Abstract
We consider the critical dissipative SQG equation in bounded domains, with the square root of the Dirichlet Laplacian dissipation. We prove global a priori interior \(C^{\alpha }\) and Lipschitz bounds for large data.
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Introduction
The Surface Quasigeostrophic equation (SQG) of geophysical origin [18] was proposed as a two dimensional model for the study of inviscid incompressible formation of singularities [5, 9]. While the global regularity of all solutions of SQG whose initial data are smooth is still unknown, the original blow-up scenario of [9] has been ruled out analytically [13] and numerically [11], and nontrivial examples of global smooth solutions have been constructed [4]. Solutions of SQG and related equations without dissipation and with non-smooth (piece-wise constant) initial data give rise to interface dynamics [3, 17] with potential finite time blow up [15].
The addition of fractional Laplacian dissipation produces globally regular solutions if the power of the Laplacian is larger or equal than one half. When the linear dissipative operator is precisely the square root of the Laplacian, the equation is commonly referred to as the “critical dissipative SQG”, or “critical SQG”. This active scalar equation [5] has been the object of intensive study in the past decade. The solutions are transported by divergence-free velocities they create, and are smoothed out and decay due to nonlocal diffusion. Transport and diffusion do not add size to a solution: the solution remains bounded, if it starts so [22]. The space \(L^{\infty }({\mathbb R}^2)\) is not a natural phase space for the nonlinear evolution: the nonlinearity involves Riesz transforms and these are not well behaved in \(L^{\infty }\). Unfortunately, for the purposes of studies of global in time behavior of solutions, \(L^{\infty }\) is unavoidable: it quantifies the most important information freely available. The equation is quasilinear and \(L^{\infty }\)-critical, and there is no “ wiggle room”, nor a known better (smaller) space which is invariant for the evolution. One must work in order to obtain better information. A pleasant aspect of criticality is that solutions with small initial \(L^{\infty }\) norm are smooth [10]. The global regularity of large solutions was obtained independently in [2] and [20] by very different methods: using harmonic extension and the De Georgi methodology of zooming in, and passing from \(L^2\) to \(L^{\infty }\) and from \(L^{\infty }\) to \(C^{\alpha }\) in [2], and constructing a family of time-invariant moduli of continuity in [20]. Several subsequent proofs were obtained (please see [12] and references therein). All the proofs are dimension-independent, but are in either \({\mathbb R}^d\) or on the torus \({\mathbb {T}}^d\). The proofs of [7] and [12] were based on an extension of the Córdoba–Córdoba inequality [14]. This inequality states that
pointwise. Here \(\Lambda = \sqrt{-\Delta }\) is the square root of the Laplacian in the whole space \({\mathbb R}^d\), \(\Phi \) is a real valued convex function of one variable, normalized so that \(\Phi (0) = 0\) and f is a smooth function. The fractional Laplacian in the whole space has a (very) singular integral representation, and this can be used to obtain (1). In [7] specific nonlinear maximum principle lower bounds were obtained and used to prove the global regularity. A typical example is
pointwise, for \(f=\partial _i\theta \) a component of the gradient of a bounded function \(\theta \). This is a useful cubic lower bound for a quadratic expression, when \(\Vert \theta \Vert _{L^{\infty }}\le \Vert \theta _0\Vert _{L^{\infty }}\) is known to be bounded above. The critical SQG equation in \({\mathbb R}^2\) is
where
and \(\nabla ^{\perp } = (-\partial _2, \partial _1)\) is the gradient rotated by \(\frac{\pi }{2}\). Because of the conservative nature of transport and the good dissipative properties of \(\Lambda \) following from (1), all \(L^p\) norms of \(\theta \) are nonincreasing in time. Moreover, because of properties of Riesz transforms, u is essentially of the same order of magnitude as \(\theta \). Differentiating the equation we obtain the stretching equation
