Abstract
We consider the Euler–Korteweg system with space periodic boundary conditions \( x \in {\mathbb {T}}^d\). We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this paper we consider the compressible Euler–Korteweg (EK) system
which is a modification of the Euler equations for compressible fluids to include capillary effects, under space periodic boundary conditions \( x \in {\mathbb {T}}^d {:}{=} ({\mathbb {R}}/2 \pi {\mathbb {Z}})^d \). The scalar variable \( \rho (t,x) > 0 \) is the density of the fluid and \( \vec {u} (t,x) \in {\mathbb {R}}^d \) is the time dependent velocity field.
The functions \( K (\rho ) \), \( g (\rho ) \) are defined on \( {\mathbb {R}}^+ \), smooth, and \( K (\rho ) \) is positive.
The quasi-linear equations (1.1) appear in a variety of physical contexts modeling phase transitions [17], water waves [14], quantum hydrodynamics where \( K(\rho ) = \kappa / \rho \) [4], see also [15].
Local well posedness results for the (EK)-system have been obtained in Benzoni-Gavage, Danchin and Descombes [8] for initial data sufficiently localized in the space variable \( x \in {\mathbb {R}}^d \). Then, for \( d \ge 3 \), thanks to dispersive estimates, global in time existence results have been obtained for small irrotational data by Audiard–Haspot [7], assuming the sign condition \(g'( \rho ) > 0\). The case of quantum hydrodynamics corresponds to \(K(\rho ) = \kappa /\rho \) and, in this case, the (EK)-system is formally equivalent, via Madelung transform, to a semilinear Schrödinder equation on \( {\mathbb {R}}^d \). Exploiting this fact, global in time weak solutions have been obtained by Antonelli–Marcati [4, 5] also allowing \(\rho (t,x) \) to become zero (see also the recent paper [6]).
In this paper we prove a local in time existence result for the solutions of (1.1), with space periodic boundary conditions, under natural minimal regularity assumptions on the initial datum in Sobolev spaces, see Theorem 1.1. Relying on this result, in a forthcoming paper [10], we shall prove a set of long time existence results for the (EK)-system in 1-space dimension, in the same spirit of [11, 12].
We consider irrotational velocity fields \( \vec {u} \), i.e. with vorticity \( \Omega := {\text {curl}}(\vec {u} ):= \nabla \vec {u} - (\nabla \vec {u} )^\top \) equal to zero. Note that, since \( \partial _t \Omega = -{\text {curl}}( \Omega \vec {u} ) \), if \( \Omega \) is initially zero, then \( \Omega \) remains zero under the evolution of (1.1). An irrotational vector field on \( {\mathbb {T}}^d \) reads (Helmotz decomposition)
where \( \phi : {\mathbb {T}}^d \rightarrow {\mathbb {R}}\) is a scalar potential. By the second equation in (1.1) and \({\text {curl}}\, \vec {u} = 0 \), we get
is independent of time. Note that if the dimension \(d = 1\), the average \( {\frac{1}{2\pi }} \int _{{\mathbb {T}}} u(t,x) {\text {d}}x\) is an integral of motion for (1.1), and thus any solution u(t, x), \( x \in {\mathbb {T}}\), of the (EK)-system (1.1) has the form (1.2) with \(c(t) = c(0) \) independent of time, that is \( u (t,x) = c(0) + \phi _x (t,x) \).
The (EK) system (1.1) is Galilean invariant: if \( (\rho (t,x), \vec {u} (t,x)) \) solves (1.1) then
solve (1.1) as well. Thus, regarding the Euler–Korteweg system in a frame moving with a constant speed \( \vec {c} (0) \), we may always consider in (1.2) that
The Euler–Korteweg equations (1.1) read, for irrotational fluids,
The main result of the paper proves local well posedness for the solutions of (1.3) with initial data \((\rho _0, \phi _0) \) in Sobolev spaces
where \( \langle j \rangle := \max \{1,|j|\} \), under the natural mild regularity assumption \( s > 2 + (d/2)\). Along the paper, \( H^s ({\mathbb {T}}^d) \) may denote either the Sobolev space of real valued functions \( H^s ({\mathbb {T}}^d, {\mathbb {R}}) \) or the complex valued ones \( H^s ({\mathbb {T}}^d, {\mathbb {C}}) \).
Theorem 1.1
(Local existence on \({\mathbb {T}}^d\)) Let \(s > 2 + \frac{d}{2} \) and fix \(\frac{d}{2}<s_0\le s-2\). For any initial data
there exists \(T:= T(\Vert (\rho _0, \phi _0)\Vert _{s_0+2}, \min _x \rho _0(x)) >0 \) and a unique solution \( (\rho , \phi )\) of (1.3) such that
and \(\rho (t,x) > 0 \) for any \(t \in [-T, T]\). Moreover, for \(|t| \le T\), the solution map \( (\rho _0, \phi _0) \mapsto (\rho (t, \cdot ), \phi (t, \cdot ) ) \) is locally defined and continuous in \( H^{s}({\mathbb {T}}^d,{\mathbb {R}})\times H^{s}({\mathbb {T}}^d,{\mathbb {R}}) \).
We remark that it is sufficient to prove the existence of a solution of (1.3) on [0, T] because system (1.3) is reversible: the Euler–Korteweg vector field X defined by (1.3) satisfies \( X\circ {\mathcal {S}}= - {\mathcal {S}}\circ X \), where \( {\mathcal {S}}\) is the involution
Thus, denoting by \( (\rho , \phi )(t,x) = \Omega ^t (\rho _0, \phi _0) \) the solution of (1.3) with initial datum \((\rho _0, \phi _0)\) in the time interval [0, T], we have that \( {\mathcal {S}}\Omega ^{-t} ({\mathcal {S}}(\rho _0, \phi _0) ) \) solves (1.3) with the same initial datum but in the time interval \([-T,0]\).
Let us make some comments about the phase space of system (1.3). Note that the average \( {\frac{1}{(2\pi )^d}} \int _{{\mathbb {T}}^d} \rho (x) \, {\text {d}}x \) is a prime integral of (1.3) (conservation of the mass), namely
remains constant along the solutions of (1.3). Note also that the vector field of (1.3) depends only on \( \phi - \frac{1}{(2 \pi )^d}\int _{{\mathbb {T}}^d} \phi \, {\text {d}}x \). As a consequence, the variables \( (\rho -{{\mathtt {m}}}, \phi ) \) belong naturally to some Sobolev space \( H^s_0({\mathbb {T}}^d) \times \dot{H}^s ({\mathbb {T}}^d) \), where \(H^s_0({\mathbb {T}}^d) \) denotes the Sobolev space of functions with zero average
and \(\dot{H}^s({\mathbb {T}}^d)\), \(s \in {\mathbb {R}}\), the corresponding homogeneous Sobolev space, namely the quotient space obtained by identifying all the \(H^s({\mathbb {T}}^d)\) functions which differ only by a constant. For simplicity of notation we denote the equivalent class \( [u] := \{ u + c, c \in {\mathbb {R}}\} \), just by u. The homogeneous norm of \( u \in \dot{H}^s ({\mathbb {T}}^d) \) is \( \Vert u \Vert _s^2 := \sum _{j \in {\mathbb {Z}}^d {\setminus } \{0\}} |u_j|^2 |j |^{2s}\). We shall denote by \( \Vert \ \Vert _s \) either the Sobolev norm in \( H^s \) or that one in the homogenous space \( \dot{H}^s \), according to the context.
Let us make some comments about the proof. First, in view of (1.5), we rewrite system (1.3) in terms of \( \rho \leadsto {{\mathtt {m}}}+ \rho \) with \( \rho \in H^s_0 ({\mathbb {T}}^d) \), obtaining
Then Theorem 1.1 follows by the following result, that we are going to prove
Theorem 1.2
Let \(s > 2 + \frac{d}{2} \), \( \frac{d}{2}< s_0 < s- 2 \) and \( 0< {{\mathtt {m}}}_1 < {{\mathtt {m}}}_2 \). For any initial data of the form \(({{\mathtt {m}}}+ \rho _0, \phi _0)\) with \( (\rho _0, \phi _0) \in H^s_0({\mathbb {T}}^d) \times \dot{H}^s({\mathbb {T}}^d) \) and \( {{\mathtt {m}}}_1< {{\mathtt {m}}}+ \rho _0(x) < {{\mathtt {m}}}_2 \), \( \forall x \in {\mathbb {T}}^d \), there exist \( T= T\big (\Vert (\rho _0, \phi _0)\Vert _{s_0+2}, \min _x ({{\mathtt {m}}}+ \rho _0(x)) \big ) >0 \) and a unique solution \(({{\mathtt {m}}}+ \rho , \phi )\) of (1.6) such that
and \( {{\mathtt {m}}}_1< {{\mathtt {m}}}+ \rho (t,x) < {{\mathtt {m}}}_2 \) holds for any \(t \in [0, T]\). Moreover, for \(|t| \le T\), the solution map \( (\rho _0, \phi _0) \mapsto (\rho (t, \cdot ), \phi (t, \cdot ) ) \) is locally defined and continuous in \(H^s_0({\mathbb {T}}^d) \times \dot{H}^s({\mathbb {T}}^d) \).
We consider system (1.6) on the homogeneous space \( \dot{H}^s \times \dot{H}^s \), that is we study
where \(\Pi _0^\perp \) is the projector onto the Fourier modes of index \(\ne 0\). For simplicity of notation we shall not distinguish between systems (1.7) and (1.6), which are equivalent via the isomorphism \( \Pi _0^\bot : \dot{H}^s({\mathbb {T}}^d) \rightarrow H_0^s({\mathbb {T}}^d)\). In Sect. 3, we paralinearize (1.6), i.e. (1.7), up to bounded semilinear terms (for which we do not need Bony paralinearization formula). Then, introducing a suitable complex variable, we transform it into a quasi-linear type Schrödinger equation, see system (3.4), defined in the phase space
We use paradifferential calculus in the Weyl quantization, because it is quite convenient to prove energy estimates for this system. Since (3.4) is a quasi-linear system, in order to prove local well posedness (Proposition 4.1) we follow the strategy, initiated by Kato [20], of constructing inductively a sequence of linear problems whose solutions converge to the solution of the quasilinear equation. Such a scheme has been widely used, see e.g. [1, 8, 18, 22] and reference therein.
The equation (1.3) is a Hamiltonian PDE. We do not exploit explicitly this fact, but it is indeed responsible for the energy estimate of Proposition 4.4. The method of proof of Theorem 1.1 is similar to the one in Feola–Iandoli [19] for Hamiltonian quasi-linear Schrödinger equations on \( {\mathbb {T}}^d \) (and Alazard–Burq–Zuily [1] in the case of gravity-capillary water waves in \( {\mathbb {R}}^d \)). The main difference is that we aim to obtain the minimal smoothness assumption \( s > 2 + (d/2) \). This requires to optimize several arguments, and, in particular, to develop a sharp para-differential calculus for periodic functions that we report in the Appendix in a self-contained way. Some other technical differences are in the use of the modified energy (Sect. 4.2), the mollifiers (4.17) which enables to prove energy estimates independent of \( \varepsilon \) for the regularized system, the argument for the continuity of the flow in \( H^s \). We expect that our approach would enable to extend the local existence result of [19] to initial data fulfilling the minimal smoothness assumptions \( s > 2 + (d/2) \).
We now set some notation that will be used throughout the paper. Since \( K : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}\) is positive, given \( 0< {{\mathtt {m}}}_1 < {{\mathtt {m}}}_2 \), there exist constants \(c_K, C_K > 0 \) such that
Since the velocity potential \( \phi \) is defined up to a constant, we may assume in (1.6) that
From now on we fix \(s_0\) so that
The initial datum \(\rho _0 (x) \) belongs to the open subset of \( H^{s_0}_0({\mathbb {T}}^d)\) defined by
and we shall prove that, locally in time, the solution of (1.6) stays in this set.
We write \(a\lesssim b\) with the meaning \(a \le C b\) for some constant \(C >0\) which does not depend on relevant quantities.
2 Functional Setting and Paradifferential Calculus
The Sobolev norms \( \Vert \ \Vert _s \) satisfy interpolation inequalities (see e.g. Sect. 3.5 in [9]):
-
(i)
for all \( s \ge s_0 > \frac{d}{2} \), \( u, v \in H^s \),
$$\begin{aligned} \Vert u v \Vert _s \lesssim \Vert u \Vert _{s_0} \Vert v \Vert _s + \Vert u \Vert _{s} \Vert v \Vert _{s_0}. \end{aligned}$$(2.1) -
(ii)
Let \(s_0 > \frac{d}{2} \). For all \( 0 \le s \le s_0 \), \( v \in H^s \), \( u \in H^{s_0} \),
$$\begin{aligned} \Vert u v \Vert _s \lesssim \Vert u \Vert _{s_0} \Vert v \Vert _s. \end{aligned}$$(2.2) -
(iii)
For all \( s_1 < s_2 \), \( \theta \in [0,1] \) and \( u \in H^{s_2} \),
$$\begin{aligned} \Vert u \Vert _{\theta s_1 + (1- \theta ) s_2} \le \Vert u \Vert _{s_1}^\theta \Vert u \Vert _{s_2}^{1-\theta }. \end{aligned}$$(2.3) -
(iv)
For all \(a\le \alpha \le \beta \le b\), with \(\alpha + \beta = a + b\), \(u,v\in H^b \),
$$\begin{aligned} \Vert u \Vert _{\alpha } \Vert v \Vert _{\beta } \le \Vert u \Vert _{a} \Vert v \Vert _{b} + \Vert u \Vert _{b} \Vert v \Vert _{a}. \end{aligned}$$(2.4)
Paradifferential calculus. We now introduce the notions of paradifferential calculus that will be used in the proof of Theorem 1.1. We develop it in the Weyl quantization since it is more convenient to get the energy estimates of Sect. 4. The main results are the continuity Theorem 2.4 and the composition Theorem 2.5, which require mild regularity assumptions of the symbols in the space variable (they are deduced by the sharper results proved in Theorems A.7 and A.8 in the Appendix). This is needed in order to prove the local existence Theorem 1.1 with the natural minimal regularity on the initial datum \( (\rho _0, \phi _0) \in H^s \times H^s \) with \( s > 2 + \frac{d}{2} \).
