through (2.5) for some $\mathbf {w}=\mathbf {w}(y_1,y_2), q=q(y_1,y_2)$, then $\mathbf {u}$ is a helical vector field and $p$ is a helical scalar function.

We omit the proof, which is a standard application of vector calculus.

In what follows, for notational convenience we set

$$\begin{aligned} m^{\sigma }(x_3) = \left[ \begin{array}{cc} \cos (2\pi x_3/\sigma ) &{} -\sin (2\pi x_3/\sigma ) \\ \\ \sin (2\pi x_3/\sigma ) &{} \cos (2\pi x_3/\sigma ) \end{array}\right] ,\end{aligned}$$

so that

$$\begin{aligned} y(x_1,x_2,x_3) = \left[ \begin{array}{c} y_1\\ y_2 \end{array} \right] = m^{\sigma }(x_3)\left[ \begin{array}{c} x_1\\ x_2 \end{array} \right] , \end{aligned}$$

and

$$\begin{aligned} M^{\sigma } (x_3) \equiv M(2\pi x_3/\sigma ) = \left[ \begin{array}{cc} (m^{\sigma }(x_3))^T &{} 0 \\ \\ 0 &{} 1 \end{array}\right] . \end{aligned}$$

It is clear, from Proposition 2.1 above that any smooth helical flow is periodic in $x_3$, both velocity and pressure, with period the step $\sigma$. We can therefore state the initial-boundary-value problem for the Navier–Stokes equations in the fundamental domain $\Omega ^\sigma : = D \times (0,\sigma )$:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t \mathbf {u} + (\mathbf {u}\cdot \nabla )\mathbf {u} = -\nabla p + \nu \Delta \mathbf {u} +\mathbf {f}, &{} \quad \text{ in } (0,+\infty ) \times \Omega ^\sigma ; \\ \text{ div }\, \mathbf {u} = 0, &{} \quad \text{ in } [0,+\infty ) \times \Omega ^\sigma ;\\ \mathbf {u}(t,x',x_3) = 0, &{} \quad \text{ for } t \in [0,+\infty ), \quad |x'|=1, \; 0\le x_3\le \sigma ;\\ \mathbf {u}(t,x',x_3) =\mathbf {u}(t,x',x_3+\sigma ) &{} \quad \text{ for } t \in [0,+\infty ), \quad x'\in D;\\ p(t,x',x_3) =p(t,x',x_3+\sigma ) &{} \quad \text{ for } t \in [0,+\infty ), \quad x'\in D;\\ \mathbf {u}(0,x) = \mathbf {u}_0, &{} \quad x\in \Omega ^\sigma , \end{array}\right. \end{aligned}$$

(2.7)

where we set $x'=(x_1,x_2)$, so that $x= (x',x_3)$.

The Euler equations are formally obtained by setting $\nu =0$ above and by replacing the no-slip boundary condition $\mathbf {u}\vert _{\partial \Omega ^\sigma } =0$ with the no-penetration condition $\mathbf {u}\cdot x' =0$ on $\partial \Omega ^\sigma$. We discuss Euler solutions in Sect. 5.

In what follows, for simplicity we set any body forcing $\mathbf {f}\equiv 0$, and take the viscosity coefficient $\nu =1$, as we do not contend ourselves with the vanishing viscosity limit in this work. We plan to study the interplay between the limits $\nu \rightarrow 0$ and $\sigma \rightarrow 0$ in future work.

We denote by $C^\alpha _{per}(\overline{\Omega ^\sigma })$ the subspace of $C^\alpha (\overline{\Omega }), \alpha \in \mathbb {R}_+$ of functions that are $\sigma$-periodic in $x_3$, and by $C^{\infty }_{c,per}(\Omega ^{\sigma })$ the space of smooth functions which are $\sigma$-periodic in $x_3$ and compactly supported in $D$ for each fixed $x_3 \in [0,\sigma ]$. We also denote by $H^1_{0,per}(\Omega ^{\sigma })$ the closure of $C^{\infty }_{c,per}(\Omega ^{\sigma })$ in $H^1(\Omega ^{\sigma })$, and by $H^{-1}_{per}(\Omega ^\sigma )$ its dual. We note that the closure of the subspace of $C^{\infty }_{c,per}(\Omega ^{\sigma })$ of divergence-free vector field is the subspace $\{ \mathbf {u}\in H^1_{0,per}(\Omega ^{\sigma }) \; | \; \text{ div }\, u =0\}$, where derivatives are taken in the weak sense.

In the remainder of the paper we will consider solutions to (2.7) and the corresponding inviscid system (5.1) with initial data $u_0$ of limited regularity. More specifically, $u_0$ will be taken in $H^1_{0,per}(\Omega ^{\sigma })$ for Navier–Stokes and in $H^1_{per}(\Omega ^\sigma )$ with initial vorticity $\text {curl}\, u_0\in L^\infty (\Omega ^\sigma )$ for Euler. We now briefly discuss helical symmetry in this context.

Definition 2.1

Let $p \in H^1_{per}(\Omega ^\sigma )$. We say that $p$ has helical symmetry if there exists a sequence of smooth, helical functions $p_n$ such that $p=\lim _{n\rightarrow \infty } p_n$ in $H^1_{per}(\Omega ^\sigma )$. Similarly, we say that a vector field $\mathbf {u}$ in $H^{1}_{per}(\Omega ^{\sigma })^3$ has helical symmetry if $\mathbf {u}$ is a strong limit in $H^{1}_{per}(\Omega ^{\sigma })^3$ of a sequence of smooth, helical vector fields $\mathbf {u}_n$.

We next show that the characterization of helical symmetry given in Proposition 2.1 carries over to functions and vector fields in $H^1$.

Proposition 2.2

Let $\mathbf {u}\in (H^1_{per}(\Omega ^\sigma ))^3, p\in H^1_{per}(\Omega ^\sigma )$ be, respectively, a helical vector field and a helical function. Then, there exist a unique $\mathbf {w}\in H^1(D)^3$ and $q\in H^1(D)$, where $D$ is the unit disk in $\mathbb {R}^2$, such that

$$\begin{aligned} \begin{aligned}&\mathbf {u}(x) = M^\sigma (x_3) \mathbf {w}(m^\sigma (x_3) x'),\\&p(x) = q(m^\sigma (x_3) x')), \end{aligned} \qquad \text {a. e. } x'\in D, \quad \forall \; 0\le x_3\le \sigma . \end{aligned}$$

(2.8)

Conversely, given $\mathbf {w}\in H^1(D)^3$ and $q\in H^1(D)$, if $\mathbf {u}$ and $p$ are defined through (2.8), then $\mathbf {u}\in (H^1_{per}(\Omega ^\sigma ))^3, p \in (H^1_{per}(\Omega ^\sigma ))$, and they have helical symmetry.

Proof

We only consider the case of a helical vector field $\mathbf {u}$. The case of a helical function is similar and simpler. By definition, there exist helical vector fields $\mathbf {u}_n \in C^\infty (\bar{\Omega }^\sigma )$ such that $u_n \rightarrow u$ strongly in $H^1_{per}(\Omega ^\sigma )$. By Proposition 2.1, for each $\mathbf {u}_n$ there exists a unique, smooth $\mathbf {w}_n$ such that $\mathbf {w}_n(x')= (M^\sigma (x_3))^T \mathbf {u}_n((m^\sigma (x_3))^T x')$, for all $x'=(x_1,x_2)\in D$. Therefore, the expression on the right-hand side is independent of $x_3$ and $\nabla _x (M^\sigma (x_3))^T \mathbf {u}((m^\sigma (x_3))^T x') = (\nabla _{x'} \mathbf {w}(x'),0)$. If we define

$$\begin{aligned} \mathbf {w}(x',x_3) := (M^\sigma (x_3))^T \mathbf {u}((m^\sigma (x_3))^T x'), \end{aligned}$$

then $\partial _{x_3} \mathbf {w}(x',x_3)=0$ in weak sense, since it is true for $\mathbf {w}_n$ and $\nabla _x\mathbf {u}_n\rightarrow \nabla \mathbf {u}$ strongly in $L^2(\Omega ^\sigma )$. Consequently, $\mathbf {w}$ is independent of $x_3$ for almost all $x'\in D$ (functions with vanishing weak derivatives are constant, see e.g. [11, Theorem 6.11]) and $\mathbf {w}\in H^1(D)$. Furthermore,

$$\begin{aligned} \Vert \mathbf {w}_n -\mathbf {w}\Vert _{H^1(D)}\le C \sqrt{\sigma } \, \Vert \mathbf {u}_n-\mathbf {u}\Vert _{H^1(\Omega ^\sigma )}, \end{aligned}$$

by a simple change of variables, so that $\mathbf {w}_n\rightarrow \mathbf {w}$ strongly in $H^1(D)$. The converse statement is a direct consequence of (2.8).$\square$

Remark 2.1

The proof of Proposition 2.2 shows that if $\mathbf {u}\in H^m(\Omega ^\sigma ), m\in \mathbb {N}$, then $\mathbf {w}\in H^m(D)$ and the $H^m$ norm of $\mathbf {w}$ on $D$ is bounded by the $H^m$ norm of $\mathbf {u}$ on $\Omega ^\sigma$ with constants that depend on $\sigma$. The same result holds in $L^p$-Sobolev spaces $W^{m,p}_{per}(\Omega ^\sigma )$ for $1\le p <\infty$. These spaces are defined in a manner totally analogous to $H^m_{per}(\Omega ^\sigma )$.

We next recall the notion of weak and strong Navier–Stokes solutions. By a classical solution of (2.7) on the time interval $[0,T]$, we mean a vector field $\mathbf {u}\in C^1([0,T];C^2(\overline{\Omega ^\sigma }))$, together with a function $p \in C^1([0,T),C^1(\Omega _\sigma ))$ such that the equations, and the initial and boundary conditions are met pointwise in $t$ and $x$. By a weak solution on the time interval $[0,T)$, we mean a divergence-free vector field $\mathbf {u}:[0,T) \times \Omega ^\sigma \rightarrow \mathbb {R}^3$ such that $\mathbf {u}\in C_{w}([0,T);L^2(\Omega ^\sigma ))\cap L^2((0,T);H^1_{0,per}(\Omega ^\sigma ))$ and $\partial _t \mathbf {u}\in L^1((0,T),H^{-1}_{per}(\Omega ^\sigma ))$, satisfying the equations in the sense of distributions and the initial condition $\mathbf {u}(0) = \mathbf {u}_0\in L^2(\Omega ^\sigma )$. Here, $C_w([0,T);L^2)$ is the space of all functions of $t$ with values in $L^2$ that are continuous w.r.t. the weak topology on $L^2$. We remark that weak solutions satisfy the Dirichlet (no-slip) boundary conditions at least in trace sense on the boundary for almost all $0<t<T$. By a strong solution we mean a weak solution that satisfies in addition $u \in L^\infty ([0,T);H^1_{0,per}(\Omega ^\sigma ))\cap L^2((0,T);H^2_{per}(\Omega ^\sigma )\cap H^1_{0,per}(\Omega ^\sigma ))$ and the condition $\mathbf {u}_0\in H^1_{0,per}(\Omega ^\sigma ).$ It then follows that there exists an associated pressure function $p\in L^2((0,T); H^1(\Omega ^\sigma ))$. A strong helical solution will denote a strong solution that is a helical field in the sense of Definition 2.1. We recall that any strong solution of the Navier–Stokes equations is unique and smooth for $t>0$ (see e.g., [16, Theorem 1.8.2]). Hence, strong solutions are actually classical solutions on any time interval $[\delta ,T], \delta >0$. It was shown in [12, Theorem 3.4] that weak solutions of (2.7) with helical symmetry are unique, global in time, and agree with a strong solution, if the initial data belongs to $H^1_{0,per}(\Omega ^\sigma )$ and the associated pressure $p$ is also a helical function. (See also [1] for more elaborate discussion regarding this matter.)

3 Symmetry Reduction for the Navier–Stokes Equations

In this section we derive a set of symmetry-reduced equations that completely capture the dynamics of the original system under the hypothesis of helical symmetry.

We begin by deriving the symmetry-reduced system under the hypothesis that $(\mathbf {u}, p)$ are classical solutions of (2.7) and have helical symmetry. Let $\mathbf {w}=\mathbf {w}(t,y_1,y_2)$ be given in terms of $\mathbf {u}$ by Proposition 2.1. We will derive from the Navier–Stokes system the equations satisfied by $\mathbf {w}$. Smoothness of $\mathbf {u}$ and $\mathbf {w}$ justifies all the algebraic manipulations. For ease of notation, in this proof we write $M^T$ for $[(M^\sigma )(x_3)]^T$. We multiply the momentum equation in (2.7) by $M^T$ and identify each term in the resulting expression as follows to obtain:

$$\begin{aligned} M^T \partial _t \mathbf {u}&= \partial _t \mathbf {w}, \end{aligned}$$

(3.1a)

$$\begin{aligned} M^T[(\mathbf {u}\cdot \nabla _x)\mathbf {u}]&= (\mathbf {w}_H \cdot \nabla _y)\mathbf {w} + \left( \frac{2\pi }{\sigma }\right) w^3 E\mathbf {w} - \left( \frac{2\pi }{\sigma }\right) w^3 \mathbf {w}_H^{\perp }, \end{aligned}$$

(3.1b)

$$\begin{aligned} M^T\nabla _x p&= (\nabla _y q)_H + \left( \frac{2\pi }{\sigma }\right) Eq\mathbf{e}_\mathbf{3}, \end{aligned}$$

(3.1c)

$$\begin{aligned} M^T\Delta _x\mathbf {u}&= \Delta _y\mathbf {w} + \left( \frac{4\pi ^2}{\sigma ^2}\right) [E^2\mathbf {w} - 2E\mathbf {w}_H^{\perp } - \mathbf {w}_H], \end{aligned}$$

(3.1d)

where $E$ is the operator defined in (2.4). We similarly perform the symmetry reduction on the incompressibility condition for $\mathbf {u}$ to obtain

$$\begin{aligned} \mathrm {div}_x \mathbf {u} = \mathrm {div}_y \mathbf {w}_H + \left( \frac{2\pi }{\sigma }\right) Ew^3. \end{aligned}$$

(3.2)

Therefore, we find that $\mathbf {w}$ and $q$ satisfy the following initial-boundary-value problem:

$$\begin{aligned}&\partial _t \mathbf {w} + (\mathbf {w}_H\cdot \nabla _y)\mathbf {w} + \displaystyle {\frac{2\pi }{\sigma }}w^3[E\mathbf {w} - \mathbf {w}_H^{\perp }] = -(\nabla _y q)_H \nonumber \\&\qquad \qquad -\displaystyle {\frac{2\pi }{\sigma }}Eq\mathbf{e}_\mathbf{3} +\Delta _y \mathbf {w} + \displaystyle {\frac{4\pi ^2}{\sigma ^2}} [E^2\mathbf {w} - 2 E \mathbf {w}_H^{\perp } - \mathbf {w}_H], \end{aligned}$$

(3.3a)

$$\begin{aligned}&\mathrm {div}_y \mathbf {w}_H + \displaystyle {\frac{2\pi }{\sigma }} E w^3 = 0, \qquad \qquad \qquad t>0, \quad y \in D, \end{aligned}$$

(3.3b)

$$\begin{aligned}&\mathbf {w}(t,y) = 0, \qquad \qquad \qquad \qquad \qquad \qquad \; t>0, \quad |y|=1, \end{aligned}$$

(3.3c)

$$\begin{aligned}&\mathbf {w}(0,y) = \mathbf {w}_0(y), \qquad \quad \qquad \qquad \qquad \; y \in D, \end{aligned}$$

(3.3d)

where $\mathbf {w}_0$ is related to $\mathbf {u}_0$ via (2.5).