(In the absence of \(\Lambda \) this is the same as the stretching equation for three dimensional vorticity in incompressible Euler equations, one of the main reasons SQG was considered in [5, 9] in the first place.) Taking the scalar product with \(\nabla ^{\perp }\theta \) we obtain
for \(q^2 = |\nabla ^{\perp }\theta |^2\), with
The operator \(\partial _t + u\cdot \nabla + \Lambda \) is an operator of advection and fractional diffusion: it does not add size. Using the pointwise bound (2) we already see that the dissipative lower bound is potentially capable of dominating the cubic term Q, but there are two obstacles. The first obstacle is that constants matter: the two expressions are cubic, but the useful dissipative cubic lower bound \(D(q)\ge K |q|^{3}\) has perhaps too small a prefactor K if the \(L^{\infty }\) norm of \(\theta _0\) is too large. The second obstacle is that although
has the same size as \(\nabla ^{\perp }\theta \) (modulo constants) in all \(L^p\) spaces \(1<p<\infty \), it fails to be bounded in \(L^{\infty }\) by the \(L^{\infty }\) norm of \(\nabla ^{\perp }\theta \). In order to overcome these obstacles, in [7] and [12], instead of estimating directly gradients, the proof proceeds by estimating finite differences, with the aim of obtaining bounds for \(C^{\alpha }\) norms first. In fact, in critical SQG, once the solution is bounded in any \(C^{\alpha }\) with \(\alpha >0\), it follows that it is \(C^{\infty }\). More generally, if the equation has a dissipation of order s, i.e., \(\Lambda \) is replaced by \(\Lambda ^s\) with \(0<s\le 1\) then if \(\theta \) is bounded in \(C^{\alpha }\) with \(\alpha >1-s\), then the solution is smooth [8]. (This condition is sharp, if one considers general linear advection diffusion equations, [23]. In [12] the smallness of \(\alpha \) is used to show that the term corresponding to Q in the finite difference version of the argument is dominated by the term corresponding to D(q).
In this paper we consider the critical SQG equation in bounded domains. We take a bounded open domain \(\Omega \subset {\mathbb R}^d\) with smooth (at least \(C^{2,\alpha }\)) boundary and denote by \(\Delta \) the Laplacian operator with homogeneous Dirichlet boundary conditions and by \(\Lambda _D\) its square root defined in terms of eigenfunction expansions. Because no explicit kernel for the fractional Laplacian is available in general, our approach, initiated in [6] is based on bounds on the heat kernel.
The critical SQG equation is
with
and smooth initial data. We obtain global regularity results, in the spirit of the ones in the whole space. There are quite significant differences between the two cases. First of all, the fact that no explicit formulas are available for kernels requires a new approach; this yields as a byproduct new proofs even in the whole space. The main difference and additional difficulty in the bounded domain case is due to the lack of translation invariance. The fractional Laplacian is not translation invariant, and from the very start, differentiating the equation (or taking finite differences) requires understanding the respective commutators. For the same reason, the Riesz transforms \(R_D\) are not spectral operators, i.e., they do not commute with functions of the Laplacian, and so velocity bounds need a different treatment. In [6] we proved using the heat kernel approach the existence of global weak solutions of (7) in \(L^2(\Omega )\). A proof of local existence of smooth solutions is provided in the present paper in \(d=2\). The local existence is obtained in Sobolev spaces based on \(L^2\) and uses Sobolev embeddings. Because of this, the proof is dimension dependent. A proof in higher dimensions is also possible but we do not pursue this here. We note that for regular enough solutions (e.g. \(\theta \in H_0^1(\Omega )\)) the normal component of the velocity vanishes at the boundary \(\left( R_D^{\perp }\theta \cdot N\right) _{\left| \right. \partial \Omega }=0\) because the stream function \(\psi = \Lambda _D^{-1}\theta \) vanishes at the boundary and its gradient is normal to the boundary. Let us remark here that even in the case of a half-space and \(\theta \in C_0^{\infty }(\Omega )\), the tangential component of the velocity need not vanish: there is tangential slip.