Along the paper \( {\mathscr {W}} \) may denote either the Banach space \( L^\infty ({\mathbb {T}}^d)\), or the Sobolev spaces \( H^s ({\mathbb {T}}^d)\), or the Hölder spaces \( W^{\varrho ,\infty } ({\mathbb {T}}^d)\), introduced in Definition A.3. Given a multi-index \( \beta \in {\mathbb {N}}_0^d \) we define \( |\beta | := \beta _1 + \cdots + \beta _d \).
Definition 2.1
(Symbols with finite regularity) Given \( m \in {\mathbb {R}}\) and a Banach space \({\mathscr {W}} \in \{ L^\infty ({\mathbb {T}}^d), H^s ({\mathbb {T}}^d), W^{\varrho ,\infty } ({\mathbb {T}}^d)\} \), we denote by \(\Gamma ^m_{\mathscr {W}}\) the space of functions \( a : {\mathbb {T}}^d \times {\mathbb {R}}^d \rightarrow {\mathbb {C}}\), \(a(x, \xi )\), which are \(C^\infty \) with respect to \(\xi \) and such that, for any \( \beta \in {\mathbb {N}}_0^d \), there exists a constant \(C_\beta >0\) such that
We denote by \(\Sigma ^m_{\mathscr {W}}\) the subclass of symbols \(a\in \Gamma ^m_{\mathscr {W}}\) which are spectrally localized, that is
where \({\widehat{a}}(j, \xi ) := (2 \pi )^{-d} \int _{{\mathbb {T}}^d} a(x,\xi ) e^{- {{\text {i}}}j \cdot x} {\text {d}}x\), \( j \in {\mathbb {Z}}^d \), are the Fourier coefficients of the function \(x \mapsto a(x, \xi )\).
We endow \(\Gamma ^m_{\mathscr {W}}\) with the family of norms defined, for any \(n \in {\mathbb {N}}_0\), by
When \({\mathscr {W}} = H^s \), we also denote \(\Gamma ^m_s \equiv \Gamma ^m_{H^s}\) and \(\left| a \right| _{m, s, n} \equiv \left| a \right| _{m,H^s, n}\). We denote by \( \Gamma _s^m \otimes {{{\mathcal {M}}}}_2 ({\mathbb {C}}) \) the \( 2 \times 2 \) matrices \( A = \begin{pmatrix} a_{1} &{} \quad a_{2} \\ a_{3} &{} \quad a_{4} \end{pmatrix} \) of symbols in \( \Gamma _s^m \) and \( | A |_{m, {\mathscr {W}}, n} := \max _{i=1, \ldots , 4}\{ | a_{i} |_{m, {\mathscr {W}}, n}\} \). Similarly we denote by \( \Gamma _s^m \otimes {\mathbb {R}}^d \) the d-dimensional vectors of symbols in \( \Gamma _s^m \).
Let us make some simple remarks:
-
(\(\mathrm{i}\)) given a function \( a(x) \in {\mathscr {W}} \) then \(a(x) \in \Gamma ^0_{{\mathscr {W}}} \) and
$$\begin{aligned} \left| u \right| _{0, {\mathscr {W}}, n} = \left\| u \right\| _{{\mathscr {W}}}, \forall n \in {\mathbb {N}}_0. \end{aligned}$$(2.8) -
(\(\mathrm{ii}\)) For any \(s_0 > \frac{d}{2}\) and \(0 \le \varrho ' \le \varrho \), we have that
$$\begin{aligned} \left| a \right| _{m, L^\infty , n} \lesssim \left| a \right| _{m, W^{\varrho ', \infty }, n} \lesssim \left| a \right| _{m, {W^{\varrho , \infty }}, n} \lesssim \left| a \right| _{m, H^{s_0+\varrho },n} , \quad \forall n \in {\mathbb {N}}_0. \end{aligned}$$(2.9) -
(\(\mathrm{iii}\)) If \( a \in \Gamma ^m_{{\mathscr {W}}}\), then, for any \( \alpha \in {\mathbb {N}}_0^d \), we have \( \partial _\xi ^\alpha a \in \Gamma ^{m-|\alpha |}_{{\mathscr {W}}} \) and
$$\begin{aligned} | \partial _\xi ^\alpha a |_{m-|\alpha |,{\mathscr {W}},n} \lesssim | a|_{m,{\mathscr {W}},n+|\alpha |}, \quad \forall n \in {\mathbb {N}}_0. \end{aligned}$$(2.10) -
(\(\mathrm{iv}\)) If \( a \in \Gamma ^m_{H^{s}} \), resp. \( a \in \Gamma ^m_{W^{\varrho , \infty }} \), then \( \partial _x^\alpha a \in \Gamma ^m_{H^{s-|\alpha |}} \), resp. \( \partial _x^\alpha a \in \Gamma ^m_{W^{\varrho -|\alpha |,\infty }} \), and
$$\begin{aligned} | \partial _x^\alpha a|_{m,s-|\alpha |,n} \lesssim | a|_{m,s,n}, \quad {\text {resp.}} \ |\partial _x^\alpha a|_{m,W^{\varrho -|\alpha |,\infty },n} \lesssim |a|_{m,W^{\varrho ,\infty },n}, \quad \forall n \in {\mathbb {N}}_0 \,.\nonumber \\ \end{aligned}$$(2.11) -
(\(\mathrm{v}\)) If \( a,b \in \Gamma ^m_{{\mathscr {W}}} \) then \( ab \in \Gamma ^m_{{\mathscr {W}}} \) with \( |ab|_{m+m',{\mathscr {W}},n} \lesssim |a|_{m,{\mathscr {W}},N} |b|_{m',{\mathscr {W}},n}\) for any \( n \in {\mathbb {N}}_0 \). In particular, if \( a,b \in \Gamma ^m_{s} \) with \( s > d / 2 \) then \( ab \in \Gamma ^{m+m'}_{s} \) and
$$\begin{aligned} | a b |_{m+m',s,n} \lesssim | a |_{m,s,n} | b |_{m',s_0,n} + | a |_{m,s_0,n} | b |_{m',s,n}, \quad \forall n \in {\mathbb {N}}_0. \end{aligned}$$(2.12)Let \(\epsilon \in (0,1)\) and consider a \( C^\infty \), even cut-off function \(\chi :{\mathbb {R}}^d \rightarrow [0,1]\) such that
$$\begin{aligned} \chi (\xi ) = {\left\{ \begin{array}{ll} 1 &{} \quad \text{ if } |\xi | \le 1.1 \\ 0 &{} \quad \text{ if } |\xi | \ge 1.9, \end{array}\right. } \qquad \chi _\epsilon (\xi ) := \chi \left( \frac{\xi }{\epsilon }\right) . \end{aligned}$$(2.13)Given a symbol a in \( \Gamma ^m_{{\mathscr {W}}} \) we define the regularized symbol
$$\begin{aligned} a_\chi (x, \xi ) := \chi _{\epsilon \left\langle \xi \right\rangle }(D) a(x, \xi ) = \sum _{j \in {\mathbb {Z}}^d} \chi _\epsilon \Big ( \frac{j}{\langle \xi \rangle } \Big ) \, {\widehat{a}} (j, \xi ) \, e^{{\text {i}}j \cdot x}. \end{aligned}$$(2.14)
Note that \(a_\chi \) is analytic in x (it is a trigonometric polynomial) and it is spectrally localized.
In order to define the Bony–Weyl quantization of a symbol \( a (x, \xi )\) we first remind the Weyl quantization formula
Definition 2.2
(Bony–Weyl quantization) Given a symbol \(a \in \Gamma ^m_{{\mathscr {W}}}\), we define the Bony–Weyl paradifferential operator \( {\text {Op}}^{BW} {(a)} = {\text {Op}}^{W} (a_\chi ) \) that acts on a periodic function u as
If \( A = \begin{pmatrix} a_{1} &{} \quad a_{2} \\ a_{3} &{} \quad a_{4} \end{pmatrix} \) is a matrix of symbols in \( \Gamma _s^m \), then \( {\text {Op}}^{BW} (A) \) is defined as the matrix valued operator \( \begin{pmatrix} {\text {Op}}^{BW} (a_{1}) &{} \quad {\text {Op}}^{BW} (a_{2}) \\ {\text {Op}}^{BW} (a_{3}) &{} {\text {Op}}^{BW} (a_{4}) \end{pmatrix} \).
Given a symbol \( a(\xi ) \) independent of x, then \( {\text {Op}}^{BW} {(a)} \) is the Fourier multiplier operator
Note that if \( \chi _\epsilon \Big ( \frac{k-j}{\langle k + j \rangle }\Big ) \ne 0 \) then \( |k-j| \le \epsilon \langle j + k \rangle \) and therefore, for \( \epsilon \in (0,1)\),
This relation shows that the action of a para-differential operator does not spread much the Fourier support of functions. In particular \( {\text {Op}}^{\text {BW}}(a) \) sends a constant function into a constant function and therefore \( {\text {Op}}^{\text {BW}}(a) \) sends homogenous spaces into homogenous spaces.
Remark 2.3
Actually, if \( \chi _\epsilon \big ( \frac{k-j}{\langle k + j \rangle }\big ) \ne 0 \), \( \epsilon \in (0,1/4) \), then \( |j| \le |j+k| \le 3 |j| \), for all \( j,k \in {\mathbb {Z}}^d \).
Along the paper we shall use the following results concerning the action of a paradifferential operator in Sobolev spaces.
Theorem 2.4
(Continuity of Bony–Weyl operators) Let \( a \in \Gamma ^m_{s_0} \), resp. \( a \in \Gamma ^m_{L^\infty } \), with \( m \in {\mathbb {R}}\). Then \({{\text {Op}}}^{\text {BW}}\!\left( a\right) \) extends to a bounded operator \(\dot{H}^{s} \rightarrow \dot{H}^{s-m}\) for any \( s \in {\mathbb {R}}\) satisfying the estimate, for any \( u \in \dot{H}^s \),
Moreover, for any \(\varrho \ge 0\), \( s \in {\mathbb {R}}\), \( u \in \dot{H}^s ({\mathbb {T}}^d)\),
Proof
Since \( {{\text {Op}}}^{\text {BW}}\!\left( a\right) = {{\text {Op}}}^W\!\left( a_\chi \right) \), the estimate (2.18) follows by (A.35), (A.21) and \( \left| a \right| _{m, L^{\infty }, N} \lesssim \left| a \right| _{m, {s_0}, N} \). Note that the condition on the Fourier support of \(a_\chi \) in Theorem A.7 is automatically satisfied provided \(\epsilon \) in (2.13) is sufficiently small. To prove (2.19) we use also (A.22). \(\square \)
The second result of symbolic calculus that we shall use regards composition for Bony–Weyl paradifferential operators at the second order (as required in the paper) with mild smoothness assumptions for the symbols in the space variable x. Given symbols \(a \in \Gamma ^m_{s_0+\varrho }\), \(b \in \Gamma ^{m'}_{s_0+\varrho }\) with \(m, m' \in {\mathbb {R}}\) and \(\varrho \in (0,2]\) we define
is the Poisson bracket between \( a (x, \xi )\) and \( b(x, \xi ) \). By (2.10) and (2.12) we have that ab is a symbol in \( \Gamma ^{m+m'}_{s_0+\varrho } \) and \( \{a, b\} \) is in \( \Gamma ^{m+m'-1}_{s_0+\varrho -1} \). The next result follows directly by Theorem A.8 and (2.9).
Theorem 2.5
(Composition) Let \(a \in \Gamma ^m_{s_0+\varrho }\), \(b \in \Gamma ^{m'}_{s_0+\varrho }\) with \(m, m' \in {\mathbb {R}}\) and \(\varrho \in (0,2]\). Then
where the linear operator \(R^{-\varrho }(a,b):\dot{H}^s \rightarrow \dot{H}^{s-(m+m')+\varrho }\), \(\forall s \in {\mathbb {R}}\), satisfies, for any \( u \in \dot{H}^s \),
where \( N \ge 3d + 4 \).
A useful corollary of Theorems 2.5 and 2.4 [using also (2.10)–(2.12)] is the following:
Corollary 2.6
Let \(a\in \Gamma ^{m}_{s_0+2}\), \(b\in \Gamma ^{m'}_{s_0+2}\), \(c \in \Gamma ^{m''}_{s_0+2}\) with \(m, m', m'' \in {\mathbb {R}}\). Then
where
satisfies \( R_1(a,b,c)=-R_1(c,b,a) \) and \(R_0(a,b,c)\) is a bounded operator \(\dot{H}^s \rightarrow \dot{H}^{s-(m+m'+m'')+2}\), \(\, \forall s\in {\mathbb {R}}\), satisfying, for any \( u \in \dot{H}^s \),
where \( N \ge 3d+ 5 \).