Before giving a weak formulation of the above initial-boundary-value problem, we note that the operator $E=y^\perp \cdot \nabla _y$ is anti-selfadjoint, i.e., $E^*= -E$, since $\text{ div }_y \, y^\perp =0$. If we write (3.3b) as $A\, \mathbf {w}=0$, for some matrix operator $A$ with $\mathbf {w}$ considered as a column vector, it follows that $A$ and its adjoint $A^*$ are given by:

$$\begin{aligned} A := \left[ \begin{matrix} \partial _{y_1},&\partial _{y_2},&\frac{2\pi }{\sigma } E \end{matrix} \right] , \qquad A^*:= \left[ \begin{matrix} - \partial _{y_1} \\ - \partial _{y_2} \\ -\frac{2\pi }{\sigma } E \end{matrix} \right] , \end{aligned}$$

assuming natural (homogeneous Neumann) boundary conditions. It can be easily checked that the (scalar) second-order operator $A\,A^*= - \Delta _y - \frac{4\pi ^2}{\sigma ^2} E^2$ is elliptic for any $\sigma \ne 0$.

We will call a vector field $\mathbf {w}$ on $[0,T)\times D$ a weak solution of (3.3) if $\mathbf {w}\in C_{w}([0,T);L^2(D))\cap L^2((0,T);H^1_0(D)), \partial _t \mathbf {w}\in L^1((0,T);H^{-1}(D))$, $\mathbf {w}(0)=\mathbf {w}_0 \in L^2(D), \mathbf {w}$ satisfies the constraint (3.3b) in the sense of distributions, and for all (vector-valued) test functions $\varvec{\Phi }\in C^\infty _c([0,T)\times D)$ that satisfy (3.3b),

$$\begin{aligned}&\int _0^t \int _D \mathbf {w}\cdot \partial _t \varvec{\Phi }\, dy\, dt + \frac{2\pi }{\sigma } \int _0^t \int _D w^3 ( \varvec{\Phi }\cdot \mathbf {w}^\perp _H + E \varvec{\Phi }\cdot \mathbf {w})\, dy \, dt \nonumber \\&\quad + \int _0^t \int _D \Delta \varvec{\Phi }\cdot \mathbf {w}\, dy\, dt + \frac{4\pi ^2}{\sigma ^2} \int _0^t \int _D \left( E^2 \varvec{\Phi }\cdot \mathbf {w}+ 2 E\varvec{\Phi }\cdot \mathbf {w}_H^\perp \right) \, dy\, dt \nonumber \\&\quad - \frac{4\pi ^2}{\sigma ^2} \int _0^t \int _D \varvec{\Phi }\cdot \mathbf {w}_H \, dy\, dt= - \int _D \varvec{\Phi }(0) \cdot \mathbf {w}(0)\, dy. \end{aligned}$$

(3.4)

A weak solution will be called a strong solution if, in addition, $\mathbf {w}\in L^\infty ([0,T);H^1_0(D)) \cap L^2((0,T);H^2(D)\cap H^1_0(D))$ and ${\mathbf {w}}_0\in H^1_0(D).$ By interpolation then, $\mathbf {w}\in C((0,T);H^1_0(D))$ (c.f. e.g. [18, Lemma 4.8 p. 570]). By projecting the momentum equation (3.3a) onto the kernel of the operator $A$, one obtains an elliptic equation for the pressure $q$:

$$\begin{aligned} A\,A^*q = A\, \left[ (\mathbf {w}_H\cdot \nabla _y)\mathbf {w} - \frac{2\pi }{\sigma } w^3\, (E\mathbf {w} - \mathbf {w}_H^{\perp }) + \frac{4\pi ^2}{\sigma ^2} (2 E \mathbf {w}_H^{\perp } + \mathbf {w}_H) \right] , \end{aligned}$$

(3.5)

using that $E$ and $\Delta _y$ (hence $E$ and $A$) commute and the fact that $A\mathbf {w}=0$. Consequently, elliptic regularity gives that $q$ is at least in $L^1([0,T); H^1(D))$ by the regularity of strong solutions.

In the following proposition we establish the relationship between strong solutions to the Navier–Stokes system (2.7) and strong solutions of the symmetry-reduced system (3.3).

Proposition 3.1

Let $\mathbf {u}_0\in H^1_{0,per}(\Omega ^\sigma )$ be a divergence-free, helical vector field. Let $\mathbf {u}$ be the unique, strong helical solution of (2.7) on $[0,T)$, for any $T>0$, with initial condition $\mathbf {u}_0$ and associated pressure function $p$. Then, the vector function $\mathbf {w}= (w^1,w^2,w^3)$ and scalar function $q$, defined through (2.8) from $\mathbf {u}$ and $p$, give a strong solution of the reduced system (3.3).

Conversely, let $\mathbf {w}$ be a strong solution of (3.3) and associated pressure $q$. Then, if $\mathbf {u}$ and $p$ are defined from $\mathbf {w}$ and $q$ via (2.8), $\mathbf {u}$ is a strong helical solution of (2.7). In particular, strong solutions of (3.3) are unique.

Proof

By Definition 2.1, there exists a sequence of smooth, helical functions $\mathbf {u}_{0,n}$ on $\Omega ^\sigma$ such that $\mathbf {u}_{0,n} \rightarrow \mathbf {u}_0$ strongly in $H^1_{0,per}(\Omega ^\sigma )$. Let $\mathbf {u}_n$ be the unique, classical helical solution of (2.7) with initial data $\mathbf {u}_{0,n}$, and pressure $p_n$. (We recall that such solutions exist globally in time by the results in [12].) The sequence $\{\mathbf {u}_n\}$ is uniformly bounded in $L^\infty ([0,T);H^1_{0,per}(\Omega ^\sigma ))\cap L^2((0,T);H^2_{per}(\Omega ^\sigma )\cap H^1_{0,per}(\Omega ^\sigma ))$ and $\{\partial _t \mathbf {u}_n\}$ is uniformly bounded in $L^1([0,T);H^{-1}_{per}(\Omega ^\sigma ))$. Therefore, by interpolation and Rellich’s theorem, there exists a subsequence converging strongly in $H^{-\epsilon }([0,T);H^1_{0,per}(\Omega ^\sigma ))\cap L^2((0,T);H^{1-\epsilon }_{per}(\Omega ^\sigma ))$, for all $\epsilon >0$, weakly in $L^2((0,T);H^2(\Omega ^\sigma ))$, and weakly-$*$ in $L^\infty ([0,T);H^1_{0,per}(\Omega ^\sigma ))$, such that $\partial _t \mathbf {u}_n$ converges weakly in $L^1((0,T);H^{-1}(D))$. The limit $\mathbf {u}$ is then a weak solution of (2.7) with initial data $\mathbf {u}_0$ (by arguments similar to those showing existence of Leray–Hopf weak solutions, cf. [18, Theorem 5.9, Chapter 17]. )

Since weak solutions agree with strong solutions as long as the latter exists, we must have that $\mathbf {u}$ is the unique, strong helical solution of (2.7) with initial data $\mathbf {u}_0$. Hence, the whole sequence $\{\mathbf {u}_m\}$ converges to $\mathbf {u}$ by uniqueness of the limit. A similar argument gives convergence of $p_n$ to $p$ in $L^1((0,T);H^1(D))$ .

Let now $\mathbf {w}_n$ be associated to $\mathbf {u}_n$ by (2.5). Then, $\mathbf {w}_n$ is a classical solution of (3.3), with associated pressure $q_n$ given by (2.5) in terms of $p_n$, by the calculations at the beginning of this section. Furthermore, the proof of Proposition 2.1 implies that all Sobolev norms of $\mathbf {w}_n$ and $q_n$ are bounded by the corresponding Sobolev norms of $\mathbf {u}_n$ with constants depending on $\sigma$. Hence, the sequence $\{\mathbf {w}_n\}$ is uniformly bounded in $L^\infty ([0,T);H^1_{0}(D))\cap L^2((0,T);H^2(D)\cap H^1_{0}(D))$. From the equations, it follows that $\partial _t \mathbf {w}_n$ is uniformly bounded in $L^1((0,T);H^{-1}(D))$. Hence, by interpolation and Rellich’s theorem there exists a subsequence converging strongly in $H^{-\epsilon }([0,T);H^1_0(D))\cap L^2((0,T);H^{1-\epsilon }(D))$, for all $\epsilon >0$, weakly in $L^2((0,T);H^2(D))$, and weakly-$*$ in $L^\infty ([0,T);H^1_0(D))$ to a weak solution $\mathbf {w}$ of the symmetry-reduced system (3.3). Since $\mathbf {w}\in L^\infty ([0,T);H^1_{0}(D))\cap L^2((0,T);H^2(D)\cap H^1_{0}(D)), \mathbf {w}$ is a strong solution of the reduced system. Also, by refining the subsequence if needed, we can assume that $\{q_n\}$ converges weakly in $L^1((0,T);H^1(D))$. Furthermore, the convergence of $\mathbf {u}_n$ to $\mathbf {u}$ implies weak convergence of the right-hand side of (3.5) in $L^1(0,T);H^{-1}(D))$ and, hence, $q$ is a weak solution of the pressure equation. Lastly, since $\mathbf {w}$ and $q$ in (2.8) are unique, given $\mathbf {u}$ and $p$, these must agree with the limits of $\mathbf {u}_n$ and $p_n$. The first half of the theorem is established.

The converse follows by similar arguments, using again the uniqueness in the relation between $\mathbf {u}, p$ with $\mathbf {w}, q$ of Proposition 2.2. Energy estimates for strong solutions of the symmetry-reduced equations are given in Propositions 4.2 and 4.3. Uniqueness of strong solutions to the reduced equations then follows from uniqueness of helical, strong solutions of the Navier–Stokes equations.$\square$

4 The Limit $\sigma \rightarrow \infty$ for the Navier–Stokes System

The purpose of this section is to discuss the limit $\sigma \rightarrow \infty$ for helical solutions of the Navier–Stokes equations. To emphasize the dependence of the solution on the parameter $\sigma$, we will write $\mathbf {u}^\sigma$ and $p^\sigma$ for $\mathbf {u}$ and $p$.

Next, we recall that to any helical vector field $\mathbf {u}^\sigma$ in $H^1(\Omega ^\sigma )$ we can associate a three-component vector function $\mathbf {w}^\sigma$ in $H^1(D)$ by means of Proposition 2.2. The divergence-free condition on $\mathbf {u}^\sigma$ is recast as (3.3b) for $\mathbf {w}^\sigma$. In what follows, we will need to relate divergence-free vector fields in $D$ to fields satisfying the condition in (3.3b). To this end, we will exploit the following useful lemma.

Lemma 4.1

There exists a constant $C>0$ such that, for every $f \in L^2(D)$ with $\int _D f(x) \, dx = 0$, there exists a vector field $\mathbf {v} \in H^1_0(D)$ satisfying

$$\begin{aligned} \mathrm {div}_y \mathbf {v}&= f\;\;\;\text{ and }\\ \Vert \nabla \mathbf {v}\Vert _{L^2(D)}&\le C \Vert f\Vert _{L^2(D)}. \end{aligned}$$

Proof

Since $D$ is clearly star-shaped, this is a special case of Lemma III.3.1 on p 116 of [7].$\square$

We note that $\mathbf {v}$ is not uniquely determined. In fact, we can add to $\mathbf {v}$ any divergence-free vector field in $D$, satisfying the $H^1$ bound above. The vector field $\mathbf {v}$ can be made unique by assuming, for example, that it is curl free.

Next, we will state and prove several energy-type estimates for $\mathbf {w}^\sigma$. These follow from corresponding bounds for $\mathbf {u}^\sigma$ thanks to Proposition 3.1, but we derive them here keeping track of the precise dependence on the parameter $\sigma$.

Given a helical vector field $\mathbf {u}_0 \in H^1_{0,per}(\Omega ^\sigma )$, Proposition 3.1 gives a one-to-one correspondence between strong helical solutions of (2.7) and strong solutions of (3.3) with initial data $\mathbf {w}_0\in H^1_0(D)$ satisfying

$$\begin{aligned} \text{ div }_y \left[ (\mathbf {w}_0^{\sigma })_H\right] + \frac{2\pi }{\sigma }E[(w^{\sigma ,3}_0)] = 0, \end{aligned}$$

(4.1)

where $w_0^{\sigma ,3}$ refers to the third component of $\mathbf {w}_0^{\sigma }$, and $\mathbf {u}_0$ and $\mathbf {w}_0$ are related via (2.8). In particular, $\mathbf {w}\in C([0,T),H^1_0(D))$.