In order to state our main results, let
denote the distance from x to the boundary of \(\Omega \). We introduce the \(C^{\alpha }(\Omega )\) space for interior estimates:
Definition 1
Let \(\Omega \) be a bounded domain and let \(0<\alpha <1\) be fixed. We say that \(\theta \in C^{\alpha }(\Omega )\) if \(\theta \in L^{\infty }(\Omega )\) and
The norm in \(C^{\alpha }(\Omega )\) is
Our main results are the following:
Theorem 1
Let \(\theta (x,t)\) be a smooth solution of (7) on a time interval [0, T), with \(T\le \infty \), with initial data \(\theta (x,0)= \theta _0(x)\). Then the solution is uniformly bounded,
There exists \(\alpha \) depending only on \(\Vert \theta _0\Vert _{L^{\infty }(\Omega )}\) and \(\Omega \), and a constant \(\Gamma \) depending only on the domain \(\Omega \) (and in particular, independent of T) such that
holds.
The second theorem is about global interior gradient bounds:
Theorem 2
Let \(\theta (x,t)\) be a smooth solution of (7) on a time interval [0, T), with \(T\le \infty \), with initial data \(\theta (x,0)= \theta _0(x)\). There exists a constant \(\Gamma _1\) depending only on \(\Omega \) such that
holds.
Remark 1
Higher interior regularity can be proved also. In fact, once global interior \(C^{\alpha }\) bounds are obtained for any \(\alpha >0\), the interior regularity problem becomes subcritical, meaning that “there is room to spare”. This is already the case for Theorem 2 and justifies thinking that the equation is \(L^{\infty }\) interior-critical. However, we were not able to obtain global uniform \(C^{\alpha }(\bar{\Omega })\) bounds. Moreover, we do not know the implication \(C^{\alpha }(\bar{\Omega }) \Rightarrow C^{\infty }(\bar{\Omega })\) uniformly, and thus the equation is not \(L^{\infty }\) critical up to the boundary. This is due to the fact that the commutator between normal derivatives and the fractional Dirichlet Laplacian is not controlled uniformly up to the boundary. The example of half-space is instructive because explicit kernels and calculations are available. In this example odd reflection across the boundary permits the construction of global smooth solutions, if the initial data are smooth and compactly supported away from the boundary. The support of the solution remains compact and cannot reach the boundary in finite time, but the gradient of the solution might grow in time at an exponential rate.
The proofs of our main results use the following elements. First, the inequality (1) which has been proved in [6] for the Dirichlet \(\Lambda _D\) is shown to have a lower bound
with \(c>0\) depending only on \(\Omega \). Note that in \({\mathbb R}^d\), \(d(x)=\infty \), which is consistent with (1). This lower bound (valid for general \(\Phi \) convex, with c independent of \(\Phi \), see (46)) provides a strong damping boundary repulsive term, which is essential to overcome boundary effects coming from the lack of translation invariance.
The second element of proofs consists of nonlinear lower bounds in the spirit of [7]. A version for derivatives in bounded domains, proved in [6] is modified for finite differences. In order to make sense of finite differences near the boundary in a manner suitable for transport, we introduce a family of good cutoff functions depending on a scale \(\ell \) in Lemma 3. The finite difference nonlinear lower bound is
when \(f=\chi \delta _h\theta \) is large (see (48)), where \(\chi \) belongs to the family of good cutoff functions.
Once global interior \(C^{\alpha }(\Omega )\) bounds are obtained, in order to obtain global interior bounds for the gradient, we use a different nonlinear lower bound,
for large \(f=\chi \nabla \theta \) (see (61)). This is a super-cubic bound, and makes the gradient equation look subcritical. Similar bounds were obtained in the whole space in [7]. Proving the bounds (16) and (17) requires a different approach and new ideas because of the absence of explicit formulas and lack of translation invariance.