We now provide the Bony-paraproduct decomposition for the product of Sobolev functions in the Bony–Weyl quantization. Recall that \( \Pi _0^\bot \) denotes the projector on the subspace \( H^s_0 \).
Lemma 2.7
(Bony paraproduct decomposition) Let \(u \in H^s\), \(v \in H^r\) with \(s + r \ge 0\). Then
where the bilinear operator \(R:H^s \times H^r \rightarrow H^{s+r-s_0}\) is symmetric and satisfies the estimate
Moreover \( R(u,v) = R(\Pi _0^\bot u, \Pi _0^\bot v) - u_0 v_0 \) and then
Proof
Introduce the function \(\theta _\epsilon (j,k)\) by
Note that \(\left| \theta _\epsilon (j, k ) \right| \le 1\). Let \( \Sigma := \{ (j, k) \in {\mathbb {Z}}^d \times {\mathbb {Z}}^d \, : \, \theta _\epsilon (j,k) \ne 0 \} \) denote the support of \(\theta _\epsilon \). We claim that
Indeed, recalling the definition of the cut-off function \( \chi \) in (2.13), we first note thatFootnote 1
Thus, for any \( (j,k) \in \Sigma \),
proving (2.30). Using (2.29) we decompose
By (2.30), \( s+ r \ge 0 \), and the Cauchy–Schwartz inequality, we get
proving (2.27). Finally, since on the support of \( \theta _\epsilon \) we have or \( (j,k) = (0,0)\) or \( j - k \ne 0 \) and \( k \ne 0 \), we deduce that
and we deduce (2.28). \(\square \)
Composition estimates.
We will use the following Moser estimates for composition of functions in Sobolev spaces.
Theorem 2.8
Let \(I \subseteq {\mathbb {R}}\) be an open interval and \(F\in C^\infty (I; {\mathbb {C}})\) a smooth function. Let \(J \subset I\) be a compact interval. For any function \(u, v \in H^s({\mathbb {T}}^d, {\mathbb {R}})\), \(s>\frac{d}{2} \), with values in J, we have
Proof
Take an extension \({\tilde{F}}\in C^\infty ({\mathbb {R}};{\mathbb {C}})\) such that \({\tilde{F}}_{| I}= F \). Then \( F(u) = {\tilde{F}}(u)\) for any \(u \in H^s({\mathbb {T}}^d;{\mathbb {R}})\) with values in J, and apply the usual Moser estimate, see e.g. [3], replacing the Littlewood–Paley decomposition on \({\mathbb {R}}^d\) with the one on \({\mathbb {T}}^d\) in (A.12). \(\square \)
3 Paralinearization of (EK)-System and Complex Form
In this section we paralinearize the Euler–Korteweg system (1.6) and write it in terms of the complex variable
The variable \( u \in \dot{H}^s \). We denote this change of coordinates in \( \dot{H}^s \times \dot{H}^s \) by
We also define the matrices
Proposition 3.1
(Paralinearized Euler–Korteweg equations in complex coordinates) The (EK)-system (1.6) can be written in terms of the complex variable \( U := {\begin{pmatrix}u \\ {\overline{u}}\end{pmatrix}} \) with u defined in (3.1), in the paralinearized form
where, for any function \(U\in \dot{\mathbf{H}}^{s_0+2} \) such that
one has
-
(i)
\(A_2(U;x,\xi ) \in \Gamma _{s_0+2}^2 \otimes {\mathcal {M}}_2({\mathbb {C}}) \) is the matrix of symbols
$$\begin{aligned} A_2(U; x, \xi ) := \sqrt{{{\mathtt {m}}}K({{\mathtt {m}}})} |\xi |^2 \begin{bmatrix} 1 + {\mathtt {a}}_+(U; x) &{} {\mathtt {a}}_-(U; x) \\ {\mathtt {a}}_-(U; x) &{} 1+ {\mathtt {a}}_+(U; x) \end{bmatrix} \end{aligned}$$(3.6)where \({\mathtt {a}}_\pm (U; x) \in \Gamma _{s_0+2}^0\) are the \(\xi \)-independent functions
$$\begin{aligned} \begin{aligned}&{\mathtt {a}}_\pm (U; x) := \frac{1}{2} \left( \frac{K(\rho + {{\mathtt {m}}})- K({{\mathtt {m}}}) }{K({{\mathtt {m}}})} \pm \frac{\rho }{{{\mathtt {m}}}} \right) . \end{aligned} \end{aligned}$$(3.7) -
(ii)
\(A_1(U; x, \xi ) \in \Gamma _{s_0+1}^1 \otimes {\mathcal {M}}_2({\mathbb {C}}) \) is the diagonal matrix of symbols
$$\begin{aligned} A_1(U; x, \xi ) := \begin{bmatrix} {\mathtt {b}}(U; x) \cdot \xi &{} 0 \\ 0 &{} -{\mathtt {b}}(U; x) \cdot \xi \end{bmatrix},\qquad {\mathtt {b}}(U; x) := \nabla \phi \in \Gamma _{s_0+1}^0\otimes {\mathbb {R}}^d.\nonumber \\ \end{aligned}$$(3.8)Moreover for any \(\sigma \ge 0 \) there exists a non decreasing function \({\mathtt {C}}( \ ) : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+ \) (depending on K) such that, for any \(U,V\in \dot{\mathbf{H}}^{s_0} \) with \(\rho (U), \rho (V) \in {\mathcal {Q}}\), \(W \in {\dot{{\mathbf {H}}}}^{{\sigma }+2}\) and \(j=1,2\), we have
$$\begin{aligned}&\Vert {{\text {Op}}}^{\text {BW}}\!\left( A_j(U)\right) W\Vert _{{\sigma }} \le {\mathtt {C}}\left( \Vert U\Vert _{s_0}\right) \Vert W\Vert _{{\sigma }+2} \end{aligned}$$(3.9)$$\begin{aligned}&\Vert {{\text {Op}}}^{\text {BW}}\!\left( A_j(U)-A_j(V)\right) W\Vert _{{\sigma }}\le {\mathtt {C}}\left( \Vert U\Vert _{s_0}, \Vert V\Vert _{s_0}\right) \Vert W\Vert _{{\sigma }+2} \Vert U-V\Vert _{s_0} \end{aligned}$$(3.10)where in (3.10) we denoted by \({\mathtt {C}}(\cdot , \cdot ):= {\mathtt {C}}\left( \max \{\cdot ,\cdot \}\right) \).
-
(iii)
The vector field R(U) satisfies the following “semilinear” estimates: for any \(\sigma \ge s_0 > d / 2 \) there exists a non decreasing function \({\mathtt {C}}( \ ) : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+ \) (depending also on g, K) such that, for any \(U,V\in \dot{\mathbf{H}}^{\sigma +2} \) such that \(\rho (U), \rho (V) \in {\mathcal {Q}}\), we have
$$\begin{aligned}&\Vert R(U) \Vert _{ \sigma } \le {\mathtt {C}}\left( \Vert U\Vert _{ {s_0+2}} \right) \Vert U\Vert _{ \sigma }, \qquad \Vert R(U)\Vert _{ {\sigma }} \le {\mathtt {C}}\left( \Vert U\Vert _{ {s_0}}\right) \Vert U\Vert _{ {\sigma +2}}, \end{aligned}$$(3.11)$$\begin{aligned}&\Vert R(U)-R(V) \Vert _{ \sigma } \le {\mathtt {C}}\left( \Vert U\Vert _{ {s_0+2}}, \, \Vert V\Vert _{ {s_0+2}}\right) \Vert U-V\Vert _{{\sigma }}\nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad \ + {\mathtt {C}}\left( \Vert U\Vert _{ {\sigma }}, \, \Vert V\Vert _{ {\sigma }}\right) \Vert U-V\Vert _{{s_0+2}} \end{aligned}$$(3.12)$$\begin{aligned}&\Vert R(U)-R(V) \Vert _{ s_0} \le {\mathtt {C}}\left( \Vert U\Vert _{ {s_0+2}}, \, \Vert V\Vert _{ {s_0+2}}\right) \Vert U-V\Vert _{{s_0}} \,, \end{aligned}$$(3.13)where in (3.12) and (3.13) we denoted again by \({\mathtt {C}}(\cdot , \cdot ):= {\mathtt {C}}\left( \max \{\cdot ,\cdot \}\right) \).
Proof
We first paralinearize the original equations (1.6), then we switch to complex coordinates.
Step 1: paralinearization of (1.6). We apply several times the paraproduct Lemma 2.7 and the composition Theorem 2.5. In the following we denote by \(R^p\) the remainder that comes from Lemma 2.7, and by \(R^{-\varrho }\), \(\varrho =1,2\), the remainder that comes from Theorem 2.5. We shall adopt the following convention: given \({\mathbb {R}}^d\)-valued symbols \( a = (a_j)_{j=1,\ldots , d} \), \( b = (b_j)_{j=1,\ldots , d} \) in some class \(\Gamma ^m_s\otimes {\mathbb {R}}^d\), we denote \( R^p(a, b) := \sum _{j=1}^d R^p(a_j, b_j) \),
We paralinearize the terms in the first line of (1.6). We have \( \Delta \phi = - {{\text {Op}}}^{\text {BW}}\!\left( |\xi |^2\right) \phi \) and \( {\text {div}}(\rho \nabla \phi ) = \nabla \rho \cdot \nabla \phi + \rho \Delta \phi \) can be written as
Then we paralinearize the terms in the second line of (1.6). We have
Using (1.10) we regard the semilinear term
directly as a remainder. Moreover, writing \( \Delta \rho = - {{\text {Op}}}^{\text {BW}}\!\left( |\xi |^2\right) \rho \), we get
Finally, using for \( \frac{1}{2} | \nabla \rho |^2 \) the expansion (3.16) for \(\rho \) instead of \( \phi \), we obtain
where
Collecting all the above expansions and recalling the definition of the symplectic matrix J in (3.3), the system (1.6) can be written in the paralinearized form
where we collected in \(R(\rho , \phi )\) all the terms in lines (3.14)–(3.21).
Step 2: complex coordinates. We now write system (3.22) in the complex coordinates \(U = {\textit{C}}^{-1}{\begin{pmatrix}\rho \\ \phi \end{pmatrix}}\). Note that \( {\textit{C}}^{-1}\) conjugates the Poisson tensor J to \({\mathbb {J}}\) defined in (3.3), i.e. \( {\textit{C}}^{-1} \, J = {\mathbb {J}}\, {\textit{C}}^* \) and therefore system (3.22) is conjugated to
Using (3.2), system (3.23) reads as system (3.4)–(3.8) with \( R(U) := {\textit{C}}^{-1} R({\textit{C}}U) \).
We note also that estimates (3.9) and (3.10) for \(j = 2\) follow by (2.18) and (2.31), whereas in case \(j = 1\) follow by (2.19) applied with \(m=1\), \(\varrho =1\).
Step 3: Estimate of the remainder R(U) . We now prove (3.11)–(3.13). Since \( \Vert \rho \Vert _\sigma , \Vert \phi \Vert _\sigma \sim \Vert U \Vert _\sigma \) for any \( \sigma \in {\mathbb {R}}\) by (3.2), the estimates (3.11)–(3.13) directly follow from those of \( R(\rho , \phi ) \) in (3.22). We now estimate each term in (3.14)–(3.21). In the sequel \(\sigma \ge s_0 > d / 2 \).
Estimate of the term in line (3.14). Applying first (2.18) with \( m = 0 \), and then (2.19) with \(\varrho =2\), we have
By (2.27), the smoothing remainder in line (3.14) satisfies the estimates
and, by (2.22) with \( \varrho = 2 \), and the interpolation estimate (2.4),
By (3.24)–(3.26) and \( \Vert \rho \Vert _\sigma , \Vert \phi \Vert _\sigma \sim \Vert U \Vert _\sigma \) we deduce that the terms in line (3.14), written in function of U, satisfy (3.11). Next we write
and, applying (2.18) with \( m = 0\), and (2.19) with \(\varrho = 2\) to \({{\text {Op}}}^{\text {BW}}\!\left( \Delta \phi _1-\Delta \phi _2\right) \rho _2\), we get
Concerning the remainder \(R^p(\rho , \Delta \phi )\), we write \(R^p(\rho _1,\Delta \phi _1)-R^p(\rho _2,\Delta \phi _2) = R^p(\rho _1-\rho _2,\Delta \phi _1) +R^p(\rho _2,\Delta \phi _2- \Delta \phi _1)\) and, applying (2.27), we get
Finally we write \( R^{-2}( \rho _1, |\xi |^2)\phi _1- R^{-2}( \rho _2, |\xi |^2)\phi _2 = R^{-2}( \rho _1-\rho _2, |\xi |^2)\phi _1 + R^{-2}( \rho _2, |\xi |^2)[\phi _1- \phi _2] \). Using (2.22) we get
We also claim that
Indeed, we bound
and, to control \(R^{-2}( \rho _1-\rho _2, |\xi |^2)\phi _1\), we use that, by definition, it equals
and we estimate the first two terms using (2.19) with \(\varrho = 0\) and the last term with \(\varrho = 1\), by \(\Vert R^{-2}( \rho _1-\rho _2, |\xi |^2)\phi _1\Vert _{\sigma }\lesssim \Vert \rho _1-\rho _2\Vert _{s_0} \Vert \phi _1\Vert _{\sigma +2} \), proving (3.30). By (3.27)–(3.30) and \( \Vert \rho \Vert _\sigma , \Vert \phi \Vert _\sigma \sim \Vert U \Vert _\sigma \) we deduce that the terms in line (3.14), written in function of U, satisfy (3.12)–(3.13).