We remark that for any helical vector field $\mathbf {u}^\sigma _0$ for which the component along the axis of the pipe, $u^{\sigma ,3}_0$, is a radial function, the symmetry-reduced constraint on the divergence is in fact simply the divergence-free constraint in 2D for $(\mathbf {w}^\sigma _0)_H$, since in this case $E\,w^{\sigma ,3}_0\equiv 0$. In this special case, the analysis is considerably simplified. We may now state our next results, consisting of energy estimates for $\mathbf {w}^{\sigma }$. We split these into two propositions, the first one valid for all $\sigma > 0$ and the second one valid for large $\sigma$.

Proposition 4.2

Given $\sigma >0$, let $\mathbf {w}^\sigma$ be a strong solution of (3.3) on the time interval $[0,T)$. Then, for all $t\in (0,T)$, we have that

$$\begin{aligned} \begin{aligned} \int _D |\mathbf {w}_0^{\sigma }(y)|^2\,dy&= \int _D |\mathbf {w}^{\sigma }(t,y)|^2\,dy + 2\int _0^t\int _D|\nabla \mathbf {w}^{\sigma }(s,y)|^2\,dy\,ds \\&\quad +2\int _0^t\int _D \frac{4\pi ^2}{\sigma ^2} \left[ (E (w^{\sigma ,3}) )^2 + |E\mathbf {w}_H^{\sigma } - (\mathbf {w}_H^{\sigma })^{\perp }|^2 \right] \,dy\,ds. \end{aligned} \end{aligned}$$

(4.2)

Proof

We simply observe that $\mathbf {w}^{\sigma }$ has enough regularity to be a test function in the weak formulation of (3.3), so we are justified in multiplying (3.3) by $\mathbf {w}^{\sigma }$ and integrating over the domain $D$ and, subsequently, in time. This easily yields the desired identity.$\square$

Remark 4.1

The following scaling relations hold:

$$\begin{aligned} \Vert \mathbf {u}_0^{\sigma }\Vert _{L^2(\Omega ^{\sigma })}&= \sqrt{\sigma } \Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}, \end{aligned}$$

(4.3)

$$\begin{aligned} \Vert \nabla _H\mathbf {u}_0^{\sigma }\Vert _{L^2(\Omega ^{\sigma })}&\le \sqrt{\sigma } \Vert \nabla \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}, \nonumber \\ \Vert \partial _{x_3}\mathbf {u}_0^{\sigma }\Vert _{L^2(\Omega ^{\sigma })}&\le \frac{C}{\sqrt{\sigma }} \Vert \mathbf {w}_0^{\sigma }\Vert _{H^1(D)}, \end{aligned}$$

(4.4)

where $C$ is a universal constant. Indeed, a straightforward change of variables from the relation

$$\begin{aligned} \mathbf {u}_0^{\sigma }(x) = M^{\sigma }(x_3)\mathbf {w}_0^{\sigma }(m^{\sigma }(x_3) x'), \end{aligned}$$

gives by the chain rule

$$\begin{aligned} \nabla _H \mathbf {u}_0^{\sigma } = M^{\sigma }(x_3) [(D\mathbf {w}_0^{\sigma })(m^{\sigma }(x_3) x')] [m^{\sigma }(x_3)], \end{aligned}$$

and

$$\begin{aligned} \partial _{x_3}\mathbf {u}_0^{\sigma }&= \frac{2\pi }{\sigma }\left[ \partial _{\rho } M^{\sigma }(x_3) \mathbf {w}_0^{\sigma }(m^{\sigma }(x_3) x') \right. \\&\quad \left. + M^{\sigma }(x_3)[(D\mathbf {w}_0^{\sigma })(m^{\sigma }(x_3) x')][\partial _{\rho }m^{\sigma }(x_3)] x'\right] . \end{aligned}$$

Proposition 4.3

Let $1 \le \sigma < \infty ,$ and fix $T>0$. Let $\mathbf {u}^\sigma$ be a strong helical solution of (2.7) on the interval $[0,T)$. Let $\mathbf {w}^\sigma$ be the corresponding symmetry-reduced flow, which solves (3.3). Then the following hold:

• (1) There exists $C>0$, independent of $\sigma$, such that

  $$\begin{aligned} \begin{array}{l} \Vert \partial _t\mathbf {w}^{\sigma }\Vert _{L^2((0,T);H^{-1}(D))} \le C(\Vert \mathbf {w}^{\sigma }\Vert _{L^{\infty }((0,T);L^2(D))} + 1) \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2((0,T);L^2(D))} . \end{array} \end{aligned}$$

• (2) There exists $C>0$, independent of $\sigma$, such that

  $$\begin{aligned} \begin{array}{l} \Vert q^{\sigma }\Vert _{L^2((0,T);L^2(D))} \le C \left( (\Vert \mathbf {w}^{\sigma }\Vert _{L^{\infty }((0,T);L^2(D))} + 1) \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2((0,T);L^2(D))} \right) . \end{array} \end{aligned}$$

Remark 4.2

As a result of Propositions 4.2 and 4.3 it follows that

$$\begin{aligned} \Vert \mathbf {w}^{\sigma }(t)\Vert _{L^2(D)}&\le \Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}, \quad \text{ for } \text{ each } t\in [0,T], \nonumber \\ \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2((0,T);L^2(D))}&\le C\Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}, \nonumber \\ \Vert \partial _t\mathbf {w}^{\sigma }\Vert _{L^2((0,T);H^{-1}(D))}&\le C_1\Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}^2 + C_2\Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}, \nonumber \\ \Vert q^{\sigma }\Vert _{L^2((0,T);L^2(D))}&\le C_1\Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}^2 + C_2\Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}, \end{aligned}$$

(4.5)

with constants that are uniform in $\sigma$ on $[1,+\infty )$.

Proof

We begin with estimate (1). We recall that $\mathbf {u}^{\sigma }(x) = M^{\sigma }(x_3)\mathbf {w}^{\sigma }(t,m^{\sigma }(x_3) x')$, where $x' = (x_1, x_2)$. We exploit the duality between $H^{-1}$ and $H^1_0$ to compute

$$\begin{aligned} \Vert \partial _t \mathbf {w}^\sigma \Vert _{H^{-1}(D)} = \sup _{\varvec{\Psi }\ne \mathbf {0}, \varvec{\Psi } \in H^1_0(D)} \frac{\langle \varvec{\Psi }, \partial _t \mathbf {w}^\sigma \rangle }{\Vert \varvec{\Psi }\Vert _{H^1_0}}. \end{aligned}$$

To this end, we test the symmetry-reduced equations (3.3) against a (vector) test function $\varvec{\Psi } \in H^1_0(D)^3$ and relate the weak form of the reduced equations to that of the Navier–Stokes equations by constructing an appropriate test function $\varvec{\Phi }$ in $H_{0,per}^1(\Omega ^{\sigma })^3$ from$\varvec{\Psi }$, as follows:

$$\begin{aligned} \varvec{\Phi }(x) \equiv \displaystyle {\frac{1}{\sigma }}M^{\sigma }(x_3)\varvec{\Psi }(m^{\sigma }(x_3) x'). \end{aligned}$$

We recall now as well that $(m^\sigma (x_3))^{-1}y= x'=(x_1,x_2)$ by (2.6). We then observe that

$$\begin{aligned} (M^{\sigma }(x_3))^{T}\varvec{\Phi }((m^{\sigma })^{-1}(x_3) y ,x_3) = \frac{1}{\sigma }\varvec{\Psi }(y) , \end{aligned}$$

by the orthogonality of $M^\sigma$.

We have that

$$\begin{aligned}&\int _{D} \varvec{\Psi } (y) \cdot \partial _t\mathbf {w}^{\sigma }(t,y)\,dy\\&\quad = \int _D \int _0^{\sigma } (M^{\sigma }(x_3))^{T}\varvec{\Phi }((m^{\sigma })^{-1}(x_3) y,x_3) \cdot \partial _t\mathbf {w}^{\sigma }(t,y)\,dx_3\, dy\\&\quad = \int _{\Omega ^{\sigma }} \varvec{\Phi }(x^{\prime },x_3) \cdot M^{\sigma }(x_3)\partial _t\mathbf {w}^{\sigma }(t,m^{\sigma }(x_3) x')\, dx = \int _{\Omega ^{\sigma }} \varvec{\Phi }(x) \cdot \partial _t\mathbf {u}^{\sigma }(t,x)\,dx. \end{aligned}$$

To bound the $H^1$ norm of $\mathbf {u}$, we calculate the derivatives of $\varvec{\Phi }$ to find

$$\begin{aligned} \nabla _H\varvec{\Phi }(x)&= \frac{1}{\sigma } M^{\sigma }(x_3)\, [(D\Psi )(m^{\sigma }(x_3) x')] \,m^{\sigma }(x_3),\\ \partial _{x_3}\varvec{\Phi } (x)&= \frac{2\pi }{\sigma ^2}\left( \partial _{\rho } M^{\sigma }(x_3) \,\varvec{\Psi }(m^{\sigma }(x_3) x') \right. \\&\left. +\, M^{\sigma }(x_3) \, [(D\varvec{\Psi })(m^{\sigma } (x_3)\,x^{\prime })]\, [(\partial _{\rho }m^{\sigma })(x_3) x'] \right) , \end{aligned}$$

where $D$ denotes differentiation of a function with respect to its variables and $\rho$ denotes the argument of $M^\sigma$ and $m^\sigma$. A simple change of variables then gives:

$$\begin{aligned} \Vert \varvec{\Phi }\Vert _{L^2(\Omega ^{\sigma })}&= \frac{1}{\sqrt{\sigma }}\Vert \varvec{\Psi }\Vert _{L^2(D)},\\ \Vert \nabla _H\varvec{\Phi }\Vert _{L^2(\Omega ^{\sigma })}&\le C\frac{1}{\sqrt{\sigma }}\Vert \nabla \varvec{\Psi }\Vert _{L^2(D)},\\ \Vert \partial _{x_3}\varvec{\Phi }\Vert _{L^2(\Omega ^{\sigma })}&\le C\frac{1}{\sigma ^{3/2}}\Vert \nabla \varvec{\Psi }\Vert _{L^2(D)}, \end{aligned}$$

with $C$ a constant independent of $\sigma$.

Hence, since $\sigma \ge 1$,

$$\begin{aligned} \Vert \partial _t\mathbf {w}^{\sigma }(t,\cdot )\Vert _{H^{-1}(D)} \le C\frac{1}{\sqrt{\sigma }}\Vert \partial _t\mathbf {u}^{\sigma } (t,\cdot )\Vert _{H^{-1}(\Omega ^{\sigma })}. \end{aligned}$$

(4.6)

Next, we estimate the $H^{-1}$ norm of $\partial _t \mathbf {u}$ directly from equations (2.7):

$$\begin{aligned} \partial _t \mathbf {u}^{\sigma } = -\mathbb {P}[(\mathbf {u}^{\sigma }\cdot \nabla )\mathbf {u}^{\sigma }]+ \mathbb {P}[\Delta \mathbf {u}^{\sigma }], \end{aligned}$$

where $\mathbb {P}$ denotes the Leray projector onto divergence-free vector fields tangent to $\partial D$ and periodic in $x_3$ with period $\sigma$, so that

$$\begin{aligned} \Vert \partial _t \mathbf {u}^{\sigma }(t,\cdot )\Vert _{H^{-1}(\Omega ^{\sigma })}&\le C_1\Vert \text{ div } (\mathbf {u}^{\sigma }\otimes \mathbf {u}^{\sigma })(t,\cdot )\Vert _{H^{-1} (\Omega ^{\sigma })} + C_2\Vert \Delta \mathbf {u}^{\sigma }(t,\cdot )\Vert _{H^{-1} (\Omega ^{\sigma })}\\&\le C_1\Vert \mathbf {u}^{\sigma }(t,\cdot )\Vert _{L^4(\Omega ^{\sigma })}^2 + C_2\Vert \nabla \mathbf {u}^{\sigma }(t,\cdot )\Vert _{L^2(\Omega ^{\sigma })}\\&= C_1\sqrt{\sigma }\Vert \mathbf {w}^{\sigma }(t,\cdot )\Vert _{L^4(D)}^2 + C_2 \left( \sqrt{\sigma } + \frac{1}{\sqrt{\sigma }} \right) \Vert \nabla \mathbf {w}^{\sigma }(t,\cdot )\Vert _{L^2(D)}, \end{aligned}$$

using the helical symmetry expressed by relation (2.8). It follows from (4.6) and the estimates above that

$$\begin{aligned} \Vert \partial _t\mathbf {w}^{\sigma }(t,\cdot )\Vert _{H^{-1}(D)}&\le C\frac{1}{\sqrt{\sigma }}\left( \sqrt{\sigma }\Vert \mathbf {w}^{\sigma }(t,\cdot )\Vert _{L^4(D)}^2 + \left( \sqrt{\sigma } + \frac{1}{\sqrt{\sigma }} \right) \Vert \nabla \mathbf {w}^{\sigma }(t,\cdot )\Vert _{L^2(D)}\right) \\&\le C\left( \Vert \mathbf {w}^{\sigma }(t,\cdot )\Vert _{L^2(D)} \Vert \nabla \mathbf {w}^{\sigma }(t,\cdot )\Vert _{L^2(D)} + \Vert \nabla \mathbf {w}^{\sigma }(t,\cdot )\Vert _{L^2(D)}\right) , \end{aligned}$$

where we have used the two-dimensional Ladyzhenskaya inequality (2.1). This concludes the proof of estimate (1).