The third element of proofs are bounds for \(R_D^{\perp }\theta \) based only on global a priori information on \(\Vert \theta \Vert _{L^{\infty }}\) and the nonlinear lower bounds on D(f) for appropriate f. Such an approach was initiated in [7] and [12]. In the bounded domain case, again, the method of proof is different because the kernels are not explicit, and reference is made to the heat kernels. The boundaries introduce additional error terms. The bound for finite differences is
for \(\rho \le cd(x)\), with \(f=\chi \delta _h \theta \) and with C a constant depending on \(\Omega \) (see 90). The bound for gradient is
for \(\rho \le cd(x)\) with \(f=\chi \nabla \theta \) with a constant C depending on \(\Omega \) (see (107)). These are remarkable pointwise bounds (clearly not valid for the case of the Laplacian even in the whole space, where \(D(f)(x) = |\nabla f(x)|^2\)).
The fourth element of the proof are bounds for commutators. These bounds
for \(\ell \le d(x)\), (see (112)), and
for \(\ell \le d(x)\), (see (115)), reflect the difficulties due to the boundaries. They are remarkable though in that the only price to pay for a second order commutator in \(L^{\infty }\) is \(d(x)^{-2}\). Note that in the whole space this commutator vanishes (\(\chi =1\)). This nontrivial situation in bounded domains is due to cancellations and bounds on the heat kernel representing translation invariance effects away from boundaries (see (37, 38)). Although the heat kernel in bounded domains has been extensively studied, and the proofs of (37) and (38) are elementary, we have included them in the paper because we have not found them readily available in the literature and for the sake of completeness.
The paper is organized as follows: after preliminary background, we prove the nonlinear lower bounds. We have separate sections for bounds for the Riesz transforms and the commutators. The proof of the main results are then provided, using nonlinear maximum principles. We give some of the explicit calculations in the example of a half-space and conclude the paper by proving the translation invariance bounds for the heat kernel (37), (38), and a local well-posedness result in two appendices.
Preliminaries
The \(L^2(\Omega )\) - normalized eigenfunctions of \(-\Delta \) are denoted \(w_j\), and its eigenvalues counted with their multiplicities are denoted \(\lambda _j\):
It is well known that \(0<\lambda _1\le \cdots \le \lambda _j\rightarrow \infty \) and that \(-\Delta \) is a positive selfadjoint operator in \(L^2(\Omega )\) with domain \({\mathcal {D}}\left( -\Delta \right) = H^2(\Omega )\cap H_0^1(\Omega )\). The ground state \(w_1\) is positive and
holds for all \(x\in \Omega \), where \(c_0, \, C_0\) are positive constants depending on \(\Omega \). Functional calculus can be defined using the eigenfunction expansion. In particular
with
for \(f\in {\mathcal {D}}\left( \left( -\Delta \right) ^{\beta }\right) = \{f\left| \right. \; (\lambda _j^{\beta }f_j)\in \ell ^2(\mathbb N)\}\). We will denote by
the fractional powers of the Dirichlet Laplacian, with \(0\le s \le 2\) and with \(\Vert f\Vert _{s,D}\) the norm in \({\mathcal {D}}\left( \Lambda _D^s\right) \):
It is well-known and easy to show that
Indeed, for \(f\in {\mathcal {D}}\left( -\Delta \right) \) we have
We recall that the Poincaré inequality implies that the Dirichlet integral on the left-hand side above is equivalent to the norm in \(H_0^1(\Omega )\) and therefore the identity map from the dense subset \({\mathcal {D}}\left( -\Delta \right) \) of \(H_0^1(\Omega )\) to \({\mathcal D}\left( \Lambda _D\right) \) is an isometry, and thus \(H_0^1(\Omega )\subset {\mathcal {D}}\left( \Lambda _D\right) \). But \({\mathcal {D}}\left( -\Delta \right) \) is dense in \({\mathcal D}\left( \Lambda _D\right) \) as well, because finite linear combinations of eigenfunctions are dense in \({\mathcal D}\left( \Lambda _D\right) \). Thus the opposite inclusion is also true, by the same isometry argument. Note that in view of the identity