The estimates (3.11)–(3.13) for the terms in lines (3.15), (3.16), (3.18) and (3.17), follow by similar arguments, using also (2.31).
Estimates of \( {\mathtt R}(\rho ) \) defined in (3.19)–(3.21).
Writing \( {{\text {Op}}}^{\text {BW}}\!\left( |\nabla \rho |^2\right) K'({{\mathtt {m}}}+\rho ) = {{\text {Op}}}^{\text {BW}}\!\left( |\nabla \rho |^2\right) (K'({{\mathtt {m}}}+\rho ) - K'({{\mathtt {m}}}))\) (in the homogeneous spaces \( \dot{H}^s \)), we have, by (2.18), the fact that \( \rho \in {\mathcal {Q}}\), Theorem 2.8, (2.27), (2.2), (2.22) with \(\varrho =1\),
Thus \( {\mathtt R} (\rho ) \), written as a function of U, satisfies (3.11). The estimates (3.12)–(3.13) follow by
Proof of (3.31). Defining \( w := \nabla ( \rho _1+\rho _2)\), \(v:= \nabla (\rho _1-\rho _2)\), then we have, by (2.1),
Let us prove (3.31) for the first term in (3.19). Remind that \( \rho _1, \rho _2 \) are in \( {\mathcal {Q}}\). We have
In the same way the second term in (3.19) is bounded by (3.35). Regarding the term in (3.20), using that \(R^p(\cdot ,\cdot )\) is bilinear and symmetric, we have
Also the terms in (3.21) are bounded by (3.35), proving that \({\mathtt R}(\rho )\) satisfies (3.31).
Proof of (3.32). Regarding the first term (3.19), we have
Similarly we deduce that the second term in (3.19) is bounded as in (3.37). Regarding the term in (3.20), note that the bound (3.32) follows from (3.36) applied for \(\sigma = s_0\). The estimate for last two terms in (3.21) follows in the same way so we analyze the last one. First we have
On the other hand, by definition, we have
Then, applying first (2.18) to the first term and then (2.19) with \(\varrho =1\), \(m=1\) and (2.31) to each term, we deduce that the \( \Vert \ \Vert _{s_0} \)-norm of (3.38) is bounded by \( {\mathtt {C}}\big (\Vert \rho _1\Vert _{s_0+2}, \Vert \rho _2\Vert _{s_0+2}\big ) \Vert \rho _1-\rho _2\Vert _{s_0}\). Thus (3.32) is proved. \(\square \)
4 Local Existence
In this section we prove the existence of a local in time solution of system (3.4). For any \(s \in {\mathbb {R}}\) and \(T >0\), we denote \(L^\infty _T {\dot{{\mathbf {H}}}}^s := L^\infty ([0,T], {\dot{{\mathbf {H}}}}^s) \). For \( \delta > 0 \) we also introduce
where \( {\mathcal {Q}}\) is defined in (1.12).
Proposition 4.1
(Local well-posedness in \({\mathbb {T}}^d\)) For any \(s > \frac{d}{2}+2\), any initial datum \(U_0 \in {\dot{{\mathbf {H}}}}^s \) with \( \rho (U_0) \in {\mathcal {Q}}_\delta \) for some \( \delta > 0 \), there exist \(T := T(\Vert U \Vert _{s_0+2}, \delta ) > 0 \) and a unique solution \(U \in C^0\big ([0, T], {\dot{{\mathbf {H}}}}^s \big ) \cap C^1\big ([0, T], {\dot{{\mathbf {H}}}}^{s-2} \big )\) of (3.4) satisfying \( \rho (U) \in {\mathcal {Q}}\), for any \( t \in [0,T] \). Moreover the solution depends continuously with respect to the initial datum in \( {\dot{{\mathbf {H}}}}^s \).
Proposition 4.1 proves Theorem 1.2 and thus Theorem 1.1.
The first step is to prove the local well-posedness result of a linear inhomogeneous problem.
Proposition 4.2
(Linear local well-posedness) Let \(\Theta \ge r > 0 \) and U be a function in \( C^0([0,T],{\dot{{\mathbf {H}}}}^{s_0+2}) \cap C^1([0,T],{\dot{{\mathbf {H}}}}^{s_0}) \) satisfying
Let \( \sigma \ge 0\) and \( t \mapsto R(t)\) be a function in \( C^0([0, T], {\dot{{\mathbf {H}}}}^\sigma ) \). Then there exists a unique solution \(V\in C^0([0,T], {\dot{{\mathbf {H}}}}^\sigma )\cap C^1([0,T], {\dot{{\mathbf {H}}}}^{\sigma -2})\) of the linear inhomogeneous system
satisfying, for some \( C_\Theta := C_{\Theta ,\sigma } > 0 \) and \(C_r := C_{r,\sigma } > 0 \), the estimate
The following two sections are devoted to the proof of Proposition 4.2. The key step is the construction of a modified energy which is controlled by the \({\dot{{\mathbf {H}}}}^{\sigma }\)-norm, and whose time variation is bounded by the \({\dot{{\mathbf {H}}}}^{\sigma }\) norm of the solution, as done e.g. in [1, 21] for linear systems. In order to construct such modified energy, the first step is to diagonalize the matrix \( {\mathbb {J}}A_2\) in (4.3).
4.1 Diagonalization at Highest Order
We diagonalize the matrix of symbols \({\mathbb {J}}A_2(U; x, \xi )\). The eigenvalues of the matrix
with \({\mathtt {a}}_\pm (U;x)\) defined in (3.7) are given by \(\pm {\text {i}}\lambda (U; x)\) with
These eigenvalues are purely imaginary because \(\rho (U) \in {\mathcal {Q}}\) [see (1.12)] and (1.9), which guarantees that \(\lambda (U;x) \) is real valued and fulfills
A matrix which diagonalizes (4.5) is
Note that F(U; x) is well defined because
by (1.12) and (1.9). The matrix F(U; x) has \( \det F(U;x) = {f^2 - g^2} = 1 \) and its inverse is
We have that
By (2.31) and (4.9) we deduce the following estimates: for any \( N \in {\mathbb {N}}_0 \), \( s \ge 0 \) and \({\sigma }>\frac{d}{2}\),
For any \(\varepsilon > 0 \), consider the regularized matrix symbol
where \( \chi \) is the cut-off function in (2.13) and \(\lambda (U;x)\) is the function defined in (4.6). In what follows we will denote by \(\chi _\varepsilon := \chi ( \varepsilon \lambda (U; x) |\xi |^2)\). Note that, by (2.31), (4.7) and by the fact that the function \(y \mapsto \langle \xi \rangle ^{|\alpha |}\partial _\xi ^\alpha \chi ( \varepsilon y |\xi |^2)\) is bounded together with its derivatives uniformly in \(\varepsilon \in (0,1)\), \( \xi \in {\mathbb {R}}^d\) and \(y\in [\lambda _{\min }, \lambda _{\max }]\), the symbol \( \chi _\varepsilon \) satisfies, for any \( N\in {\mathbb {N}}_0 \), \( \sigma > d/2 \)
The diagonalization (4.11) has the following operatorial consequence.
Lemma 4.3
We have
where \({\mathcal {F}}(U):= {\mathcal {F}}_\varepsilon (U):{\dot{{\mathbf {H}}}}^\sigma \rightarrow {\dot{{\mathbf {H}}}}^\sigma \), \(\forall \sigma \ge 0\), satisfies, uniformly in \( \varepsilon \),
Proof
We have that
where
By Corollary 2.6 we obtain
where \( {\mathcal {F}}_1,{\mathcal {F}}_2 \) satisfy (4.16) by (2.25), (4.12), and (4.14) and since, by the definition of f and g in (4.8) and \( \lambda \) in (4.6), we have \( (f^2+g^2)(1+{\mathtt {a}}_+) + 2fg{\mathtt {a}}_- = \lambda \) and \( (f^2+g^2){\mathtt {a}}_-+2fg(1+{\mathtt {a}}_+)=0 \). Moreover
where
Applying Theorem 2.5, (4.12), (4.14), using that \(f^2-g^2=1\) we obtain \( D_1= {{\text {Op}}}^{\text {BW}}\!\left( {\mathtt {b}}\cdot \xi \, \chi _\varepsilon \right) + {\mathcal {F}}_1(U) \) and \( B_1= {\mathcal {F}}_2(U) \) with \({\mathcal {F}}_1,{\mathcal {F}}_2\) satisfying (4.16). \(\square \)
4.2 Energy Estimate for Smoothed System
We first solve (4.3) in the case \(R(t)=0\) and \(V_0 \in \dot{C}^\infty := \cap _{{\sigma }\in {\mathbb {R}}} {\dot{{\mathbf {H}}}}^{\sigma }\). Consider the regularized Cauchy problem
where \(A^\varepsilon (U; x, \xi ) \) is defined in (4.13). As the operator \( {{\text {Op}}}^{\text {BW}}\!\left( A^\varepsilon (U;x,\xi )\right) \) is bounded for any \( \varepsilon >0\), and U(t) satisfies (4.2), the differential equation (4.17) has a unique solution \(V^\varepsilon (t)\) which belongs to \( C^2 ([0,T],{\dot{{\mathbf {H}}}}^\sigma )\) for any \( \sigma \ge 0\). The important fact is that it admits the following \(\varepsilon \)-independent energy estimate.
Proposition 4.4
(Energy estimate) Let U satisfy (4.2). For any \(\sigma \ge 0\), there exist constants \( C_r, C_{\Theta } > 0 \) (depending also on \(\sigma \)), such that for any \( \varepsilon > 0 \), the unique solution of (4.17)
As a consequence, there are constants \( C_r, C_\Theta \) independent of \(\varepsilon \), such that
In order to prove Proposition 4.4, we define, for any \(\sigma \ge 0\), the modified energy
where we introduce the real scalar product
Lemma 4.5
Fix \(\sigma \ge 0\), \(r >0\). There exists a constant \(C_{r}>0\) (depending also on \({\sigma }\)) such that for any \(U \in {\dot{{\mathbf {H}}}}^{s_0}\) with \(\left\| U \right\| _{s_0} \le r\) and \(\rho (U) \in {\mathcal {Q}}\) we have
Proof
We first prove the upper bound in (4.21). We note that, by (4.12), \(\lambda ^{\sigma }(U;x) |\xi |^{2{\sigma }}\in \Gamma _{s_0}^{2{\sigma }}\) and \(F^{-1}(U;x) \in \Gamma _{s_0}^0\otimes {\mathcal {M}}_2({\mathbb {C}})\) and, by Theorem 2.4 and (4.12) we have
In order to prove the lower bound, we fix \(\delta \in (0,1)\) such that \(s_0-\delta >\frac{d}{2}\) and, due to (4.7), we have \( \lambda ^{-\frac{{\sigma }}{2}}\in \Gamma _{s_0-\delta }^{0} \). So, applying Theorem 2.5 and (4.12) with \(s_0-\delta \) instead of \(s_0\) and with \(\varrho =\delta \), we have
where for any \(\sigma ' \in {\mathbb {R}}\) there exists a constant \(C_{r, \sigma '}>0\) such that
Again, applying Theorem 2.5 with \(s_0-\delta \) instead of \(s_0\) and with \(\varrho =\delta \), we have also
where for any \(\sigma ' \in {\mathbb {R}}\) there exists a constant \(C_{r,\sigma '}>0\) such that
By (4.22)–(4.25), Theorem 2.4 and (4.12) and using also that \({{\text {Op}}}^{\text {BW}}\!\left( \lambda ^{\frac{{\sigma }}{2}}\right) \) is symmetric with respect to \(\langle \cdot , \cdot \rangle \), we have
Now we use (2.3) and the asymmetric Young inequality to get, for any \(\epsilon > 0\),
we choose \(\epsilon \) so small so that \( \epsilon ^{\frac{2({\sigma }+2)}{2({\sigma }+2)-\delta }}C_{r}=\frac{1}{2}\) and we get \( \Vert V\Vert _{{\sigma }}^2\le 2C_{r} \big ( \Vert V\Vert _{{\sigma },U}^2+\Vert V\Vert _{{-2}}^2 \big ) \). This proves the lower bound in (4.21). \(\square \)
Proof of Proposition 4.4
The time derivative of the modified energy (4.20) along a solution \( V^\varepsilon (t) \) of (4.17) is
By Theorem 2.4 and using that \(\forall \sigma \ge 0 \), \( N \in {\mathbb {N}}_0 \),
and the assumption (4.2), there exists a constant \(C_{\Theta }>0\) (depending also on \({\sigma }\)) such that
We now estimate (4.28). By Theorem 2.5 with \( \varrho = 2 \) and (4.2) we have
where \( {\mathcal {F}}_\pm ^{-2}(U)\) are bounded operators from \({\dot{{\mathbf {H}}}}^{\sigma '}\) to \({\dot{{\mathbf {H}}}}^{\sigma '+2}\), \(\forall \sigma ' \in {\mathbb {R}}\), satisfying
Thus, denoting \( {\widetilde{V}}^\varepsilon := {{\text {Op}}}^{\text {BW}}\!\left( F^{-1}\right) V^\varepsilon \), by (4.30), we have
Recalling (4.17) we have
and by Lemma 4.3 we get
where in line (4.35) the operator \({\mathcal {F}}(U)\) is the bounded remainder of Lemma 4.3. We estimate each contribution. First we consider line (4.33). Using Theorem 2.5 with \( \varrho = 2 \), the principal symbol of the commutator is
and, using (4.14), (4.12) and assumption (4.2), we get
Similarly, using Theorem 2.5 with \(\varrho =1\), Theorem 2.4, (4.12) and estimates (4.31) and (4.16), we obtain
In conclusion, by (4.29), (4.37), (4.38), we deduce the bound \(\frac{d}{dt} \Vert V^\varepsilon (t)\Vert _{{\sigma },U(t)}^2 \le C_{\Theta } \left\| V^\varepsilon (t) \right\| _{\sigma }^2 \), that gives, for any \( t \in [0,T] \)
Since \(V^\varepsilon (t)\) solves (4.17), by Theorem 2.4, (4.12), (4.14) there exists a constant \(C_{\Theta }>0\) (independent on \(\varepsilon \)) such that \( \Vert \partial _t V^\varepsilon (t)\Vert _{{-2}}^2\le C_{\Theta }\Vert V^\varepsilon (t)\Vert _{0}^2\le C_{\Theta }\Vert V^\varepsilon (t)\Vert _{{\sigma }}^2 \) and therefore
We finally deduce (4.18) by (4.39), the lower bound in (4.21) and (4.40). The estimate (4.19) follows by Gronwall inequality. \(\square \)
Proof of Proposition 4.2. By Proposition 4.4, Ascoli–Arzelá theorem ensures that, for any \( \sigma \ge 0 \), \(V^\varepsilon \) converges up to subsequence to a limit V in \( C^1([0,T],{\dot{{\mathbf {H}}}}^{\sigma })\), as \(\varepsilon \rightarrow 0\) that solves (4.3) with \(R(t)=0 \), initial datum \(V_0\in \dot{C}^\infty \), and satisfies \( \Vert V(t)\Vert _{{{\sigma }}}\le C_r e^{C_\Theta t} \Vert V_0\Vert _{{{\sigma }}} \), for any \( \sigma \ge 0 \). The case \(V_0\in {\dot{{\mathbf {H}}}}^{\sigma }\) follows by a classical approximation argument with smooth initial data. This shows that the propagator of \({\mathbb {J}}{{\text {Op}}}^{\text {BW}}\!\left( A_2(U(t);x, \xi )+A_1(U(t);x, \xi )\right) \) is, for any \( \sigma \ge 0 \), a well defined bounded linear operator
In the inhomogeneous case \(R\not =0 \), the solutions of (4.3) is given by the Duhamel formula \( V(t)= \Phi (t) V_0 + \Phi (t)\int _0^t \Phi ^{-1}(\tau ) R(\tau )\, {\text {d}}\tau , \) and the estimate (4.4) follows.