To prove estimate (2), we deal directly with the equations for $\mathbf {w}^{\sigma }, q^{\sigma }$. Since $p^\sigma$, and hence $q^{\sigma }$, is chosen up to a constant, we can assume that

$$\begin{aligned} \int _D q^{\sigma }(y) \, dy = 0. \end{aligned}$$

We again use duality and interpret the $L^2$-norm of $q^\sigma$ as the dual norm in $(L^2(D))^*$. Consequently, we pick an arbitrary $f \in L^2(D)$ such that $\int _D f(y)\,dy = 0$ and $\Vert f\Vert _{L^2}=1$. By virtue of Lemma 4.1 there exists $\mathbf {v}_H \in H^1_0(D)$ such that

$$\begin{aligned} \text{ div } \mathbf {v}_H&= f, \nonumber \\ \Vert \mathbf {v}_H\Vert _{H^1(D) }&\le C \Vert f\Vert _{L^2(D)} = C. \end{aligned}$$

(4.7)

We multiply (3.3) by $\mathbf {v}_H$ and integrate over $D$ to find:

$$\begin{aligned}&\displaystyle {\int _D \mathbf {v}_H \cdot \left( \partial _t \mathbf {w}^{\sigma }_H + (\mathbf {w}_H^{\sigma }\cdot \nabla _y)\mathbf {w}^{\sigma }_H + \displaystyle {\frac{2\pi }{\sigma }}w^{\sigma ,3}[E\mathbf {w}^{\sigma }_H - (\mathbf {w}_H^{\sigma })^{\perp }] \right) \,dy} \nonumber \\&\quad \displaystyle {= \int _D \mathbf {v}_H }\cdot \left( - (\nabla _y q^{\sigma })_H + \Delta _y \mathbf {w}^{\sigma }_H + \displaystyle {\frac{4\pi ^2}{\sigma ^2}} [E^2\mathbf {w}^{\sigma }_H \right. \nonumber \\&\left. \quad - 2 E (\mathbf {w}_H^{\sigma })^{\perp } - \mathbf {w}_H^{\sigma }]\right) \,dy. \end{aligned}$$

(4.8)

We next perform several integrations by parts, using the divergence constraint for $\mathbf {w}^\sigma$:

$$\begin{aligned} \mathrm {div}_y \mathbf {w}_H + \displaystyle {\frac{2\pi }{\sigma }} E w^3 = 0, \end{aligned}$$

together with (4.7), to find

$$\begin{aligned}&\displaystyle {\int _D \mathbf {v}_H \cdot \partial _t \mathbf {w}\,dy - \int _D \mathbf {w}^{\sigma }\cdot [(\mathbf {w}_H^{\sigma }\cdot \nabla _y)\mathbf {v}_H ]\,dy} \nonumber \\&\quad - \displaystyle {\frac{2\pi }{\sigma }}\int _D \mathbf {w}_H^{\sigma }\cdot w^{3,\sigma } E\mathbf {v}_H + w^{3,\sigma }\mathbf {v}_H \cdot (\mathbf {w}_H^{\sigma })^{\perp } \,dy \nonumber \\&\qquad \displaystyle { = \int _D f(y)\,q^{\sigma }(y)\,dy -\int _D \nabla _y \mathbf {v}_H\cdot \nabla _y \mathbf {w}^{\sigma }\,dy } - \displaystyle {\frac{4\pi ^2}{\sigma ^2}} \displaystyle {\int _D E\mathbf {v}_H\cdot E\mathbf {w}^{\sigma }_H \,dy}\nonumber \\&\qquad \quad + \; \displaystyle {\frac{8\pi ^2}{\sigma ^2}} \int _D E\mathbf {v}_H \cdot (\mathbf {w}_H^{\sigma })^{\perp }\,dy - \displaystyle {\frac{4\pi ^2}{\sigma ^2}}\int _D \mathbf {v}_H\cdot \mathbf {w}_H^{\sigma }\,dy . \end{aligned}$$

(4.9)

By Poincaré’s inequality for functions with zero average on $D$, we deduce that

$$\begin{aligned} \begin{aligned} \left| \int _D f \,q^{\sigma }\,dy \right| \le C\Vert \mathbf {v}_H\Vert _{H^1(D)}(\Vert \partial _t\mathbf {w}^{\sigma }\Vert _{H^{-1}(D)} + \Vert \mathbf {w}^{\sigma }\Vert _{L^4(D)}^2 + \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}), \end{aligned} \end{aligned}$$

(4.10)

for $C$ a constant independent of $f$ or $\sigma$. Above we exploit that the operator $E=y^\perp \cdot \nabla _y$ is first order and $\sigma \ge 1$.

Hence, using that $\Vert \mathbf {v}_H\Vert _{H^1(D)} \le C\Vert f\Vert _{L^2(D)}= C$ from (4.7) and the Ladyzhenskaya inequality again, we find

$$\begin{aligned} \Vert q^{\sigma }\Vert _{L^2(D)} \equiv \left| \int _D f \,q^{\sigma }\,dy \right| \le C(\Vert \partial _t\mathbf {w}^{\sigma }\Vert _{H^{-1}} + \Vert \mathbf {w}^{\sigma }\Vert _{L^2}\Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2} + \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2}). \end{aligned}$$

(4.11)

Finally, squaring both sides of the inequality (4.11) and using Young’s inequality, subsequently integrating in time, we arrive at

$$\begin{aligned} \begin{aligned} \Vert q^{\sigma }&\Vert _{L^2((0,T);L^2(D))}^2 \le C \left( \Vert \partial _t\mathbf {w}^{\sigma } \Vert _{L^2((0,T);H^{-1}(D))}^2 \right. \\&\left. \quad + \, (\Vert \mathbf {w}^{\sigma }\Vert _{L^{\infty }((0,T);L^2(D))}^2 + 1) \Vert \nabla \mathbf {w}^{\sigma } \Vert _{L^2((0,T);L^2(D))}^2\right) . \end{aligned} \end{aligned}$$

(4.12)

$\square$

With these estimates at hand, we are now ready to discuss the limit $\sigma \rightarrow \infty$. We observe that $\sigma$ is not a parameter appearing explicitly in the Navier–Stokes system (2.7). Therefore it is not clear what the limit equations are even at a formal level. The dependence on $\sigma$ is elucidated however in the symmetry-reduced system (3.3), which is equivalent to the original system at the level of strong solutions thanks to Proposition 3.1.

For the reduced system (3.3), formally setting $\sigma =\infty$ produces the following system of equations for a three-component vector function $\mathbf {w}^\infty : (0,+\infty ) \times D\rightarrow \mathbb {R}^3$, with associated pressure $q^\infty$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t w^{\infty ,1} + (w^{\infty ,1}\partial _{y_1}+w^{\infty ,2} \partial _{y_2}) w^{\infty ,1} = -\partial _{y_1} q^\infty + (\partial _{y_1}^2 +\partial _{y_2}^2) w^{\infty ,1 },\\ \partial _t w^{\infty ,2} + (w^{\infty ,1} \partial _{y_1}+w^{\infty ,2} \partial _{y_2}) w^{\infty ,2}= -\partial _{y_2} q^\infty + (\partial _{y_1}^2 +\partial _{y_2}^2) w^{\infty ,2} ,\\ \partial _t w^{\infty ,3} + \left( w^{\infty ,1} \partial _{y_1}+w^{\infty ,2} \partial _{y_2}\right) \, w^{\infty ,3} = (\partial _{y_1}^2 +\partial _{y_2}^2) w^{\infty ,3},\\ \partial _{y_1} w^{\infty ,1} + \partial _{y_2} w^{\infty ,2} = 0, &{} \text{ in } \; [0,+\infty ) \times D ; \\ \mathbf {w}^\infty = 0, &{} \text{ on } \; [0,+\infty ) \times \partial D ;\\ \mathbf {w}^\infty (0,y) = \mathbf {w^\infty }_0(y), &{} y\in D. \end{array}\right. } \end{aligned}$$

(4.13)

The initial condition $\mathbf {w}^\infty _0$ will be taken in $H^1_0(D)$ and assumed to satisfy:

$$\begin{aligned} \partial _{y_1} w^{\infty ,1}_0 + \partial _{y_2} w^{\infty ,2}_0 = 0. \end{aligned}$$

(4.14)

The first two momentum equations are independent of $w^{\infty ,3}$ and together with the fourth equation give precisely the two-dimensional Navier–Stokes equations in $D$, where the fluid velocity is identified with $\mathbf {w}^\infty _H:= (w^{\infty ,1},w^{\infty ,2},0)$. The third component $w^{\infty ,3}$ is simply advected by the first two and diffused. For this reason, we refer to this flow as a planar flow. Existence and regularity results for the 2D Navier–Stokes equations immediately give existence and uniqueness of the divergence-free vector field $\mathbf {w}^\infty _H\in C([0,T);H^1_0(D))\cap L^2((0,T);H^2(D)\cap H^1_0(D))$ and associated pressure $q^\infty \in L^2((0,T);H^1(D))\cap C^\infty ((0,T)\times D)$ for any initial condition $\mathbf {w}^\infty _H(0)\in H^1_0(D)$ satisfying (4.14), and any $T>0$. In fact, $\mathbf {w}_H^\infty$ is smooth for $t>0$. Consequently, the advection-diffusion equation for $w^{\infty ,3}$ admits a unique solution, which belongs to the same class (see e.g. Proposition 2.7 in [14] and Theorem 3.10 in [19].) We refer to the three-component vector function

$$\begin{aligned} \mathbf {w}^\infty \in C([0,T);H^1_0(D))\cap L^2((0,T);H^2(D)\cap H^1_0(D))\cap C^\infty ((0,T)\times D), \end{aligned}$$

as the unique strong solution of problem (4.13).

The System (4.13) gives the so-called two-dimensional, three-component Navier–Stokes equations (also known as the $2\frac{1}{2}$ D Navier–Stokes equations, see [14].) We can uniquely associate to $\mathbf {w}^\infty$ a solution $\mathbf {u}^\infty$ of the Navier–Stokes equations in $\Omega$ with initial data $\mathbf {u}_0^\infty$ by:

$$\begin{aligned} \begin{aligned}&\mathbf {u}^\infty (t,x) := \mathbf {w}^\infty (t, x'),&\quad x' \in D, \; t>0, \\&\mathbf {u}^\infty _0(x) := \mathbf {w}^\infty _0(x'),&\quad x' \in D \\&p^\infty (t,x) := q^\infty (t,x',0),&\quad x' \in D, \; t>0, \end{aligned} \end{aligned}$$

(4.15)

with $x'=(x_1,x_2)$. It is immediate to see that $\mathbf {u}^\infty$ and $p^\infty$ have at least the same regularity as $\mathbf {w}^\infty$ and $q^\infty$. We will refer to $\mathbf {u}^\infty$ as the $2\frac{1}{2}$D solution of the Navier–Stokes system (2.7) in $\Omega$ with associated pressure $p^\infty$.

To obtain a relationship with the original problem (2.7), at least at a formal level, we observe that, if we take the limit $\sigma \rightarrow \infty$ in (2.5), thanks to (2.2) and (2.6), we have the identification:

$$\begin{aligned} \mathbf {u}^\infty (t,x) = \mathbf {w}^\infty (t,x') \equiv \lim _{\sigma \rightarrow \infty } \mathbf {u}^\sigma (t,x). \end{aligned}$$

(4.16)

Above, we have naturally identified the cross section of the cylinder $\Omega$ at height $x_3=0$ with $D$ and $x'=(x_1,x_2)$ with $y$. We will use the identities and estimates established in Proposition 4.3, valid for all $1 \le \sigma < \infty$, to establish an estimate for the difference between $\mathbf {w}^{\sigma }$ and $\mathbf {w}^{\infty }$. One difficulty in studying the limit $\sigma \rightarrow \infty$ is that $\mathbf {w}^\infty _H$ is divergence free, while $\mathbf {w}^\sigma _H$ satisfies a divergence constraint that is $\sigma$ dependent.

Proposition 4.4

Let $\mathbf {w}^\infty _0\in H^1_0(D)$ satisfy (4.14). Given $\sigma \ge 1$, let $\mathbf {w}_0^{\sigma } \in H^1_0(D)$ satisfy (4.1). Let $\mathbf {w}^\infty$ be the unique strong solution of (4.13) with initial data $\mathbf {w}^\infty _0$, and let $\mathbf {w}^{\sigma }$ be the unique regular solution of (3.3) with initial data $\mathbf {w}_0^{\sigma }$ on the time interval $(0,T), T>0$. Set:

$$\begin{aligned} \varvec{\varvec{\Theta }}^{\sigma } \equiv \mathbf {w}^{\sigma } - \mathbf {w}^\infty . \end{aligned}$$

(4.17)

Then, for all $0<t<T$,

$$\begin{aligned} \begin{aligned}&\int _D |\varvec{\varvec{\Theta }}^{\sigma }(t,y)|^2 \, dy + \int _0^t\int _D |\nabla \varvec{\varvec{\Theta }}^{\sigma }(s,y)|^2\,ds\,dy \\&\quad \le C\left( t,\Vert \mathbf {w}^\infty _0\Vert _{L^2}^2,\Vert \mathbf {w}_0^{\sigma }\Vert _{L^2}^2\right) \left( \Vert \varvec{\varvec{\Theta }}^{\sigma }_0\Vert _{L^2}^2 + \frac{1}{\sigma }\right) . \end{aligned} \end{aligned}$$

(4.18)

Proof

Since $\mathbf {w}^\sigma$ is a strong solution of (3.3) and $\mathbf {w}^\infty$ is a strong solution of (4.13) on the interval $[0,T)$, there exist functions $q^\sigma$ and $q^\infty \in L^1((0,T);H^1(D))$ enforcing the divergence constraints. If we set $r^\sigma =q^\infty -q^\sigma$, then $\varvec{\Theta }^\sigma$ satisfies the following set of equations on $(0,T)\times D$:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t \varvec{\Theta }^{\sigma } + (\mathbf {w}^{\sigma }_H\cdot \nabla _y)\varvec{\Theta }^{\sigma } + (\varvec{\Theta }^{\sigma }_H\cdot \nabla _y)\mathbf {w}^\infty + \displaystyle {\frac{2\pi }{\sigma }}w^{\sigma ,3}[E\mathbf {w}^{\sigma } - (\mathbf {w}^{\sigma }_H)^{\perp }] \\ \\ \;\;\;\;\;\;\;= -(\nabla _y r^{\sigma })_H -\displaystyle {\frac{2\pi }{\sigma }}Eq^{\sigma }\mathbf{e}_\mathbf{3} + \Delta _y \varvec{\Theta }^{\sigma } + \displaystyle {\frac{4\pi ^2}{\sigma ^2}} [E^2\mathbf {w}^{\sigma } - 2 E (\mathbf {w}^{\sigma }_H)^{\perp } - \mathbf {w}^{\sigma }_H], \\ \\ \mathrm {div}_y \varvec{\Theta }^{\sigma }_H + \displaystyle {\frac{2\pi }{\sigma }} E w^{\sigma ,3} = 0, \end{array}\right. \end{aligned}$$

(4.19)

where $E$ is again the differential operator $y^\perp \cdot \nabla _y$ defined in (2.4). These equations are complemented by the initial condition

$$\begin{aligned} \varvec{\Theta }^\sigma _0:= \varvec{\Theta }^\sigma (0) = \mathbf {w}^\sigma _0 - \mathbf {w}^\infty _0 \in H^1_0(D) \end{aligned}$$

and no-slip boundary conditions on $\partial D$.