4.3 Iterative Scheme
In order to prove that the nonlinear system (3.4) has a local in time solution we consider the sequence of linear Cauchy problems
for \( n \ge 2 \), where \( A := A_2 + A_1 \), cfr. (3.6), (3.8). The strategy is to prove that the sequence of solutions \(U_n\) of the approximated problems \({\mathcal {P}}_n \) converges to a solution U of system (3.4).
Lemma 4.6
Let \( U_0\in {\dot{{\mathbf {H}}}}^s \), \( s > 2 + \frac{d}{2}\), such that \( \rho (U_0) \in {\mathcal {Q}}_\delta \) for some \(\delta >0\) (recall (3.5) and (4.1)) and define \(r:= 2\Vert U_0\Vert _{{s_0}} \). Then there exists a time \(T:=T(\Vert U_0\Vert _{{s_0+2}},\delta )>0\) such that, for any \(n\in {\mathbb {N}}\):
- \((S0)_n\)::
-
The problem \({\mathcal {P}}_n\) admits a unique solution \(U_n\in C^0([0,T], {\dot{{\mathbf {H}}}}^s)\cap C^1 ([0,T], {\dot{{\mathbf {H}}}}^{s-2})\).
- \((S1)_n\)::
-
For any \( t\in [0,T]\), \( \rho (U_n(t)) \) belongs to \({\mathcal {Q}}_{\frac{\delta }{2}}\).
- \((S2)_n\)::
-
There exists a constant \({C_r\ge 1}\) (depending also on s) such that, defining \(\Theta :=4C_r \Vert U_0 \Vert _{{s_0+2}}\) and \( M:= 4C_r \Vert U_0\Vert _{{s}}\), for any \(1\le m\le n\) one has
$$\begin{aligned}&\Vert U_m\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s_0}} \le r \, ;\end{aligned}$$(4.41)$$\begin{aligned}&\Vert U_m\Vert _{L^{\infty }_T {\dot{{\mathbf {H}}}}^{s_0+2}}\le \Theta , \quad \Vert \partial _t U_m\Vert _{L^{\infty }_T {\dot{{\mathbf {H}}}}^{s_0}}\le C_r \Theta \, ;\end{aligned}$$(4.42)$$\begin{aligned}&\Vert U_m\Vert _{L^{\infty }_T {\dot{{\mathbf {H}}}}^{s}}\le M, \quad \Vert \partial _t U_m\Vert _{L^{\infty }_T {\dot{{\mathbf {H}}}}^{s-2}}\le C_r M . \end{aligned}$$(4.43) - \((S3)_n\)::
-
For \(1\le m\le n\) one has
$$\begin{aligned} \Vert U_1\Vert _{L^{\infty }_T{\dot{{\mathbf {H}}}}^{s_0}} = r / 2, \qquad \Vert U_m -U_{m-1}\Vert _{L^{\infty }_T{\dot{{\mathbf {H}}}}^{s_0}}\le 2^{-m} {r}, \ \ m \ge 2. \end{aligned}$$
Proof
We prove the statement by induction on \(n\in {\mathbb {N}}\). Given \(r >0\), we define
where \(C_{r, \sigma }\) is the constant in Proposition 4.2 (where we stress that it depends also on \({\sigma }\)) and \({\mathtt {C}}(\cdot )\) is the function in (3.9) and (3.11). In the following we shall denote by \( C_\Theta \) all the constants depending on \(\Theta \), which can vary from line to line.
Proof of \((S0)_1\): The problem \({\mathcal {P}}_1\) admits a unique global solution which preserves Sobolev norms.
Proof of \((S1)_1\): We have \(\rho (U_0) \in {\mathcal {Q}}_{\delta }\). In addition
for \(T := T(\Vert U_0\Vert _{s_0+2}, \delta )>0\) sufficiently small, which implies \(\rho (U_1(t)) \in {\mathcal {Q}}_{\frac{\delta }{2}}\), for any \( t \in [0, T]\).
Proof of \((S2)_1\) and \((S3)_1\): The flow of \({\mathcal {P}}_1\) is an isometry and \(M\ge \Vert U_0\Vert _{ s}\), \( \Theta \ge \Vert U_0\Vert _{{s_0+2}}\).
Suppose that \((S0)_{n-1}\)–\((S3)_{n-1}\) hold true. We prove \((S0)_{n}\)–\((S3)_{n}\).
Proof of \((S0)_n\): We apply Proposition 4.2 with \(\sigma = s \), \( U \leadsto U_{n-1} \) and \(R(t) := R(U_{n-1}(t))\). By \((S1)_{n-1}\) and \((S2)_{n-1}\), the function \(U_{n-1}\) satisfies assumption (4.2) with \(\Theta \leadsto (1+C_r)\Theta \). In addition \( R(U_{n-1}(t))\) belongs to \(C^0([0, T], {\dot{{\mathbf {H}}}}^{s})\) thanks to (3.12) and \( U_{n-1} \in C^0 ([0,T]; {\dot{{\mathbf {H}}}}^s)\). Thus Proposition 4.2 with \(\sigma = s\) implies \((S0)_n \). In particular \( U_n \) satisfies the estimate (4.4).
Proof of \((S2)_n\): We first prove (4.42). The estimate (4.4) with \({\sigma }= s_0+2\), the bound (3.11) of \(R(U_{n-1}(t))\) and (4.42) at the step \( n - 1 \), imply
As \(\Theta = 4C_r \Vert U_0\Vert _{{s_0+2}} \), we take \(T>0\) small such that
which, by (4.44), gives \(\Vert U_n\Vert _{L^\infty _T{\dot{{\mathbf {H}}}}^{s_0+2}}\le \Theta \). This proves the first estimate of (4.42). Regarding the control of \(\partial _t U_n\), we use the equation \({\mathcal {P}}_n\), the second estimate in (3.11) and (3.9) with \({\sigma }=s_0\) to obtain
which proves the second estimate of (4.42).
Next we prove (4.43). Applying estimate (4.4) with \({\sigma }= s\), we have
for \( M= 4C_r \Vert U_0\Vert _{s} \) and since \(T>0\) is chosen as in (4.45). The estimate for \(\left\| \partial _t U_n \right\| _{s-2}\) is similar to (4.46), and we omit it. Estimate (4.41) is a consequence of \((S3)_n\), which we prove below.
Proof of \((S1)_n\): We use estimate (4.46) to get
provided that \(T< \delta /{(2 C C_r \Theta )}\). This shows that \(\rho (U_n(t)) \in {\mathcal {Q}}_{\frac{\delta }{2}}\).
Proof of \((S3)_n\): Define \(V_n:= U_n-U_{n-1}\) if \( n \ge 2 \) and \( V_1 := U_1 \). Note that \( V_n\), \( n \ge 2 \), solves
where \(A:= A_2+A_1\) and
Applying estimates (3.13), (3.10), (3.11) and (4.42) we obtain, for \( n \ge 2 \),
We apply Proposition 4.2 to (4.47) with \({\sigma }=s_0\). Thus by (4.4) and (4.48) we get
provided \( C_\Theta e^{C_\Theta T} T\le \frac{1}{2}\). The proof of Lemma 4.6 is complete. \(\square \)
Corollary 4.7
With the same assumptions of Lemma 4.6, for any \(s_0+2 \le s'<s\):
-
(i)
\((U_n)_{n \ge 1}\) is a Cauchy sequence in \(C^0([0,T],{\dot{{\mathbf {H}}}}^{s'})\cap C^1([0,T],{\dot{{\mathbf {H}}}}^{s'-2})\) with \( T = T(\Vert U_0\Vert _{{s_0+2}},\delta ) \) given by Lemma 4.6. It converges to the unique solution U(t) of (3.4) with initial datum \(U_0\), U(t) is in \( C^0([0,T],{\dot{{\mathbf {H}}}}^{s'})\cap C^1([0,T],{\dot{{\mathbf {H}}}}^{s'-2}) \). Moreover \(\rho (U(t)) \in {\mathcal {Q}}\), \( \forall t \in [0, T]\).
-
(ii)
For any \(t\in [0,T]\), \(U(t) \in {\dot{{\mathbf {H}}}}^s \) and \(\left\| U(t) \right\| _s \le 4 C_r \left\| U_0 \right\| _s\) where \(C_r\) is the constant of \((S2)_n\).
Proof
- \(\mathrm{(i)}\) :
-
If \(s' = s_0\) it is the content of \((S3)_n\). For \( s' \in (s_0, s)\), we use interpolation estimate (2.3), (4.43) and \((S3)_n\) to get, for \( n \ge 2 \),
$$\begin{aligned} \Vert U_n-U_{n-1}\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s'}} \le \Vert U_n-U_{n-1}\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s_0}}^{\theta } \Vert U_n-U_{n-1}\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s}}^{1-\theta } \le 2^{-n\theta } C_M, \end{aligned}$$where \(\theta \in (0,1) \) is chosen so that \(s' = \theta s_0 + (1-\theta )s\). Thus \((U_n)_{n \ge 1}\) is a Cauchy sequence in \(C^0([0,T], {\dot{{\mathbf {H}}}}^{s'})\); we denote by \(U(t)\in C^0([0,T],{\dot{{\mathbf {H}}}}^{s'})\) its limit. Similarly using that \( \partial _t U_n \) solves \( {\mathcal {P}}_n \), one proves that \(\partial _tU_n\) is a Cauchy sequence in \(C^0([0,T],{\dot{{\mathbf {H}}}}^{s'-2})\) that converges to \(\partial _t U\) in \(C^0([0,T],{\dot{{\mathbf {H}}}}^{s'-2})\). In order to prove that U(t) solves (3.4), it is enough to show that
$$\begin{aligned}&{\mathcal {R}}(U, U_{n-1}, U_n) := {\mathbb {J}}{{\text {Op}}}^{\text {BW}}\!\left( A(U_{n-1})\right) U_n - {\mathbb {J}}{{\text {Op}}}^{\text {BW}}\!\left( A(U)\right) U + R(U_{n-1}) - R(U) \end{aligned}$$converges to 0 in \( L^\infty _T {\dot{{\mathbf {H}}}}^{s'-2} \). This holds true because by estimates (2.18), (3.10), (3.12), \( (S2)_n \), and the fact that \(U(t) \in C^0([0,T],{\dot{{\mathbf {H}}}}^{s'})\), we have
$$\begin{aligned}&\Vert {\mathcal {R}}(U, U_{n-1}, U_n) \Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s'-2}}\le C_M\left( \Vert U - U_{n-1}\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s'}} + \left\| U - U_{n-1} \right\| _{L^\infty _T {\dot{{\mathbf {H}}}}^{s_0+2}} + \Vert U - U_n\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s'}} \right) \end{aligned}$$which converges to 0 as \(n \rightarrow \infty \). Let us now prove the uniqueness. Suppose that \(V_1,V_2\in C^0([0,T],{\dot{{\mathbf {H}}}}^{s'})\cap C^1([0,T],{\dot{{\mathbf {H}}}}^{s'-2})\) are solutions of (3.4) with initial datum \(U_0\). Then \(W:= V_1-V_2\) solves
$$\begin{aligned} \partial _tW = {\mathbb {J}}{{\text {Op}}}^{\text {BW}}\!\left( A(V_1)\right) W + \varvec{R}(t), \quad W(0)=0, \end{aligned}$$where \({\varvec{R}}(t):={\mathbb {J}}{{\text {Op}}}^{\text {BW}}\!\left( A(V_1)-A(V_2)\right) V_2+ R(V_1)-R(V_2). \) Applying Proposition 4.2 with \(\sigma = s_0\) and \(\Theta , r\) defined by
$$\begin{aligned} \Theta := \max _{j=1,2}\big ( \Vert V_j\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s_0+2}} + \Vert \partial _t V_j\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s_0}} \big ), \quad r := \max _{j = 1,2} \Vert V_j\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^{s_0}}, \end{aligned}$$together with estimates (3.13) and (3.10) we have, for any \( t \in [0,T] \),
$$\begin{aligned} \Vert W\Vert _{{L^\infty _t {\dot{{\mathbf {H}}}}^{s_0}}}&\le C_\Theta e^{C_\Theta t }t \Vert {\varvec{R}}\Vert _{L^\infty _t {\dot{{\mathbf {H}}}}^{s_0}} \le C_\Theta e^{C_\Theta t }t \Vert W\Vert _{L^\infty _t {\dot{{\mathbf {H}}}}^{s_0}}. \end{aligned}$$Therefore, provided t is so small that \(C_\Theta e^{C_\Theta t }t< 1\), we get \(V_1(\tau )=V_2(\tau )\) \(\, \forall \tau \in [0,t]\). As (3.4) is autonomous, actually one has \(V_1(t)=V_2(t)\) for all \(t\in [0,T]\). This proves the uniqueness. Finally, as \(\rho (U_n(t)) \in {\mathcal {Q}}_{\frac{\delta }{2}}\) and \(U_n(t) \rightarrow U(t)\) in \({\dot{{\mathbf {H}}}}^{s_0}\), then \(\rho (U(t)) \in {\mathcal {Q}}_{\frac{\delta }{2}} \subset {\mathcal {Q}}\).