We observe that $\varvec{\Theta }^\sigma$ has enough regularity to be a test function for the weak formulation of (4.19). In particular, $\partial _t \varvec{\Theta }^\sigma \in L^2((0,T),L^2(D))$. The weak form, after rearranging the terms and integrating by parts, gives:

$$\begin{aligned}&\displaystyle {\frac{1}{2}\frac{d}{dt}\int _D |\varvec{\Theta }^{\sigma }|^2\,dy + \int _D |\nabla \varvec{\Theta }^{\sigma }|^2\,dy = -\int _D \varvec{\Theta }^{\sigma }\cdot \left[ (\mathbf {w}^{\sigma }_H\cdot \nabla _y)\varvec{\Theta }^{\sigma }\right] \,dy} \nonumber \\&\quad \displaystyle {- \int _D \varvec{\Theta }^{\sigma }\cdot \left[ (\varvec{\Theta }^{\sigma }_H\cdot \nabla _y)\mathbf {w}^\infty \right] \,dy -\frac{2\pi }{\sigma } \int _D \varvec{\Theta }^{\sigma }\cdot \left[ w^{\sigma ,3}(E\mathbf {w}^{\sigma } - (\mathbf {w}^{\sigma }_H)^{\perp }) \right] \,dy}\nonumber \\&\quad \displaystyle {+\frac{4\pi ^2}{\sigma ^2} \int _D \varvec{\Theta }^{\sigma }\cdot \left[ E^2\mathbf {w}^{\sigma } - 2 E (\mathbf {w}^{\sigma }_H)^{\perp } - \mathbf {w}^{\sigma }_H \right] \,dy} \\&\quad \displaystyle {-\int _D \varvec{\Theta }^{\sigma }\cdot \left[ (\nabla _y r^{\sigma })_H + \frac{2\pi }{\sigma }Eq^{\sigma }\mathbf{e}_\mathbf{3} \right] \,dy \equiv -I_1-I_2 -I_3+I_4-I_5.}\nonumber \end{aligned}$$

(4.20)

We estimate each of the five integrals on the right-hand side. Since $\mathbf {w}^\sigma$ is a strong solution and in view of estimates (4.5) for $\mathbf {w}^\sigma$, all norms appearing in the bounds to follow are finite and all constants $C$ are uniform in $\sigma \in [1,+\infty )$. We have:

$$\begin{aligned} \begin{aligned}&|I_1| \le \frac{2\pi }{\sigma }\int _D \frac{1}{2} |\varvec{\Theta }^{\sigma }|^2|\nabla \mathbf {w}^{\sigma }|\,dy, \\&|I_2| \le \int _D |\varvec{\Theta }^{\sigma }|^2|\nabla \mathbf {w}^\infty |\,dy, \\&|I_3| \le \frac{2\pi }{\sigma }\int _D |\varvec{\Theta }^{\sigma }||\mathbf {w}^{\sigma }|(|\nabla \mathbf {w}^{\sigma }| + |\mathbf {w}^{\sigma }|)\,dy, \\&|I_4| \le \frac{4\pi ^2}{\sigma ^2} \left[ \int _D |\nabla \varvec{\Theta }^{\sigma }| |\nabla \mathbf {w}^{\sigma }|\,dy + \int _D |\varvec{\Theta }^{\sigma }|(|\nabla \mathbf {w}^{\sigma }| + |\mathbf {w}^{\sigma }|)\,dy \right] , \\&|I_5| = \left| \frac{2\pi }{\sigma } \int _D [q^\infty \, \,E(w^{\sigma ,3}) - q^{\sigma }\,E(w^{\infty ,3})]\,dy \right| \le \frac{2\pi }{\sigma }\int _D ( |\nabla \mathbf {w}^{\sigma }|\,|q^\infty | + |\nabla \mathbf {w^\infty }| \,|q^{\sigma }|)\,dy. \end{aligned} \end{aligned}$$

We bound further each integral $I_i, i=1,\ldots ,5$, using repeatedly the Ladyzhenskaya inequality (2.1), Cauchy-Schwartz and Young’s inequalities:

$$\begin{aligned} |I_1|&\le \frac{C}{\sigma } \Vert \varvec{\Theta }^{\sigma }\Vert _{L^4(D)}^2\Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)} \le \frac{C}{\sigma } (\Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2\Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2), \end{aligned}$$

(4.21)

$$\begin{aligned} |I_2|&\le \Vert \varvec{\Theta }^{\sigma }\Vert _{L^4(D)}^2\Vert \nabla \mathbf {w}^\infty \Vert _{L^2(D)} \le \frac{1}{2} \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2\Vert \nabla \mathbf {w}^\infty \Vert _{L^2(D)}^2 + \frac{1}{2} \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2, \end{aligned}$$

(4.22)

$$\begin{aligned} |I_3|&\le \displaystyle {\frac{C}{\sigma }(\Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}\Vert \mathbf {w}^{\sigma }\Vert _{L^4(D)}^2 + \Vert \varvec{\Theta }^{\sigma }\Vert _{L^4(D)} \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)} \Vert \mathbf {w}^{\sigma }\Vert _{L^4(D)})} \nonumber \\&\le \displaystyle \!\frac{C}{\sigma }(\Vert \!\varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 \!\Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 \!+\! \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 \!+\! \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^{1/2}\! \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^{1/2} \nonumber \\&\quad \quad \, \times \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^{3/2} \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^{1/2}) \nonumber \\&\le \displaystyle \frac{C}{\sigma }(\Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2\nonumber \\&\quad \quad \, + \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2) \nonumber \\&\le \displaystyle {\frac{C}{\sigma }}[\Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 \nonumber \\&\quad \quad \, + (\Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \Vert {\mathbf {w}}^{\infty }\Vert _{L^2(D)}^2) \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2], \end{aligned}$$

(4.23)

$$\begin{aligned} |I_4|&\le \frac{C}{\sigma ^2} ( \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2}\Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)} + \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}\Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)} + \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}\Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)})\nonumber \\&\le \frac{C}{\sigma ^2} ( \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 + \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 + \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2), \end{aligned}$$

(4.24)

$$\begin{aligned} |I_5|&\le \frac{C}{\sigma } (\Vert q^\infty \Vert _{L^2(D)}\Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)} + \Vert q^{\sigma }\Vert _{L^2(D)}\Vert \nabla \mathbf {w}^\infty \Vert _{L^2}(D)) \nonumber \\&\le \frac{C}{\sigma } (\Vert q^\infty \Vert _{L^2(D)}^2 + \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \Vert q^{\sigma }\Vert _{L^2(D)}^2 + \Vert \nabla \mathbf {w}^\infty \Vert _{L^2(D)}^2). \end{aligned}$$

(4.25)

Inserting estimates (4.21) — (4.25) into identity (4.20) yields:

$$\begin{aligned}&\frac{d}{dt}\Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 + \Vert \nabla \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2 \le \Vert \nabla \varvec{\Theta }^{\sigma } \Vert _{L^2(D)}^2 \nonumber \\&\quad \cdot \displaystyle {\left( \frac{C}{\sigma } + \frac{1}{2} + \frac{C}{\sigma }\Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^4 + \frac{C}{\sigma }\Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2\Vert \mathbf {w}^\infty \Vert _{L^2(D)}^2 + \frac{C}{\sigma ^2} \right) + \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2(D)}^2} \nonumber \\&\quad \displaystyle {\cdot \left( \frac{C}{\sigma } \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \frac{C}{\sigma }\Vert \nabla \mathbf {w}^\infty \Vert _{L^2(D)}^2 + \frac{C}{\sigma } \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \frac{1}{2} \Vert \nabla \mathbf {w}^\infty \Vert ^2_{L^2(D)} + \frac{C}{\sigma ^2} \right) } \nonumber \\&\qquad \displaystyle {+ \left( \frac{C}{\sigma } \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \frac{C}{\sigma ^2} \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \frac{C}{\sigma ^2} \Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2 + \frac{C}{\sigma } \Vert \nabla \mathbf {w}^\infty \Vert _{L^2(D)}^2 \right. } \nonumber \\&\quad \qquad \qquad \displaystyle { + \left. \frac{C}{\sigma } \Vert q^\infty \Vert _{L^2(D)}^2 + \frac{C}{\sigma }\Vert q^{\sigma }\Vert _{L^2(D)}^2 \right) .} \end{aligned}$$

(4.26)

Thanks again to the regularity of $\mathbf {w}^\infty$, i.e.,

$$\begin{aligned} \mathbf {w}^\infty \in C([0,T);H^1_0(D))\cap L^2((0,T);H^2(D)\cap H^1_0(D)), \end{aligned}$$

and estimates (4.5) for $\mathbf {w}^\sigma$, we can now choose $\sigma$ large enough such that

$$\begin{aligned} \frac{C}{\sigma } + \frac{1}{2} + \frac{C}{\sigma }\Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^4 + \frac{C}{\sigma }\Vert \mathbf {w}^{\sigma }\Vert _{L^2(D)}^2\Vert \mathbf {w}^\infty \Vert _{L^2(D)}^2 + \frac{C}{\sigma }^2 < \frac{3}{4}. \end{aligned}$$

We will rewrite (4.26) as a differential inequality in order to apply Grönwall’s Lemma. To this end, we introduce the functions

$$\begin{aligned} f(t) = \frac{C}{\sigma } \Vert \nabla \mathbf {w}^{\sigma }\Vert _{L^2}^2 + \left( \frac{C}{\sigma }+\frac{1}{2}\right) \Vert \nabla \mathbf {w}^\sigma \Vert _{L^2}^2 + \frac{C}{\sigma } \Vert \mathbf {w}^{\sigma }\Vert _{L^2}^2 + \frac{C}{\sigma ^2} \end{aligned}$$

and

$$\begin{aligned} g(t)&= \frac{C}{\sigma } \Vert \nabla \mathbf {w}^{\sigma }(t)\Vert _{L^2}^2 + \frac{C}{\sigma ^2} \Vert \nabla \mathbf {w}^{\sigma } (t)\Vert _{L^2}^2 + \frac{C}{\sigma ^2} \Vert \mathbf {w}^{\sigma }(t)\Vert _{L^2}^2 \\&+ \frac{C}{\sigma } \Vert \nabla \mathbf {w}^\infty (t)\Vert _{L^2}^2 + \frac{C}{\sigma } \Vert q^\infty (t)\Vert _{L^2}^2 + \frac{C}{\sigma }\Vert q^{\sigma }(t)\Vert _{L^2}^2. \end{aligned}$$

We also set

$$\begin{aligned} z(t) = \Vert \varvec{\Theta }^{\sigma }(t)\Vert _{L^2}^2. \end{aligned}$$

With this notation the differential inequality above becomes

$$\begin{aligned} \frac{d}{dt}z \le f(t)z + g(t), \end{aligned}$$

so that, by Grönwall’s Lemma we conclude that

$$\begin{aligned} z(t) \le \exp \left\{ \int _0^t f(s)\,ds\right\} \, z(0) + \int _0^t \exp \left\{ \int _s^t f(\tau )\,d\tau \right\} \, g(s) \, ds. \end{aligned}$$

(4.27)

Next, standard energy estimates for the 2D Navier–Stokes equations along with similar energy estimates for advection-diffusion equations, using that $\mathbf {w}^\infty _H$ is divergence-free, give:

$$\begin{aligned} \int _0^t \Vert \nabla \mathbf {w}^\infty (s)\Vert _{L^2(D)}^2\,ds \le \Vert \mathbf {w}^\infty _0\Vert _{L^2(D)}^2. \end{aligned}$$

We employ once again the estimates (4.5) in Remark 4.2 to deduce that

$$\begin{aligned}&\int _0^t f(s) \, ds \le C\, \left( \Vert \mathbf {w}^\infty _0\Vert _{L^2}^2 + \frac{1}{\sigma } \Vert \mathbf {w}_0^{\sigma }\Vert _{L^2}^2\right) ,\\&\int _0^t g(s) \, ds \le \frac{C}{\sigma }\left( \Vert \mathbf {w}^\infty _0\Vert _{L^2}^2 + \Vert \mathbf {w}_0^{\sigma }\Vert _{L^2}^2\right) . \end{aligned}$$

Hence we arrive at the estimate, using that $\sigma \ge 1$,

$$\begin{aligned} \Vert \varvec{\Theta }^{\sigma }\Vert _{L^2}^2 \le C\left( \Vert \mathbf {w}^\infty _0\Vert _{L^2}^2,\Vert \mathbf {w}_0^{\sigma }\Vert _{L^2}^2\right) \Vert \varvec{\Theta }^{\sigma }_0\Vert _{L^2}^2 + \frac{C}{\sigma }\left( \Vert \mathbf {w}^\infty _0\Vert _{L^2}^2 + \Vert \mathbf {w}_0^{\sigma }\Vert _{L^2}^2\right) . \end{aligned}$$

This estimate, together with the choice of $\sigma$, produces, upon integrating the differential inequality (4.26) in time, the desired result.$\square$

Before formulating our main results concerning the limit $\sigma \rightarrow \infty$ , we note a consequence of Proposition 4.4; namely, there may be more than one $2\frac{1}{2}$D flow within a certain distance to a given helical flow $\mathbf {w}^\sigma$. This non-uniqueness will be apparent later, since a correction to the initial data $\mathbf {w}^\sigma _0$ will be needed to enforce the divergence-free condition for $\mathbf {w}^\infty _0$.

We start with a simpler result, describing a way in which solutions of the two-dimensional, three-component Navier–Stokes equations can be approximated by suitable helical solutions of the three-dimensional Navier–Stokes equations. More precisely, suppose we are initially given a vector function

$$\begin{aligned} \mathbf {w}^\infty _0 = (w_0^{\infty ,1},w_0^{\infty ,2},w_0^{\infty ,3}) \in H^1_0 (D) \end{aligned}$$

that satisfies the divergence-free constraint (4.14). Let $\mathbf {w}^\infty$ be the unique strong solution of (4.13) with pressure $q^\infty$. Recall that we can uniquely associate to $\mathbf {w}^\infty$ a solution $\mathbf {u}^\infty$ of the Navier–Stokes equations in $\Omega$ with initial data $\mathbf {u}_0^\infty$ via (4.15). We will construct a $\sigma$-dependent correction to $\mathbf {w}^\infty _0, \mathbf {v}_0^{\sigma }$, using Lemma 4.1, so that the resulting field $\mathbf {w}^\sigma _0$, given in (4.30) below, satisfies (4.1) and, hence, can be taken as initial data for the reduced helical equations (3.3).

We first observe that, since $\mathbf {w}^\infty _0\vert _{\partial D}\equiv 0$ and $\text{ div }_y y^\perp \equiv 0$,

$$\begin{aligned} \int _D y^\perp \cdot \nabla _y w_0^{\infty ,3} \,dy = \int _D \text{ div }_y \, (y^{\perp }\,w_0^{\infty ,3})\,dy = 0. \end{aligned}$$

Therefore, by Lemma 4.1, there exists a solution $\mathbf {v}_0^{\sigma }=(v_0^{\sigma ,1},v_0^{\sigma ,2}) \in H^{1}_0(D)$ to the problem

$$\begin{aligned} \text{ div }_y \, \mathbf {v}_0^{\sigma } = -\frac{2\pi }{\sigma } E\,w_0^{\infty ,3}, \end{aligned}$$

(4.28)

such that

$$\begin{aligned} \Vert \mathbf {v}_0^{\sigma }\Vert _{H^1} \le C\frac{1}{\sigma }\Vert E \, w^{\infty ,3}_0\Vert _{L^2(D)}\le \frac{C}{\sigma }\Vert \mathbf {w}^\infty _0\Vert _{H^1(D)}, \end{aligned}$$

(4.29)

where we recall that $E=y^\perp \cdot \nabla _y$.

Next, we introduce the three-component vector function

$$\begin{aligned} \mathbf {w}_0^{\sigma } = \mathbf {w}^\infty _0 + (\mathbf {v}_0^{\sigma },0) \in H^1_0(D), \end{aligned}$$

(4.30)

which by construction satisfies (4.1), since $w^{\sigma ,3}_{0} = w^{\infty ,3}_{0}$. We will take $\mathbf {w}^\sigma _0$ so constructed as initial data for (3.3). We are now ready to state our first theorem.

Theorem 4.5

Fix $\sigma \ge 1$. Let $\mathbf {w}^\infty _0\in H^1_0(D)$ satisfy (4.14). Let $\mathbf {w}^\infty$ be the unique strong solution of (4.13) with initial data $\mathbf {w}^\infty _0$. Let $\mathbf {u}^\infty$ be the unique $2\frac{1}{2}$D solution of the Navier–Stokes equations (2.7) associated to $\mathbf {w}^\infty$ via (4.15). Let $\mathbf {w}^\sigma _0$ be given by (4.30) for a choice of $\mathbf {v}^\sigma _0$ solution of (4.28), and denote by $\mathbf {w}^\sigma$ the strong solution of (3.3) with initial data $\mathbf {w}^\sigma _0$. Let $\mathbf {u}_0^{\sigma }$ be the associated strong helical solution of the Navier–Stokes equations (2.7) given by Proposition 3.1. Then, for any fixed $T>0$,

$$\begin{aligned} \begin{aligned} \Vert \mathbf {u}^{\sigma }(t,\cdot ,x_3=0) - \mathbf {u}^\infty (t,\cdot ,x_3=0)\Vert _{L^2(D)}&\le \frac{C}{\sqrt{\sigma }}, \qquad \text { for all } \; 0<t<T, \\ \Vert \nabla _H\mathbf {u}^{\sigma }\vert _{x_3=0} - \nabla _H\mathbf {u}^\infty \vert _{x_3=0}\Vert _{L^2(0,T;L^2(D))}&\le \frac{C}{\sqrt{\sigma }}, \end{aligned} \end{aligned}$$

(4.31)

where $C$ is independent of $\sigma \in [1,\infty )$.

Proof

Since by hypothesis, both $\mathbf {u}^\sigma$ and $\mathbf {u}^\infty \in C([0,T);H^1_{0,per}(\Omega ^\sigma )$

$\cap L^2((0,T);H^2_{per}(\Omega ^\sigma )\cap H^1_{0,per}(\Omega ^\sigma ))$, the traces $\mathbf {u}^\sigma \vert _{x_3=0}(t)$ and $\mathbf {u}^\infty \vert _{x_3=0}(t)$ are well defined as elements of $L^2(D)$ for all $0\le t<T$, while the traces $\nabla \mathbf {u}^\sigma \vert _{x_3=0}$ and $\nabla \mathbf {u}^\infty \vert _{x_3=0}$ are well defined as elements of $L^2((0,T);L^2(D))$.

We continue by showing that

$$\begin{aligned} \Vert \mathbf {w}_0^{\sigma }- \mathbf {w}^\infty _0\Vert _{L^2(D)}&\le \frac{C}{\sqrt{\sigma }} \;\;\; \text{ and }\end{aligned}$$

(4.32)

$$\begin{aligned} \Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}&\le C, \end{aligned}$$

(4.33)

with constants $C$ uniform in $\sigma \in [1,\infty )$. To see that the first statement (4.32) holds true, we note that

$$\begin{aligned} \mathbf {w}_0^{\sigma }- \mathbf {w}^\infty _0 = (\mathbf {v}_0^{\sigma },0), \end{aligned}$$

where $\mathbf {v}_0^{\sigma }$ is a solution of (4.28) and satisfies (4.29).

Hence,

$$\begin{aligned} \Vert \mathbf {w}_0^{\sigma }- \mathbf {w}^\infty _0\Vert _{L^2(D)} = \Vert \mathbf {v}_0^{\sigma }\Vert _{L^2(D)} \le \Vert \mathbf {v}_0^{\sigma }\Vert _{H^1(D)} \le \frac{C}{\sigma }\Vert \mathbf {w}^\infty _0\Vert _{H^1(D)}. \end{aligned}$$

The second statement (4.33) follows immediately from the first.

Then, Proposition 4.4 gives that

$$\begin{aligned} \Vert \mathbf {w}^{\sigma }(t,\cdot ) - \mathbf {w}^\infty (t,\cdot )\Vert _{L^2(D)} \le \frac{C}{\sqrt{\sigma }},\;\;\; \text{ for } \text{ almost } \text{ all } 0<t<T, \end{aligned}$$

(4.34)

and

$$\begin{aligned} \Vert \nabla _H\mathbf {w}^{\sigma } - \nabla _H\mathbf {w}^\infty \Vert _{L^2(0,T;L^2(D))} \le \frac{C}{\sqrt{\sigma }}, \end{aligned}$$

(4.35)

again with constants $C$ that do not depend on $\sigma \ge 1$.

Next, the proof of Proposition 3.1 shows that the helical solution $\mathbf {u}^\sigma$ of (2.7) with initial condition $\mathbf {u}^\sigma _0$ related to $\mathbf {w}^\sigma _0$ via (2.8) is given by

$$\begin{aligned} \mathbf {u}^{\sigma }(t,x',x_3) = M^{\sigma }(x_3)\,\mathbf {w}^{\sigma }(t,m^{\sigma }(x_3) x'), \end{aligned}$$

for $t>0$, where $x'=(x_1,x_2)$, so that in particular:

$$\begin{aligned} \mathbf {u}^{\sigma }(t,x_H)= \mathbf {w}^{\sigma }(t, x'), \qquad \nabla _H \mathbf {u}^{\sigma }(t,x_H)= \nabla _{x'}\mathbf {w}^{\sigma }(t, x'). \end{aligned}$$

From (4.15), it also immediately follows that

$$\begin{aligned} \mathbf {u}^\infty (t,x_H)= \mathbf {w}^\infty (t,x'), \qquad \nabla _x\mathbf {u}^{\infty }(t,x_H)= \nabla _H \mathbf {u}^{\infty }(t,x_H)= \nabla _{x'}\mathbf {w}^{\sigma }(t, x'). \end{aligned}$$

Then, estimate (4.31) is a straightforward consequence of (4.34) and (4.35).$\square$

Remark 4.3

It is natural to derive bounds of traces at $x_3=0$ in view of (4.16). In fact, recalling that $\mathbf {u}^\sigma$ is smooth in $x\in \Omega ^\sigma$ for $t>0$, a simple argument, using a Taylor’s expansion for $\mathbf {u}^\sigma$ in $0\le x^3/\sigma <1$, centered at $0$ with $x'\in D$ fixed, shows that for a given fixed $t$,

$$\begin{aligned} |\mathbf {u}^\sigma (t,x)-\mathbf {u}^\infty (t,x)|=|\mathbf {w}^\sigma (t,x')-\mathbf {w}^\infty (t,x')| + O\left( \frac{|x_3|}{\sigma }\right) , \end{aligned}$$

with bounds that depend on $|\mathbf {w}^\sigma (x')|$ and $|\nabla _y \mathbf {w}^\sigma (x')|$. Therefore, an argument similar to that of the proof of Theorem 4.5 above gives:

$$\begin{aligned} \begin{aligned}&\Vert \mathbf {u}^{\sigma }(t) - \mathbf {u}^\infty (t)\Vert _{L^2(U)}\underset{\sigma \rightarrow \infty }{\rightarrow 0}, \qquad \text { for all } \; 0<t<T, \\&\Vert \nabla \mathbf {u}^{\sigma } - \nabla \mathbf {u}^\infty \Vert _{L^2(0,T;L^2(U))} \underset{\sigma \rightarrow \infty }{\rightarrow 0}, \end{aligned} \end{aligned}$$

(4.36)

for any cylinder $U\subset \Omega$ of the form

$$\begin{aligned} U = \{x=(x',x_3) \;|\; x'\in D, \; x_3 \in [0,\delta ], \; \delta /\sigma \underset{\sigma \rightarrow \infty }{\longrightarrow } 0 \}. \end{aligned}$$

On the other hand, $|x_3|/\sigma$ is $O(1)$ in $\Omega ^\sigma$. Hence, it seems difficult to obtain any convergence estimate of $\mathbf {u}^\sigma$ to $\mathbf {u}^\infty$ globally in $\Omega ^\sigma$ as $\sigma \rightarrow \infty$.

The previous result is not exactly what we aimed at, as it represents a way of approximating a general two-dimensional flow by a well-chosen helical flow. What we want, instead, is to show that helical flows with large $\sigma$ are nearly two-dimensional. This adjustment is expressed in our next result.

Theorem 4.6

Fix $\sigma \ge 1$ and $T>0$. Let $\mathbf {u}_0^{\sigma } \in H^1_{0,per}(\Omega ^{\sigma })$ be a divergence-free, helical vector field. Let $\mathbf {u}^{\sigma }$ be the unique, strong helical solution of (2.7) on $[0,T)$ with initial velocity $\mathbf {u}_0^{\sigma }$. There exists a (not necessarily unique) $\widetilde{\mathbf {w}_0^{\infty }} \in H^1(D)$, such that, if $\widetilde{\mathbf {u}^\infty }$ is the unique $2\frac{1}{2}$D solution of the Navier–Stokes equations (2.7) with initial data $\widetilde{\mathbf {u}^{\infty }_{0}}(\cdot ,x_3) = \widetilde{\mathbf {w}^{\infty }_{0}}$, then for all $0<t<T$,

$$\begin{aligned}&\Vert \mathbf {u}^{\sigma }(t,\cdot ,x_3=0) - \widetilde{\mathbf {u}^{\infty }}(t,\cdot ,x_3=0)\Vert _{L^2(D)}\nonumber \\&\quad + \Vert \nabla _H \mathbf {u}^{\sigma }\vert _{x_3=0} - \nabla _H \widetilde{\mathbf {u}^{\infty }}\vert _{x_3=0} \Vert _{L^2(0,T;L^2(D))} \le C(T) \frac{1}{\sqrt{\sigma }}, \end{aligned}$$

(4.37)

where $C$ is independent of $\sigma \in [1,\infty )$.

We use the notation $\widetilde{\mathbf {u}^\infty }$ to emphasize that, while this is a solution of the limit problem, it is still dependent on $\sigma$ due to the correction to the initial condition to enforce the divergence-free condition.

Proof

As in the proof of Theorem 4.5, the traces of $\mathbf {u}^\sigma$ and $\mathbf {u}^\infty$ are well defined at the level of strong solutions. Furthermore, as in that theorem we will introduce a correction to the initial data $\mathbf {u}^\sigma _0$ to enforce the divergence-free condition on the initial data $\widetilde{\mathbf {u}^\infty }_0$ we take for the limit problem. Let $\mathbf {w}^\sigma _0\in H^1_0(D)$ be associated to the helical field $\mathbf {u}^\sigma _0\in H^1_{0,per}(\Omega ^\sigma )$ by (2.8), satisfying (4.1). Let $\mathbf {w}^\sigma$ be the regular solution of (3.3) with this initial data.