- \(\mathrm{(ii)}\) :
-
Since \( \Vert U_n(t)\Vert _s \le 4 C_r \left\| U_0 \right\| _s \) and \(U_n(t) \rightarrow U(t)\) in \({\dot{{\mathbf {H}}}}^{s'}\) then \(\Vert U (t)\Vert _s \le 4 C_r \left\| U_0 \right\| _s \). \(\square \)
Let \(\displaystyle { \Pi _{N} U := \Big ({\sum _{1\le |j|\le N} {u}_j e^{{\text {i}}j\cdot x}},{ \sum _{1\le |j|\le N} \overline{{ u}_{j}} e^{-{\text {i}}j \cdot x}}\Big )} \). We need below the following technical lemma.
Lemma 4.8
Let \( U_0\in {\dot{{\mathbf {H}}}}^s\), \( s > 2 + \frac{d}{2} \), with \(\rho (U_0) \in {\mathcal {Q}}_\delta \) for some \(\delta >0\). Then there exists a time \({\tilde{T}}:= {\tilde{T}}( \Vert U_0\Vert _{s_0+2}, \delta )>0\) and \(N_0 >0\) such that for any \(N > N_0\):
-
(i)
system (3.4) with initial datum \( \Pi _{N} U_0\) has a unique solution \(U_N \in C^0([0, {\tilde{T}}], {\dot{{\mathbf {H}}}}^{s+2})\).
-
(ii)
Let U be the unique solution of (3.4) with initial datum \(U_{0}\) defined in the time interval [0, T] (which exists by Corollary 4.7). Then there is \( \breve{T} < \min \{T, {\tilde{T}} \} \), depending on \( \Vert U_0 \Vert _s \), independent of N, such that
$$\begin{aligned} \left\| U - U_N \right\| _{ L^\infty _{\breve{T}} {\dot{{\mathbf {H}}}}^s} \le {\mathtt {C}}(\left\| U_0 \right\| _s) \, \left( \left\| U_0 - \Pi _{N} U_{0} \right\| _s + N^{s_0 +2 -s} \right) . \end{aligned}$$(4.49)In particular \(U_N \rightarrow U\) in \(C^0([0, \breve{T}], {\dot{{\mathbf {H}}}}^{s})\) when \(N \rightarrow \infty \).
Proof
Clearly \( \Pi _{N} U_0 \in \dot{C}^\infty \). Moreover, as \( \Vert \rho (U_0 - \Pi _{N} U_0) \Vert _{L^\infty ({\mathbb {T}}^d)} \rightarrow 0 \) when \(N \rightarrow \infty \), one has \(\rho (\Pi _{N} U_0) \in {\mathcal {Q}}_{\frac{\delta }{2}}\) provided \(N \ge N_0 \) is sufficiently large. So we can apply Corollary 4.7 and obtain a time \({\tilde{T}} > 0 \), independent on N, and a unique solution \(U_N \in C^0([0, {\tilde{T}}], {\dot{{\mathbf {H}}}}^{s+2}) \) of (3.4) with initial datum \( \Pi _{N} U_0 \). Moreover, by item \(\mathrm{(ii)}\) of that corollary, setting \( r = 2 \left\| \Pi _{N} U_0 \right\| _{s_0} \),
This proves item \(\mathrm{(i)}\). In the following let \( {\breve{T}}\le \min \{ {\tilde{T}}, T \} \).
Let us prove \(\mathrm{(ii)}\). Let \( \Theta := \left\| U \right\| _{L^\infty _{\breve{T}}{\dot{{\mathbf {H}}}}^{s_0+2} } + \left\| \partial _t U \right\| _{L^\infty _{\breve{T}}{\dot{{\mathbf {H}}}}^{s_0} }\) and \( r := \Vert U\Vert _{L^\infty _{\breve{T}}{\dot{{\mathbf {H}}}}^{s_0}} \). The function \(W_N(t):=U(t)-U_N(t)\) satisfies \( \left\| W_N(t) \right\| _s \le \left\| U(t) \right\| _s + \left\| U_N(t) \right\| _s \le {\mathtt {C}}(\left\| U_{0} \right\| _{s}) \), \(\, \forall t \in [0, {\breve{T}}]\), by Corollary 4.7-(ii). Moreover, \(W_N\) solves
where \( {\varvec{R}}(t):= {\mathbb {J}}{{\text {Op}}}^{\text {BW}}\!\left( A(U)-A(U_N)\right) U_N + R(U)-R(U_N). \) Applying Proposition 4.2 with \( \sigma = s_0 \) and estimates (3.10), (3.13), (4.50) one obtains
which, provided \({\breve{T}}\) is so small that \( {\breve{T}}C_\Theta e^{C_\Theta {\breve{T}}}\, {\mathtt {C}}(\Vert U_0\Vert _{s_0+2}) \le \frac{1}{2}\) (eventually shrinking it), gives
Similarly one estimates \(\left\| W_N(t) \right\| _{{\dot{{\mathbf {H}}}}^s}\), getting
from which (4.49) follows provided \( {\breve{T}}\) (depending on \(\Vert U_0\Vert _{s}\)) is sufficiently small. \(\square \)
Proof of Proposition 4.1
Given an initial datum \( U_0\in {\dot{{\mathbf {H}}}}^s\) with \(\rho (U_0) \in {\mathcal {Q}}\), choose \(\delta >0\) so small that \( \rho (U_0) \in {\mathcal {Q}}_\delta \). Then Corollary 4.7 gives us a time \(T = T(\left\| U_0 \right\| _{s_0+2}, \delta ) > 0 \) and a unique solution \(U \in C^0([0,T],{\dot{{\mathbf {H}}}}^{s'})\cap C^1([0,T],{\dot{{\mathbf {H}}}}^{s'-2})\), \(\forall s_0+2 \le s' < s\), of (3.4) with initial datum \(U_0\). Now take an open neighborhood \({\mathcal {U}}\subset {\dot{{\mathbf {H}}}}^{s}\) of \(U_0\) such that \(\forall V \in {\mathcal {U}}\) one has \(\rho (V) \in {\mathcal {Q}}_{\frac{\delta }{2}} \) and \(\left\| V \right\| _s \le 2 \left\| U_0 \right\| _s\). Then there exists \({\tilde{T}} \in (0, T] \) such that the flow map of (3.4),
is well defined for any \(t \in [0, {\tilde{T}}]\), it satisfies the group property
and \( \left\| \Omega ^t(U_0) \right\| _s \le {\mathtt {C}}(\left\| U_0 \right\| _s) \) for all \( U_0 \in {\mathcal {U}}\), \( t\in [0, {\tilde{T}}] \). For simplicity of notation in the sequel we denote by T a time, independent of N, smaller than \( {\tilde{T}} \).
Continuity of \(t \mapsto U(t)\): We show that \( U \in C^0([0,T], {\dot{{\mathbf {H}}}}^s )\). By (4.53), it is enough to prove that \(t\mapsto U(t)\) is continuous in a neighborhood of \(t=0\). This follows by Lemma 4.8, as U is the uniform limit of continuous functions.
Continuity of the flow map: We shall follow the method by [8, 13]. Let \( U^n_0 \rightarrow U_0\in {\dot{{\mathbf {H}}}}^s\) and pick \(\delta >0\) such that \(\rho (U^n_0)\), \(\rho (U_0) \), \(\rho (\Pi _{N} U^n_0)\), \(\rho (\Pi _{N} U_0) \in {\mathcal {Q}}_{\delta }\), for any \( n \ge n_0 \), \( N \ge N_0 \) sufficiently large. Denote by \(U^n,U \in C^0([0,T], {\dot{{\mathbf {H}}}}^s)\) the solutions of (3.4) with initial data \( U^n_0 \), respectively \( U_0 \), and \( U_N(t):= \Omega ^t(\Pi _{ N} U_0)\), \( U^n_N(t):= \Omega ^t(\Pi _{ N} U^n_0)\). Note that these solutions are well defined in \({\dot{{\mathbf {H}}}}^s\) up to a common time \(T' \in (0,T]\), depending on \( \Vert U_0 \Vert _s \), thanks to Lemma 4.8. By triangular inequality we have, by (4.49), for any \( n \ge n_0 \), \( N \ge N_0 \),
For any \(\varepsilon > 0 \), since \(s>s_0+2\), there exists \(N_\varepsilon \in {\mathbb {N}}\) (independent of n) such that
Consider now the term \( \Vert U^n_N - U_N\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^s} \). As \(\Pi _{ N} U_0, \Pi _{ N} U_0^n \in \dot{C}^\infty \), the solutions \(U_N(t)\), \(U_N^n(t) \) belong actually to \( {\dot{{\mathbf {H}}}}^{s+2}\). By interpolation and by item \(\mathrm{(ii)}\) of Corollary 4.7 applied with \( s \leadsto s+2\) one has, for \( s+ 2 = \theta s_0 + (1-\theta ) (s+2) \),
Arguing in the same way of the proof of (4.52) one obtains
By (4.54)–(4.57), we have \( \limsup _{n\rightarrow \infty } \Vert U^n-U\Vert _{L^\infty _T {\dot{{\mathbf {H}}}}^s}\le \varepsilon \), \( \forall \varepsilon \in (0,1).\) \(\square \)
Notes
For \(\delta \) sufficiently small, if \( |j-k| \le \delta \left\langle j + k \right\rangle \) and \( |k| \le \delta \left\langle 2j - k \right\rangle \) then \( (j,k) = (0,0)\).