Next, let $\mathbf {v}_0^{\sigma }=(v_0^{\sigma ,1},v_0^{\sigma ,2}) \in H^{1}_0(D)$ be a solution, given by Lemma 4.1, to the problem

$$\begin{aligned} \text{ div }_y \mathbf {v}_0^{\sigma } = -\frac{2\pi }{\sigma }Ew_0^{\sigma ,3}, \end{aligned}$$

(4.38)

where again $E$ is the differential operator defined in (2.4), such that

$$\begin{aligned} \Vert \mathbf {v}_0^{\sigma }\Vert _{H^1} \le C\frac{1}{\sigma }\Vert E \,w_0^{\sigma ,3}\Vert _{L^2(D)}\le \frac{C}{\sigma }\Vert \mathbf {w}_0^{\sigma }\Vert _{H^1(D)}. \end{aligned}$$

(4.39)

Its existence is justified exactly as before.

We then set

$$\begin{aligned} \widetilde{\mathbf {u}_0^{\infty }}(x) = \widetilde{\mathbf {w}^\infty }_0(x'):= \mathbf {w}_0^{\sigma } (x')- (\mathbf {v}_0^{\sigma }(x'),0), \end{aligned}$$

(4.40)

which is divergence free by (4.38). Let $\widetilde{\mathbf {w}^{\infty }}$ be the solution of (4.13) with initial data $\widetilde{\mathbf {w}^\infty }_0$. The $2\frac{1}{2}$D solution of the Navier–Stokes equations is given by $\widetilde{\mathbf {u}^{\infty }}(t,x)= \widetilde{\mathbf {w}^{\infty }}(t,x')$. In particular, the trace $\widetilde{\mathbf {u}^{\infty }}(t,\cdot ,x_3=0)=\widetilde{w^\infty }(t,\cdot )$.

By Proposition 4.4, estimate (4.37) now follows from

$$\begin{aligned} \Vert \mathbf {w}_0^{\sigma } - \widetilde{\mathbf {w}_0^{\infty }} \Vert _{L^2(D)}&= \Vert \mathbf {v}_0^{\sigma }\Vert _{L^2(D)} \le \Vert \mathbf {v}_0^{\sigma }\Vert _{H^1(D)} \le \frac{C}{\sigma }\Vert \mathbf {w}_0^{\sigma }\Vert _{H^1(D)}, \end{aligned}$$

(4.41)

$$\begin{aligned} \Vert \widetilde{\mathbf {w}_0^{\infty }}\Vert _{L^2(D)}&\le \Vert \mathbf {w}_0^{\sigma } \Vert _{L^2(D)} + \Vert \mathbf {v}_0^{\sigma }\Vert _{L^2(D)} \le \left( 1+\frac{C}{\sigma }\right) \Vert \mathbf {w}_0^{\sigma }\Vert _{H^1(D)},\end{aligned}$$

(4.42)

$$\begin{aligned} \Vert \mathbf {w}_0^{\sigma }\Vert _{L^2(D)}&\le \Vert \mathbf {w}_0^{\sigma }\Vert _{H^1(D)} = \Vert \mathbf {u}_0^{\sigma }(x_3=0)\Vert _{H^1(D)}, \end{aligned}$$

(4.43)

with constants uniform in $\sigma \in [1,\infty )$.$\square$

5 The Inviscid Case

In this section we discuss symmetry reduction and the limit $\sigma \rightarrow \infty$ for the Euler equations under an additional geometric assumption, considered already in [5, 6]. This assumption can be viewed as the analog of the no-swirl condition in axisymmetric flows and for this reason we will call it the no helical swirl or no helical stretching condition. It can be shown that the flow induced by solutions of the Euler equations preserves this condition at least when the solution is regular enough. Furthermore, vorticity has an especially simple form, being determined by its vertical component, which is advected by the flow. This observation allows to prove global existence and uniqueness of weak, helical solutions in much the same spirit as for solutions to the two-dimensional Euler equations, provided the initial velocity is bounded (cf. [9].)

We now briefly review these results, referring the reader to [5, 6] for more details. We will then discuss the limit problem as $\sigma \rightarrow \infty$ and convergence of solutions. On one hand the limit problem is simpler, being given by the 2D Euler equations. In fact, under the no-stretching constraint the symmetry-reduced helical Euler equations become a two-dimensional systems for two components of the velocity, which admits a vorticity-stream function formulation (see e.g. [14].) This system is the analog of the symmetry-reduced equations (3.3) for the Navier–Stokes. On the other hand, to circumvent the lack of smoothing in the equations for positive time we will use compactness arguments to pass to the limit in $\sigma$, which do not provide a rate of convergence.

For ease of notation, we temporarily suppress the explicit $\sigma$-dependence of solutions and write $\mathbf {u}$ for $\mathbf {u}^\sigma$ for example. We assume for now that $\mathbf {u}$ and $p$ are smooth, so that all the manipulations to follow are justified.

Given that smooth, helical vector fields and functions are $\sigma$ periodic by Proposition 2.1, we state the initial-boundary-value problem for the Euler equations in the fundamental domain $\Omega ^\sigma$:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t \mathbf {u} + (\mathbf {u}\cdot \nabla )\mathbf {u} = -\nabla p, &{} \quad \text{ in } (0,+\infty ) \times \Omega ^\sigma ; \\ \text{ div }\, \mathbf {u} = 0, &{} \quad \text{ in } [0,+\infty ) \times \Omega ^\sigma ;\\ \mathbf {u}(t,x',x_3) \cdot x'= 0, &{} \quad \text{ for } t \in [0,+\infty ), \quad |x'|=1, \; 0\le x_3\le \sigma ;\\ \mathbf {u}(t,x',0) =\mathbf {u}(t,x',\sigma ) &{} \quad \text{ for } t \in [0,+\infty ), \quad x'\in D;\\ p(t,x',0) =p(t,x',\sigma ) &{} \quad \text{ for } t \in [0,+\infty ), \quad x'\in D;\\ \mathbf {u}(0,x) = \mathbf {u}_0, &{} \quad x\in \Omega ^\sigma , \end{array}\right. \end{aligned}$$

(5.1)

where again $x=(x',x_3)$ and $x'=(x_1,x_2)$.

Let

$$\begin{aligned} \varvec{\xi }: = \left( x_2, -x_1, \frac{\sigma }{2\pi } \right) = - \mathbf {x}_H^\perp + \frac{\sigma }{2\pi } \mathbf {e}_3. \end{aligned}$$

(5.2)

We will consider flows satisfying the following no-helical-swirl or stretching condition:

$$\begin{aligned} \mathbf {u}\cdot \varvec{\xi }=0. \end{aligned}$$

(5.3)

This condition is preserved by smooth flows under the time evolution governed by the Euler equations.

There are several consequences of this condition. Firstly, the vertical component $u_3$ of the velocity field $\mathbf {u}$ is computed from the other two components, i.e., the dynamics is planar. Secondly, the vorticity $\varvec{\omega }= \text {curl}_x \mathbf {u}$ is given by

$$\begin{aligned} \varvec{\omega }(t, x) = \frac{2\pi }{\sigma } \omega (t,x) \, \varvec{\xi },\qquad \omega :=\omega ^3, \end{aligned}$$

(5.4)

where $\omega ^3$ is the component of the vorticity along the axis of the cylinder $\Omega$ (see Lemma 2.11 in [6] for a proof of this fact). Furthermore, $\omega$ is advected by the flow $\mathbf {u}$:

$$\begin{aligned} \partial _t \omega + \mathbf {u}\cdot \nabla \omega = 0. \end{aligned}$$

(5.5)

To derive the symmetry-reduced equations, we recall that $\mathbf {w}(t,y) = \mathbf {u}(t,y,0)$ from Proposition 2.1, given that the matrices $M$ and $m$ becomes the identity matrix for $x_3=0$. Consequently,

$$\begin{aligned} \varpi (t,y) := \omega (t,y,0) = \nabla _y^{\perp }\cdot \mathbf {w}_H(t,y) = \text {curl}_y \mathbf {w}_H(t,y). \end{aligned}$$

(5.6)

Above, to avoid introducing further notation, we have abused notation slightly and identified $(w^1,w^2)$ with $\mathbf {w}_H= (w^1,w^2,0)$, where $w^1, w^2$ are the horizontal components of $\mathbf {w}$ with respect to the standard Cartesian frame in $\mathbb {R}^3$.

While $\mathbf {w}_H$ is not divergence free in view of (3.3b), one observes that a divergence-free 2D flow can be constructed from $\mathbf {w}$ under the no-helical-swirl condition, which therefore admits a stream function $\psi$ on $D$. This stream function satisfies:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{y_1} \psi = \frac{4\pi ^2}{\sigma ^2} \left[ -y_1 y_2 \, w^1 + \left( \frac{\sigma ^2}{4\pi ^2} + y_1^2 \right) w^2 \right] , \\ \partial _{y_2} \psi = - \frac{4\pi ^2}{\sigma ^2} \left[ \left( \frac{\sigma ^2}{4\pi ^2} + y_2^2 \right) w^1 -y_1 y_2 \, w^2 \right] . \end{array}\right. } \end{aligned}$$

(5.7)

We define the following matrix:

$$\begin{aligned} H(y) := \frac{4\pi ^2}{\sigma ^2} \left[ \begin{matrix} \left( \frac{\sigma ^2}{4\pi ^2} + y_2^2 \right) &{} -y_1 y_2\\ -y_1 y_2 &{} \left( \frac{\sigma ^2}{4\pi ^2} + y_1^2 \right) \\ \end{matrix} \right] , \end{aligned}$$

(5.8)

and rewrite as

$$\begin{aligned} \nabla _y^\perp \psi = H(y)\, \mathbf {w}_H. \end{aligned}$$

A direct calculation, as in [6], then shows that

$$\begin{aligned} \begin{aligned} \text {curl}\, \mathbf {w}_H&= \text{ div }_y K(y)\, \nabla _y \psi , \quad \text { with}\\ K(y)&:= \frac{1}{\frac{\sigma ^2}{4\pi ^2} + |y|^2} \left[ \begin{matrix} \left( \frac{\sigma ^2}{4\pi ^2} + y_2^2 \right) &{} - y_1 y_2 \\ - y_1 y_2 &{} \left( \frac{\sigma ^2}{4\pi ^2} + y_1^2 \right) \end{matrix} \right] . \end{aligned} \end{aligned}$$

(5.9)

From (5.6) and (5.9), it follows that

$$\begin{aligned} \varpi = \mathcal {L}_H\, \psi , \end{aligned}$$

where the operator $\mathcal {L}_H$ is defined by:

$$\begin{aligned} \mathcal {L}_H := \text{ div }_y( K(y)\, \nabla _y). \end{aligned}$$

(5.10)

It is not difficult to show that $\mathcal {L}_H$ is a second-order, scalar, strongly elliptic operator. Consequently, $\nabla ^{2} \mathcal {L}_H^{-1}$ is a singular integral (see e.g. [17] for a proof).

Next, calculus equalities show that the transport equation (5.5) for $\omega$ reduces by helical symmetry (i.e., using the correspondence in Proposition 2.1) to the following equation for $\varpi$ on $(0,T)\times D$:

$$\begin{aligned} \partial _t \varpi + \mathbf {w}_H \cdot \nabla _y \varpi + \frac{4\pi ^2}{\sigma ^2} (y^\perp \cdot \mathbf {w}_H) E \varpi = 0, \end{aligned}$$

where $E$ is again the operator given in (2.4). Using (5.9), we can rewrite this equation as an equation for $\varpi$ and $\psi$ only (cf. [6, Lemma 2.17].) Furthermore, we can choose Dirichlet boundary conditions for $\psi$ from the no-penetration condition for $\mathbf {u}$ as in Corollary 2.16 of [6]. Therefore, under the no-helical swirl condition and for sufficiently regular solutions, the initial-boundary-value (5.1) for the Euler equations is equivalent to the following symmetry-reduced system:

$$\begin{aligned}&\partial _t \varpi + \partial _{y_1} \psi \,\partial _{y_2} \varpi - \partial _{y_2} \psi \,\partial _{y_1} \varpi =0,&y\in D, 0<t<T, \\&\varpi = \mathcal {L}_H \psi ,&y\in D, 0<t<T,\nonumber \end{aligned}$$

(5.11a)

$$\begin{aligned}&\psi (0,y) = \psi _0(y),&y\in D, \end{aligned}$$

(5.11b)

$$\begin{aligned}&\psi \vert _{\partial D} = 0,&0<t<T . \end{aligned}$$

(5.11c)

Since (5.11a) is a transport equation by the divergence-free vector field $\nabla ^\perp _y \psi$, the $L^\infty$ norm of the reduced vorticity $\varpi$ is preserved under the flow. By (5.4) and Proposition 2.1, the vorticity $\varvec{\omega }= \text {curl}_x \mathbf {u}$ is preserved under the flow induced by $\mathbf {u}$. By the Beale-Kato-Majda criterion ( see e.g. [14]) then, smooth helical solutions of (5.1) are global in time and agree with a weak solution with the same initial data.

We next discuss weak solutions. Given $\psi _0\in H^1_0(D)\cap H^2(D)$, we call a function $\psi \in L^1([0,T);H^1_0(D)\cap H^2(D))$ a weak solution of the above system on $[0,T)$ with initial data $\psi _0$ if, for all test function $\phi \in C^\infty _c([0,T)\times D), \psi$ satisfies:

$$\begin{aligned} \begin{aligned} \int _{D} \mathcal {L}_H \psi _0 \,\phi (0)\, dy&- \int _0^T \int _D \mathcal {L}_H \psi \,\partial _t \phi \, dy\, dt + \int _0^T \int _D \partial _{y_2} \psi \mathcal {L}_H \psi \, \partial _{y_1} \phi \, dy\, dt \\&- \int _0^T \int _D \partial _{y_1} \mathcal {L}_H \psi \, \partial _{y_2} \phi \, dy\, dt = 0. \end{aligned} \end{aligned}$$

(5.12)

Ettinger and Titi [6] proved that there exists a unique weak solution on $[0,T)$, for all $T>0$, provided in addition $\mathcal {L}_H \psi _0\in L^\infty (D)$. In this case the solution satisfies $\mathcal {L}_H \psi \in L^\infty ((0,T)\times D)$.

While there is an existence theory for weak solutions of the Euler equation in three dimensions [4, 20], we will give here a definition of weak solution to (5.1) adapted to the geometry of the problem and amenable to the analysis of the limit $\sigma \rightarrow \infty$ (for further discussion of helical weak solutions, in particular non-uniqueness of weak solutions with helical initial data and symmetry-breaking, we refer the reader to [1].) Let $\psi$ be the unique weak solution of (5.11) with initial condition $\psi _0\in H^1_0(D)\cap H^2(D)$ such that $\mathcal {L}_H \psi _0\in L^\infty (D)$. Let $\mathbf {w}=(\mathbf {w}_H,w^3)$, where $\mathbf {w}_H$ is given in the equation above (5.9) and $w^3$ is obtained from $\mathbf {w}_H$ via (5.3) as

$$\begin{aligned} w^3= \frac{2\pi }{\sigma } y^\perp \cdot \mathbf {w}_H. \end{aligned}$$

Let $\mathbf {u}$ be defined from $\mathbf {w}$ by (2.8). We will call $\mathbf {u}$ a weak, helical solution of (5.1). This definition is justified in view of the following proposition.

Proposition 5.1

Let $\{\psi _{0,n}\}$ be a sequence of functions converging to $\psi _0\in H^1_0(D)\cap H^2(D)$. Let $\psi _n$ be the smooth solution of (5.11a) with initial data $\psi _{0,n}$. Then, $\psi _n$ converges uniformly on $[0,T)\times D)$ to $\psi$ the unique weak solution of (5.11).

The proof is contained in [6]. We recall it briefly.

Proof

The sequence $\{\psi _n\}$ is uniformly bounded in $L^1([0,T);H^1_0(D)\cap H^2(D))$ and $\mathcal {L}_H \psi$ is uniformly bounded in $L^\infty ([0,T)\times D)$. Recall that the equation for $\varpi _n=\mathcal {L}_H \psi _n$ is a transport equation by $\nabla ^\perp _y \psi _n$, which is divergence free. Since $\partial _i\partial _j \mathcal {L}_H^{-1}$ is a Calderon-Zygmund singular integral, $\{\nabla ^\perp _y \psi _n\}$ is bounded in the space LLip of Log-Lipschitz vector fields. Hence, the family $\{X_n\}$, where $X_n$ is the flow generated by $\nabla ^\perp \psi _n$ is equicontinuous and hence, upon possibly passing to subsequences, $\varpi _n$ converges strongly in $L^1((0,T)\times D)$ and $\nabla ^\perp _y \psi _n$ converges uniformly to $\nabla ^\perp _y \psi _n$. In particular, $\psi _n$ converges uniformly to $\psi$. These convergence results are enough to pass to the limit in the weak formulation (5.12) (cf. [14, Section 8.2.2].) The limit $\lim _{n\rightarrow \infty } \psi _n$ must necessarily agree with $\psi$ by uniqueness of the solution, so the whole sequence converges to $\psi$.$\square$

This result also implies that, if $\varpi (0)\in L^\infty (D)$, then $\varpi (t,x)=\varpi (X^{-1}(t,x)$, where $X$ is the flow generated by $\nabla ^\perp \psi$, is the (unique) weak solution of (5.11), hence all its $L^p$ norms are constant in time.

We next discuss the limit $\sigma \rightarrow \infty$. We reinstate the explicit dependence on $\sigma$, and write for example $\mathbf {u}^\sigma$ for the solution of (5.1), $\varvec{\omega }^\sigma$ for $\text {curl}_x \mathbf {u}^\sigma$ and so on. We denote the corresponding quantities in the limit by $\mathbf {u}^\infty , \varvec{\omega }^\infty$ and so on.

Formally taking the limit $\sigma \rightarrow \infty$ in (5.3) gives $u^{\infty ,3}\equiv 0$ and, hence, $\mathbf {u}^\infty = \mathbf {u}^\infty _H$. Furthermore, $\mathbf {u}^\infty$ becomes independent of the $x_3$ variable, so that

$$\begin{aligned} \mathbf {u}^\infty (x) = \mathbf {w}^\infty (x')=\mathbf {w}^\infty _H(x') \end{aligned}$$

is divergence-free as a vector field on $D$. Also, the matrix $K^\sigma$ approaches the identity matrix in the limit, so that $\mathcal {L}^\infty _H$ is simply the Laplace operator, $\psi ^\infty$ is the stream functions associated to $\mathbf {u}^\infty _H$, and $\varpi ^\infty (x') = \omega ^\infty (x) = \text {curl}_{x'} \mathbf {u}^\infty _H(x)$. In particular, (5.11a) becomes the vorticity-stream function formulation of the 2D Euler equations. We conclude that, at least formally, helical solutions to the 3D Euler equations become planar 2D solutions of the Euler equations as $\sigma \rightarrow \infty$.

We explicitly state the limit problem:

$$\begin{aligned}&\partial _t \varpi ^\infty + \partial _{y_1} \psi ^\infty \,\partial _{y_2} \varpi ^\infty - \partial _{y_2} \psi ^\infty \,\partial _{y_1} \varpi ^\infty =0,&\quad y\in D, 0<t<T, \end{aligned}$$

(5.13a)

$$\begin{aligned}&\varpi ^\infty = \Delta _y \psi ^\infty ,&\quad y\in D, 0<t<T, \nonumber \\&\psi ^\infty (0,y) = \psi ^\infty _0(y),&\quad y\in D, \end{aligned}$$

(5.13b)

$$\begin{aligned}&\psi ^\infty \vert _{\partial D} = 0,&\quad 0<t<T. \end{aligned}$$

(5.13c)

Below we will study convergence of the corresponding stream functions $\psi ^\sigma \rightarrow \psi ^\infty$ as $\sigma \rightarrow \infty$. Since the uniqueness and regularity of weak solutions depends on an $L^\infty$ control on the vorticity, we will prescribe the initial vorticity $\varpi ^\sigma _0$ independent of $\sigma$, i.e.,

$$\begin{aligned} \varpi ^\infty _0 = \varpi ^\sigma _0=\varpi _0 \in L^\infty (D). \end{aligned}$$

This choice can be relaxed by taking a sequence $\varpi ^\sigma _0$ converging to $\varpi _0$ strongly in $L^\infty (D)$. We then obtain an initial condition for the stream function, $\psi _0^\sigma$, that is $\sigma$-dependent. We choose the initial data for the stream function as the unique solution in $H^1_0(D)$ of the following problems, respectively:

$$\begin{aligned} \begin{aligned} \Delta \psi ^\infty _0 = \varpi _0, \\ \mathcal {L}^\sigma _H \psi ^\sigma _0 = \varpi _0. \end{aligned} \end{aligned}$$

(5.14)

By elliptic regularity, $\psi _0^\infty , \psi ^\sigma _0 \in W^{2,p}$ for all $1<p<\infty$.

Next we will derive uniform bounds in $\sigma$ on the $W^{2,p}$ norm of $\psi ^\sigma$ and then use compactness arguments to pass to the limit. It is well known that, under the condition that the initial vorticity $\varpi _0$ is bounded, solutions to the 2D Euler equations are global in time and unique [9]. Therefore, it will be enough to establish convergence along subsequences.

Lemma 5.2

Let $1<p<\infty$ be fixed. Then, there exists a constant $C_p>0$ such that, for all $\sigma >1$ and for all $f \in W^{2,p}(D)$,

$$\begin{aligned} \Vert \mathcal {L}_H f\Vert _{L^p(D)}\le C_p \Vert f\Vert _{W^{2,p}(D)}. \end{aligned}$$

(5.15)

Moreover, there exists a $\sigma _0>1$ and a constant $C_p>0$, independent of $\sigma \in [\sigma _0,\infty )$ such that

$$\begin{aligned} \Vert f\Vert _{W^{2,p}(D)} \le C_p\, \Vert \mathcal {L}_H f\Vert _{L^p(D)}. \end{aligned}$$

(5.16)

Proof

We observe that we can write the matrix $K^\sigma = I_2 + F^\sigma$, where $I_2$ is the $2\times 2$-identity matrix and

$$\begin{aligned} F^\sigma (y) = \frac{1}{ 1+ \frac{4\pi ^2 \, |y|^2}{\sigma ^2}} \left[ \begin{array}{ll} \frac{4\pi ^2\, y_1^2}{\sigma ^2} &{} \frac{4\pi ^2 \, y_1 y_2 }{\sigma ^2} \\ \frac{4\pi ^2 \, y_1 y_2 }{\sigma ^2} &{} \frac{4\pi ^2 \, y_2^2}{\sigma ^2} \end{array}\right] . \end{aligned}$$

We have:

$$\begin{aligned} \Vert F^\sigma \Vert _{L^\infty (D)} \le C_1\, \frac{1}{\sigma ^2}, \qquad \Vert \nabla _y F^\sigma \Vert _{L^\infty (D)} \le C_2 \frac{1}{\sigma ^2} , \end{aligned}$$

(5.17)

for some constants $C_1, C_2$ independent of $\sigma$. The bound (5.15) then follows immediately.

To establish (5.16), we write

$$\begin{aligned} \Delta _y f = \mathcal {L}_H f - F^\sigma (y):\nabla ^2 f - (\text{ div }_y F^\sigma (y)) \cdot \nabla _y f, \end{aligned}$$

so that from elliptic regularity for the Poisson’s problem for $1<p<\infty$, Hölder’s inequality and (5.17):

$$\begin{aligned} \begin{aligned} \Vert f\Vert _{W^{2,p}}&\le C_p' \, \Vert \mathcal {L}_H f\Vert _{L^p} + \Vert F^\sigma :\nabla ^2 f \Vert _{L^p} + \Vert \text{ div }_y F^\sigma \cdot \nabla _y f\Vert _{L^p} \\&\le C_p'\, \Vert \mathcal {L}_H f\Vert _{L^p} + C_1\, \frac{1}{\sigma ^2} \Vert \nabla ^2_y f\Vert _{L^p} + C_2\, \frac{1}{\sigma ^2} \Vert \nabla _y f\Vert _{L^p}, \end{aligned} \end{aligned}$$

or equivalently:

$$\begin{aligned} (1-(C_1+C_2)/\sigma ^2) \Vert f\Vert _{W^{2,p}} \le C_p' \, \Vert \mathcal {L}_H f\Vert _{L^p}. \end{aligned}$$

So, the result follows provided we choose $\sigma _0 > 1/\sqrt{(C_1+C_2)}$.$\square$

We now state and prove our convergence result for the Euler equations. We recall that the only difference between the equations at $\sigma$ finite and in the limit is the equation expressing the relationship between the vorticity and the stream function.

Theorem 5.3

Let $\varpi _0\in L^\infty (D)$. Let $\psi ^\sigma _0$ and $\psi ^\infty _0$ be given by (5.14). Let $\psi ^\sigma$ be the unique weak solution of (5.11) with initial data $\psi _0^\sigma$. Let $\psi ^\infty$ be the unique weak solution of (5.13) with initial data $\psi ^\infty _0$. Then, $\psi ^\sigma$ converges to $\psi ^\infty$ strongly in $L^p([0,T);W^{1,p}(D)), 1<p<\infty$.

Proof

Since the initial vorticity $\varpi _0\in L^\infty (D), \nabla ^\perp \psi ^\sigma \in$ LLip$(D)$ with a bound on the Log-Lipschitz norm that is uniform in $\sigma$ for $\sigma \in [1,\infty )$ by (5.17). Therefore, we have a uniform bound on $\varpi ^\sigma$ in $L^\infty ([0,T)\times D)$, thanks to the transport equation (5.11a). In turn by (5.16), this bound implies a bound on the family $\{\psi ^\sigma \}$ of weak solutions of (5.11) in all spaces $L^\infty ([0,T);W^{2,p}), 1<p<\infty$ that is uniform in $\sigma \ge \sigma _0$ for $\sigma _0$ large enough.

Next, we recall the following a priori bound for weak solutions of (5.11) (see [6, Lemma 4.2]):

$$\begin{aligned} \Vert \partial _t \psi ^\sigma \Vert _{L^\infty ([0,T);W^{1,p}(D))} \le C_p \Vert \mathcal {L}_H^\sigma \psi ^\sigma \Vert _{L^\infty ((0,T)\times D)}\, \Vert \psi ^\sigma \Vert _{L^\infty ([0,T);W^{1,p})}, \end{aligned}$$

where $C_p$ is independent of $\sigma$ for $\sigma$ large enough as in Lemma 5.2. Therefore, $\{\psi ^\sigma \}$ is uniformly bounded in Lip$([0,T);W^{1,p}(D))$. By the Aubin compactness theorem (see e.g. [3, Lemma 8.4]) then, there is a sequence $\{\psi ^{\sigma _n}\}$ that converges strongly in $L^q([0,T);W^{1,p}), 1<q<\infty$, to a function $\psi ^\infty$. Upon passing to a subsequence if necessary, one can assume also that $\varpi ^{\sigma _n}$ converges weakly-$*$ in $L^\infty ([0,T)\times D)$ to a function $\varpi ^\infty$ from the uniform bound obtained above. It remains to show that $\varpi ^\infty =\Delta \psi ^\infty$ in $L^2(D)$. This result follows from the identity $\varpi ^\sigma =\mathcal {L}_H \psi ^\sigma$, valid for all $\sigma$, and (5.17), by writing again $K^\sigma =I_2 +F^\sigma$.

As in the proof of Proposition 5.8 in [6], these convergence results are sufficient to show that $\psi ^\infty$ and $\varpi ^\infty$ satisfy the weak formulation of the limit problem (5.13). But weak solutions of the 2D Euler equations are unique if the vorticity is bounded, hence any converging sequence of $\{\psi ^\sigma \}$ must converge to $\psi ^\infty$.$\square$