References
Alazard, T., Burq, N., Zuily, C.: On the water-wave equations with surface tension. Duke Math. J. 158(3), 413–499 (2011)
Alazard, T., Métivier, G.: Paralinearization of the Dirichlet–Neumann operator and regularity of three-dimensional water waves. Commun. PDEs 34(12), 1632–1704 (2009)
Alinhac, S., Gérard, P.: Pseudo-differential operators and the Nash–Moser theorem. AMS, Graduate Studies in Mathematics, vol. 82. ISBN-10: 0-8218-3454-1 (2007)
Antonelli, P., Marcati, P.: On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. 287(2), 657–686 (2009)
Antonelli, P., Marcati, P.: The quantum hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. 203, 499–527 (2012)
Antonelli, P., Marcati, P., Zheng, H.: genuine hydrodynamic analysis to the 1-D QHD system: existence, dispersion and stability. arXiv:1910.08104 (2019)
Audiard, C., Haspot, B.: Global well-posedness of the Euler–Korteweg system for small irrotational data. Commun. Math. Phys 351(1), 201–247 (2017)
Benzoni-Gavage, S., Danchin, R., Descombes, S.: On the well-posedness for the Euler–Korteweg model in several space dimensions. Indiana Univ. Math. J. 56, 1499–1579 (2007)
Berti, M., Bolle, P.: Quasi-periodic solutions of nonlinear wave equations on \( {{\mathbb{T}}^{d}}\). In: European Research Monographs, European Research Monographs, 374 pages. ISBN 978-3-03719-211-5, 10.4171/211 (2020)
Berti, M., Maspero, A., Murgante, F.: Long time existence results for the Euler–Korteweg equations. In Preparation
Berti, M., Delort, J.-M.: Almost global solutions of capillary-gravity water waves equations on the circle. In: UMI Lecture Notes, x+268 pages, ISBN 978-3-319-99486-4 (2018)
Berti, M., Feola, R., Franzoi, L.: Quadratic life span of periodic gravity-capillary water waves. Water Waves (2020). https://doi.org/10.1007/s42286-020-00036-8
Bona, J.L., Smith, R.: The initial-value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 278(1287), 555–601 (1975)
Bresch, D., Desjardins, B., Lin, C.: On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28(3–4), 843–868 (2003)
Carles, R., Danchin, R., Saut, J.-C.: Madelung, Gross-Pitaevskii and Korteweg. Nonlinearity 25(10), 2843–2873 (2012)
Deng, Y., Ionescu, A., Pausader, B., Pusateri, F.: Global solutions of the gravity-capillary water-wave system in three dimensions. Acta Math. 219(2), 213–402 (2017)
Dunn, J., Serrin, J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88(2), 95–133 (1985)
Feola, R., Iandoli, F.: Local well-posedness for quasi-linear NLS with large Cauchy data on the circle. Ann. l’Institut H. Poincaré (C) Anal. Non linéaire 36(1), 119–164 (2018)
Feola, R., Iandoli, F.: Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori. arXiv:2003.04815
Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)
Maspero, A., Robert, D.: On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms. J. Funct. Anal. 273(2), 721–781 (2017)
Métivier, G.: Para-differential calculus and applications to the Cauchy problem for nonlinear systems. Edizioni della Normale, Pisa, xii+140 pp. ISBN: 978-88-7642-329-1 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Walter Craig.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A Bony–Weyl Calculus in Periodic Hölder Spaces
A Bony–Weyl Calculus in Periodic Hölder Spaces
In this Appendix we develop in a self-contained manner Weyl paradifferential calculus for space periodic symbols \( a(x, \xi ) \) which belong to the Banach scale of Hölder spaces \( W^{\varrho ,\infty } ({\mathbb {T}}^d) \). We mention [2] for paradifferential calculus on \( {\mathbb {T}}^d \) using the standard quantization, see [22] for the case of \({\mathbb {R}}^d \). The convenience of Weyl calculus for energy estimates was noted in [16], and then implemented in [11, 18]. The main results are the continuity Theorem A.7 and the composition Theorem A.8, which require mild regularity assumptions of the symbols in the space variable, and imply Theorems 2.4 and 2.5. We first provide some preliminary technical results.
Technical lemmas. In the following we denote by \(\partial _m\), \(m= 1, \ldots , d\) the discrete derivative, defined for functions \(f :{\mathbb {Z}}^d \rightarrow {\mathbb {C}}\) as
where \( \vec {e}_m \) denotes the usual unit basis vector of \( {\mathbb {N}}_0^d \) with 0 components expect the m-th one. Given a multi-index \(\beta \in {\mathbb {N}}^d_0 \), we set \( \partial ^\beta f := \partial _1^{\beta _1} \cdots \partial _d^{\beta _d} f \).
We shall use the Leibniz rule for finite differences in the following form: given \( k \in {\mathbb {N}}\), \( m = 1, \ldots , d \), there exist constants \( C_{k_1,k_2} \) (binomial coefficients) such that
Moreover, when using discrete derivatives, the analogous of the integration by parts formula is given by the Abel resummation formula:
Lemma A.1
Let \(K :{\mathbb {T}}^d \rightarrow {\mathbb {C}}\) be a function satisfying, for constants A and B, the estimate
Then
Proof
If \( A \le 1 \) the bound (A.5) follows trivially integrating the first inequality in (A.4). Then we suppose \( A > 1 \). We split the integral in (A.5) as
We bound the first integral using the first inequality in (A.4), getting
To bound the second integral in (A.6) we use that, for some \(c >0\), \( \max _{1\le m \le d} \big | \sin \big (\frac{y_m}{2}\big ) \big | \ge c|y| \), \( \forall y\in [-\pi ,\pi ]^d \), and therefore the second inequality in (A.4) implies
The bounds (A.7)–(A.8) and (A.6) imply (A.5). \(\square \)
The next lemma represents a Fourier multiplier operator acting on periodic functions as a convolution integral on \( {\mathbb {R}}^d \). The key step is the use of Poisson summation formula.
Lemma A.2
Let \( \chi \in {{{\mathcal {S}}}} ({\mathbb {R}}^d) \). Then the Fourier multiplier \( \chi _\theta ( D) := \chi ( \theta ^{-1} D) \), \( \theta \ge 1 \), acting on a periodic function \( u \in L^1 ({\mathbb {T}}^d, {\mathbb {C}}) \) can represented by
where \( \psi _{\theta } (z) := \theta ^{d} \, \psi ( \theta z) \) and \( \psi \) denotes the anti-Fourier transform of \( \chi \) on \({\mathbb {R}}^d \).
Proof
For \( \theta \ge 1 \) we write
Then the Fourier transform \( {\widehat{\psi _\theta }} (\xi ) = \int _{{\mathbb {R}}^d} \, \theta ^d \, \psi (\theta z) \, e^{- {{\text {i}}}z \cdot \xi } {\text {d}}z = \int _{{\mathbb {R}}^d} \psi (y)\, e^{- {{\text {i}}}y \cdot \frac{\xi }{\theta }} {\text {d}}y = {\widehat{\psi }} \left( \frac{\xi }{\theta } \right) = \chi \left( \frac{\xi }{\theta } \right) \), and, using Poisson summation formula, we write the periodic function \( h_\theta (z) \) in (A.10) as
Therefore the integral (A.10) is
proving (A.9). \(\square \)
We now give the definition and basic properties of the Hölder spaces \( W^{\varrho , \infty } ({\mathbb {T}}^d )\).
Definition A.3
(Periodic Hölder spaces) Given \(\varrho \in {\mathbb {N}}_0\), we denote by \( W^{\varrho , \infty } ({\mathbb {T}}^d ) \) the space of continuous functions \( u : {\mathbb {T}}^d \rightarrow {\mathbb {C}}\), \( 2 \pi \)-periodic in each variable \( (x_1, \ldots , x_d) \), whose derivatives of order \(\varrho \) are in \(L^\infty \), equipped with the norm \( \Vert u \Vert _{W^{\varrho ,\infty }} := \sum _{|\alpha | \le \varrho } \Vert \partial _x^\alpha u \Vert _{L^\infty } \), \( \alpha \in {\mathbb {N}}_0^d \). In case \( \varrho >0\), \(\varrho \notin {\mathbb {N}}\), we denote \( \lfloor \varrho \rfloor \) the integer part of \( \varrho \), and we define \( W^{\varrho , \infty } ({\mathbb {T}}^d ) \) as the space of functions u in \( C^{\lfloor \varrho \rfloor }({\mathbb {T}}^d,{\mathbb {C}}) \) whose derivatives of order \( \lfloor \varrho \rfloor \) are \( (\varrho - \lfloor \varrho \rfloor ) \)-Hölder-continuous, that is
equipped with the norm
For \( \varrho = 0 \) the norm \( \Vert \ \Vert _{W^{\varrho ,\infty }} = \Vert \ \Vert _{L^{\infty }} \).
The Hölder spaces \( W^{\varrho , \infty } ({\mathbb {T}}^d) \) can be described by the Paley–Littlewood decomposition of a function. Consider the locally finite partition on unity
where \( \chi : {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) is the cut-off function defined in (2.13). It induces the decomposition of a distribution \( u \in {{{\mathcal {S}}}}' ({\mathbb {T}}^d) \) as
We also set
The Paley–Littlewood theory of the Hölder spaces \( W^{\varrho ,\infty } ({\mathbb {T}}^d)\) follows as in \( {\mathbb {R}}^d \), see e.g. [22], once we represent the Fourier multipliers \( \Delta _k \) as integral convolution operators on \( {\mathbb {R}}^d \), by Lemma A.2. In particular the following smoothing estimates hold: for any \( \alpha \in {\mathbb {N}}^d_0 \), \( \varrho \ge 0 \),
and, for any \( \varrho >0 \),
In this way it results as in \( {\mathbb {R}}^d \) that the Hölder norms \( \left\| \ \right\| _{W^{\varrho , \infty }} \) satisfy interpolation estimates. In particular we shall use that, given \(\varrho , \varrho _1, \varrho _2 \ge 0 \),
Hölder estimates of regularized symbols.
In order to prove estimates of the regularized symbol \( a_\chi \) defined in (2.14) in Hölder spaces (Lemma A.5) we represent it as a convolution integral on \( {\mathbb {R}}^d \), by Lemma A.2,
where \(\psi _\theta (z) = \theta ^d \psi ( \theta z ) \) and \( \psi \) is the anti-Fourier transform of \( \chi \).
In the proof of Lemma A.5 we shall use the following estimate.
Lemma A.4
For any \( \beta \in {\mathbb {N}}_0^d \), \( u \in L^\infty ({\mathbb {T}}^d) \), we have
Proof
By (A.17) we have, for all \( \beta \in {\mathbb {N}}^d_0 \),
By the definition \(\psi _{\epsilon \left\langle \xi \right\rangle }(y) = (\epsilon \left\langle \xi \right\rangle )^d \, \psi (\epsilon \left\langle \xi \right\rangle y)\) and Faà di Bruno formula, we have that
Then (A.18) follows by (A.19) and (A.20). \(\square \)
The next lemma provides estimates of the regularized symbol \( a_\chi \) in terms of the symbol a.
Lemma A.5
(Estimates on regularized symbols) Let \( m \in {\mathbb {R}}\), \( N \in {\mathbb {N}}_0 \).
-
1.
If \(a \in \Gamma ^m_{L^\infty }\), \( m \in {\mathbb {R}}\), then \( a_\chi \) defined in (2.14) belongs to \(\Sigma ^m_{L^\infty }\) and
$$\begin{aligned} \left| a_\chi \right| _{m, L^{\infty }, N} \lesssim \left| a \right| _{m, L^{\infty }, N}. \end{aligned}$$(A.21) -
2.
If \(a \in \Gamma _{H^{s_0-\varrho }}^m\), \(\varrho \ge 0\), \(s_0 > \frac{d}{2}\), then \(a_\chi \) belongs to \(\Gamma ^{m+\varrho }_{L^\infty }\) and
$$\begin{aligned} | a_\chi |_{m+\varrho , L^\infty , N} \lesssim | a|_{m, {H^{s_0-\varrho }},N}. \end{aligned}$$(A.22) -
3.
If \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(\varrho > 0 \), then, for any \(\beta \in {\mathbb {N}}_0^d \), \(\partial _\xi ^\beta a_\chi - (\partial _{\xi }^\beta a)_\chi \in \Sigma ^{m-|\beta | -\varrho }_{L^\infty }\) and
$$\begin{aligned} \big | \partial _\xi ^\beta a_\chi - (\partial _{\xi }^\beta a)_\chi \big |_{m-|\beta |-\varrho , L^{\infty }, N} \lesssim \left| a \right| _{m, {W^{\varrho , \infty }}, N+ |\beta |}. \end{aligned}$$(A.23) -
4.
If \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(\varrho \ge 0 \), then, for any \(\alpha \in {\mathbb {N}}_0^d\) with \(|\alpha | \ge \varrho \), \( \partial _x^\alpha a_\chi = (\partial _x^\alpha a)_\chi \in \Sigma ^{m+|\alpha | -\varrho }_{L^\infty }\) and
$$\begin{aligned} \left| \partial _x^\alpha a_\chi \right| _{m+|\alpha | -\varrho , L^{\infty }, N} \lesssim \left| a \right| _{m, {W^{\varrho , \infty }}, N}. \end{aligned}$$(A.24) -
5.
If \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(\varrho > 0 \), then, \(a- a_\chi \in \Gamma ^{m-\varrho }_{L^\infty }\) and
$$\begin{aligned} \left| a- a_\chi \right| _{m-\varrho , L^{\infty }, N} \lesssim \left| a \right| _{m, {W^{\varrho , \infty }}, N}. \end{aligned}$$(A.25)
Proof
Proof of (A.21). Differentiating (2.14) for any \( \beta \in {\mathbb {N}}_0^d \), we have
Then (2.7) and (A.18) directly imply (A.21).
Proof of (A.22) By the Cauchy–Schwartz inequality
The case \(N\ge 1\) follows in the same way.
Proof of (A.23). First, for any \( \xi \in {\mathbb {R}}^d \), we define \( k \in {\mathbb {N}}\) such that \( 2^{k-1} \le 2 \epsilon \langle \xi \rangle \le 2^k \). Then, by the properties of the cut-off function \( \chi \) in (2.13) and the projector \( S_k \) in A.13 we have
Differentiating (2.14) and using (A.26) we get
because \(\left\langle \xi \right\rangle \lesssim 2^k\). This proves (A.23) for \(N = 0\). For \(N \ge 1\) the estimate is similar.
Proof of (A.24). For any \( \xi \in {\mathbb {R}}^d \), we define \( k \in {\mathbb {N}}\) such that \( 2^{k-1} \le 2 \epsilon \langle \xi \rangle \le 2^k \). By (2.14) and (A.26) with \(\beta = 0\), we write \( a_\chi (\cdot , \xi ) = \chi _{\epsilon \langle \xi \rangle } (D) a(\cdot , \xi ) = \chi _{\epsilon \langle \xi \rangle } (D) S_k a(\cdot , \xi ) \), and then
by (2.7). This proves (A.24) with \(N = 0\). For \(N \ge 1\) the estimate is similar.
Proof of (A.25). For any \( \beta \in {\mathbb {N}}_0^d \) we write \( \partial _\xi ^\beta (a - a_\chi ) = \big [ \partial _\xi ^\beta a - (\partial _\xi ^\beta a)_\chi \big ] + \big [ (\partial _\xi ^\beta a)_\chi - \partial _\xi ^\beta a_\chi \big ] \). The first term is bounded, using (A.15) with \( \theta = \epsilon \langle \xi \rangle \), as
The second term satisfies the same bound by (A.23). This proves (A.25). \(\square \)
Change of quantization. In order to prove the boundedness Theorem A.7 and the composition Theorem A.8, it is convenient to pass from the Weyl quantization of a symbol \( a(x, \xi )\), defined in (2.15), to the standard quantization which is defined, given a symbol \( b(x, \xi ) \), as
We have the change of quantization formula
In the next lemma we estimate the norms of b in terms of those of a. We remind that \( \Sigma ^m_{\mathscr {W}}\) denotes the set of spectrally localized symbols, i.e. satisfying (2.6).
Lemma A.6
(Change of quantization) Let \(a \in \Sigma ^m_{L^\infty }\), \(m \in {\mathbb {R}}\). If \( \delta > 0 \) in (2.6) is small enough, then (cfr. A.28)
is a symbol in \( \Sigma ^m_{L^\infty }\) satisfying
Proof
Since a satisfies (2.6) with \(\delta \) small enough, it follows that b satisfies (2.6). In order to prove (A.30) we differentiate (A.29) obtaining that, for any \( \beta \in {\mathbb {N}}_0^d \),
for some \( \epsilon = \epsilon (\delta ') > 0 \), where in the last equality we used that the sum is actually restricted over the indexes for which \(|n|\le \delta ' \langle \xi \rangle \), \(\delta ' \in (0,1)\). Then we represent \(\partial _\xi ^\beta b\) as the integral
We are going to estimate the \( L^1 \)-norm of \( K(x, \cdot ) \) using Lemma A.1. First note that, since \( a \in \Sigma ^m_{L^\infty }\), we have \(\left\langle \xi + \frac{n}{2}\right\rangle \sim \left\langle \xi \right\rangle \) on the support of 
, and then we bound (A.31) as
uniformly in x. Moreover, using Abel resummation formula (A.3) and the Leibniz rule (A.2) for finite differences, we get, for any \( h = 1, \ldots , d \),
Then, using (2.7) and that \( \displaystyle { \big | \partial _h^k \chi _\epsilon \big ( \frac{n}{\left\langle \xi \right\rangle } \big ) \big | \lesssim \left\langle \xi \right\rangle ^{-k}} \), \(\forall h = 1, \ldots , d\), we estimate
uniformly in x. In view of (A.32)–(A.33) we apply Lemma A.1 with \(A = \langle \xi \rangle \) and \(B= \left\langle \xi \right\rangle ^{m - |\beta |} \, |a|_{m, L^\infty , |\beta | + d+1}\) obtaining
that proves (A.30). \(\square \)
Continuity.
We now prove boundedness estimates in Sobolev spaces of operators with spectrally localized symbols, requiring derivatives in \(\xi \) of the symbol and no derivatives in x.
Theorem A.7
(Continuity) Let \(a \in \Sigma ^m_{L^\infty }\) with \(m \in {\mathbb {R}}\). Then \({{\text {Op}}}\left( a\right) \) defined in (A.27) extends to a bounded operator from \( H^s \rightarrow H^{s-m}\), for any \( s \in {\mathbb {R}}\), satisfying
Moreover, if a fulfills (2.6) with \( \delta > 0 \) small enough, then the operator \({{\text {Op}}}^W\!\left( a\right) \) defined in (2.15) satisfies
Proof
We first recall the Littlewood–Paley characterization of the Sobolev norm
where \( \Delta _k \) are defined in (A.12). The norm \( \Vert \ \Vert _0 = \Vert \ \Vert _{L^2} \). We first prove (A.34). Step 1: according to (A.11), we perform the Littlewood–Paley decomposition of \({{\text {Op}}}\left( a\right) \),
where
In order to prove (A.34), it is sufficient to prove that
Indeed, decomposing v in Paley–Littlewood packets as in (A.12),
which are almost orthogonal in \(L^2\) (namely \( \Delta _k \Delta _j = 0 \) for any \( |j-k| \ge 3 \)), using the fact that \( {{\text {Op}}}\left( a_k\right) v = {{\text {Op}}}\left( a\right) \Delta _k v \), and since the action of \({{\text {Op}}}\left( a_k\right) \) does not spread much the Fourier support of functions being a spectrally localized, according to (2.17), we have
Step 2: By (A.38) and (A.27) we write \({{\text {Op}}}\left( a_k\right) \) as the integral operator
with kernel
We shall deduce (A.39) by applying the Schur lemma: if
then Schur lemma guarantees that the integral operator (A.41) is bounded on \( L^2 ({\mathbb {T}}^d) \) and
Let us prove (A.43) and estimate the constants \(C_1, C_2\). By (A.42) we have that
Then, applying \( (d+1)\)-times Abel resummation formula (A.3) to (A.42), we obtain, for any \( h = 1, \ldots , d \),
and we deduce, using (2.7), (A.2), \( \left| K_k(x,z) \right| \lesssim \left| 2 \sin (z_h/2) \right| ^{-d-1} |a|_{m, L^\infty ,d+1} 2^{k(m-1)} \) for any \( h = 1, \ldots , d \), thus
By (A.45), (A.46) we apply Lemma A.1 with \( A = 2^{k}\) and \(B= 2^{km} |a|_{m, L^\infty , d+1} \), deducing that
uniformly for \( x \in {\mathbb {T}}^d \). Similarly
uniformly for \( y \in {\mathbb {T}}^d \). Finally (A.47), (A.48), (A.44) prove (A.39) completing the proof of (A.34).
Proof of (A.35). By Lemma A.6 we have \({{\text {Op}}}^W\!\left( a\right) = {{\text {Op}}}\left( b\right) \) for a spectrally localized symbol \(b \in \Sigma ^m_{L^\infty }\) which fulfills estimate (A.30). Then (A.35) follows by (A.34). \(\square \)
Composition of paradifferential operators. We finally prove a composition result for paradifferential operators. The difference with respect to Theorem 6.1.1 and 6.1.4 in [22] is to have periodic symbols and the use of the Weyl quantization.
We shall use that, in view of the interpolation inequality (A.16), if \( a \in \Gamma ^m_{W^{\varrho ,\infty }} \) and \( b \in \Gamma ^{m'}_{W^{\varrho ,\infty }} \) then \( ab \in \Gamma ^{m+m'}_{W^{\varrho ,\infty }} \) and, for any \( N \in {\mathbb {N}}_0 \), any \(0 \le \varrho _1 \le \alpha \le \beta \le \varrho _2 \) such that \(\varrho _1 + \varrho _2 = \alpha + \beta \)
Theorem A.8
(Composition) Let \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(b \in \Gamma ^{m'}_{W^{\varrho , \infty }}\) with \(m, m' \in {\mathbb {R}}\) and \(\varrho \in (0,2]\). Then
where the linear operator \(R^{-\varrho }(a,b):{\dot{H}}^s \rightarrow {\dot{H}}^{s-(m+m')+\varrho }\), \(\forall s \in {\mathbb {R}}\), satisfies
with \( N \ge 3d+4\).
Proof
We give the proof in the case \( \varrho \in (1, 2] \). We first compute \({{\text {Op}}}^{\text {BW}}\!\left( a\right) {{\text {Op}}}^{\text {BW}}\!\left( b\right) \). Recalling the definition (2.16) we obtain
We now perform a Taylor expansion of \( {\widehat{a}}_\chi \big (j-k, \frac{j+k}{2} \big ) \) in the second variable, around the point \(\frac{j+ \ell }{2}\). Writing \(j+ k = j+ \ell + (k - \ell )\), we obtain
We expand analogously \( {\widehat{b}}_\chi \big (k-\ell , \frac{k+ \ell }{2} \big ) \) around the point \(\frac{j+\ell }{2}\). Writing \( k + \ell = j+ \ell - (j-k)\), we obtain
Moreover, recalling (2.20) and (2.15), we write \({{\text {Op}}}^{\text {BW}}\!\left( a\#_\varrho b \right) u = {{\text {Op}}}^W\!\left( (ab + \frac{1}{2{\text {i}}}\{a, b\})_\chi \right) u \) and, by the previous expansions,
where
We show now that the operators \(R_i(a,b)\), \( i = 1, \ldots , 4\) fulfill estimate (A.51).
Estimate of \({R_1(a,b)}\). By exchanging the role of a and b it is enough to prove that the symbols \( \partial _\xi ^\alpha a_\chi \partial _x^\alpha b_\chi - (\partial _\xi ^\alpha a \,\partial _x^\alpha b)_\chi \), \(|\alpha |\le 1\), belong to \(\Sigma _{L^\infty }^{m+m'-\varrho }\) and then apply Theorem (A.7). The spectral localization property follows because of the cut-off \(\chi _\epsilon \) and \(\epsilon \) small. As \(\partial _x^\alpha \) commutes with the Fourier multiplier \( \chi _{\epsilon \left\langle \xi \right\rangle }(D) \) we have that \( \partial _x^\alpha b_\chi = (\partial _x^\alpha b)_\chi \) and we write \( \partial _{\xi }^\alpha a_\chi \, \partial _{x}^\alpha b_\chi - (\partial _{\xi }^\alpha a \, \partial _{x}^\alpha b)_\chi \) as
Consider first the term in (A.57). By Lemma A.5, \(\partial _{\xi }^\alpha a_\chi - (\partial _{\xi }^\alpha a)_\chi \in \Gamma ^{m-\varrho -|\alpha |}_{L^\infty }\) and \((\partial _{x}^\alpha b)_\chi \in \Gamma _{L^\infty }^{m'+|\alpha |}\) and by remark (v) after Definition 2.1, for any \( n \in {\mathbb {N}}_0 \),
Next consider the terms in (A.56). By remarks (iii), (iv) after Definition 2.1, we have \( \partial _{\xi }^\alpha a \in \Gamma ^{m-|\alpha |}_{W^{\varrho , \infty }}\subset \Gamma ^{m-|\alpha |}_{W^{\varrho - |\alpha |, \infty }}\), \(\partial _{x}^\alpha b \in \Gamma ^{m'}_{W^{\varrho -|\alpha |, \infty }}\), so we can apply Lemma A.5, property (A.49) and (2.10) to obtain
where to pass from the first to the second line we used the second interpolation inequality in (A.49). Altogether we have proved that the symbol in (A.52) belongs to \(\Sigma ^{m+m'-\varrho }_{L^\infty }\) and its seminorms are bounded by (A.58). Then Theorem (A.7) proves that \(R_1(a,b)\) fulfills estimate (A.51).
Estimate of \({R_{2}(a,b)}\). First we rewrite (A.53) as
where
and \( D_{x_n} := \partial _{x_n} / {\text {i}}\) and \( D_x^\alpha := D_{x_1}^{\alpha _1} \cdots D_{x_d}^{\alpha _d} \). Then, recalling (A.27),
where
We claim that \( {f_t^\alpha } (x, \xi ) \) is spectrally localized, namely
In fact on the support of \({\widehat{b}}_\chi \big (j, \xi +\frac{j}{2} \big )\) we have, for some \( \delta ' \in (0,1) \),
whereas, on the support of \( \partial _\xi ^\alpha {\widehat{a}}_\chi \big (n -j, \xi + \frac{n + tj}{2} \big )\), \( t \in [0,1] \),
The estimates (A.61)–(A.62) then give \( | n | \le | j | +| n - j | \le \delta ' \langle \xi \rangle + (\delta + \delta \delta ' ) \langle \xi \rangle + \delta \langle n \rangle \), which implies (A.60).
In order to apply Theorem (A.7) it remains to prove that, for any \( N \ge 3d+4 \),
which implies, for any \(s\in {\mathbb {R}}\), \( u \in \dot{H}^s \), \( \Vert R_{2}(a,b)u\Vert _{{s-m-m'+\varrho }}\lesssim | b |_{m', W^{\varrho ,\infty }, N}\ | a |_{m, L^\infty , N}\, \Vert u\Vert _{s} \). Thus \(R_2(a,b)\) satisfies the estimate (A.51).
In order to prove (A.63) note that, differentiating (A.59), for any \( \beta \in {\mathbb {N}}_0^{d} \),
where \( C_{\beta _1, \beta _2} \) are binomial coefficients and
By (A.60) and (A.61) the sum over n in (A.59) is restricted to indexes satisfying
We deduce that the sum in (A.65) is bounded by
recalling that \(|\alpha | = 2 \). We also estimate \(K_t^{\beta _1, \beta _2} (x, y, z)\) applying Abel resummation formula (A.3) in the sum (A.65), in the index n and in the index j separately, obtaining, using (A.24), (A.21), (A.2) and (2.10),
In view of (A.66)–(A.67) and \( |\beta | \le d+ 1 \), we apply Lemma A.1 with \(d \leadsto 2d\), choosing \(A=\langle \xi \rangle \), \( B=\langle \xi \rangle ^{m+m'-|\beta |-\varrho } \, | b|_{m', W^{\varrho ,\infty } , 2d+1+|\beta |} \, | a |_{m, L^\infty , 2d+3+|\beta |} \) and we obtain
proving (A.63).
The proof that \(R_3(a,b)\) and \(R_4(a,b)\) satisfy the estimate (A.51) follows similarly. \(\square \)
Rights and permissions
About this article
Cite this article
Berti, M., Maspero, A. & Murgante, F. Local Well Posedness of the Euler–Korteweg Equations on \({{\mathbb {T}}^d}\). J Dyn Diff Equat 33, 1475–1513 (2021). https://doi.org/10.1007/s10884-020-09927-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-020-09927-3