with
valid for \(0\le s <2\), we have the representation
for \(f\in {\mathcal {D}}\left( \left( -\Lambda _D\right) ^{s}\right) \). We use precise upper and lower bounds for the kernel \(H_D(t,x,y)\) of the heat operator,
These are as follows [16, 24, 25]. There exists a time \(T>0\) depending on the domain \(\Omega \) and constants c, C, k, K, depending on T and \(\Omega \) such that
holds for all \(0\le t\le T\). Moreover
holds for all \(0\le t\le T\). Note that, in view of
elliptic regularity estimates and Sobolev embedding which imply uniform absolute convergence of the series (if \(\partial \Omega \) is smooth enough), we have that
for positive t, where we denoted by \(\partial _1^{\beta }\) and \(\partial _2^{\beta }\) derivatives with respect to the first spatial variables and the second spatial variables, respectively.
Therefore, the gradient bounds (32) result in
We also use a bound
valid for \(t\le cd(x)^2\) and \(0<t\le T\), which follows from the upper bounds (31), (32).
Important additional bounds we need are
and
valid for \(t\le cd(x)^2\) and \(0<t\le T\). These bounds reflect the fact that translation invariance is remembered in the solution of the heat equation with Dirichlet boundary data for short time, away from the boundary. We sketch the proofs of (36), (37) and (38) in Appendix 1.
Nonlinear Lower Bounds
We prove bounds in the spirit of [7]. The proofs below are based on the method of [6], but they concern different objects (finite differences, properly localized) or different assumptions (\(C^{\alpha }\)). Nonlinear lower bounds are an essential ingredient in proofs of global regularity for drift-diffusion equations with nonlocal dissipation.
We start with a couple lemmas. In what follows we denote by c and C generic positive constants that depend on \(\Omega \). When the logic demands it, we temporarily manipulate them and number them to show that the arguments are not circular. There is no attempt to optimize constants, and their numbering is local in the proof, meaning that, if for instance \(C_2\) appears in two proofs, it need not be the same constant. However, when emphasis is necessary we single out constants, but then we avoid the letters c, C with or without subscripts.
Lemma 1
The solution of the heat equation with initial datum equal to 1 and zero boundary conditions,
obeys \(0\le \Theta (x,t)\le 1\), because of the maximum principle. There exist constants T, c, C depending only on \(\Omega \) such that the following inequalities hold:
for all \(0\le t\le T\), and
for all \(0\le t\le T\). Let \(0<s<2\). There exists a constant c depending on \(\Omega \) and s such that
holds.
Remark 2
\(\Lambda _D^{s} 1 \) is defined by duality by the left hand side of (42) and belongs to \(H^{-1}(\Omega )\).
Proof
Indeed,
because \(H_D\) is positive. Using the lower bound in (23) we have that \(|x-y|\le \frac{d(x)}{2}\) implies
and then, using the lower bound in (31) we obtain
Integrating it follows that
If \(\frac{d(x)}{2\sqrt{kt}}\ge 1\) then the integral is bounded below by \(\int _0^1\rho ^{d-1}e^{-\rho ^2}d\rho \). If \(\frac{d(x)}{2\sqrt{kt}}\le 1\) then \(\rho \le 1\) implies that the exponential is bounded below by \(e^{-1}\) and so (40) holds. \(\square \)
Now (41) holds immediately from (23) and the upper bound in (31) because the integral
if \(d\ge 2\).
Regarding (42) we use
and choose appropriately \(\tau \). In view of (41), if
then, when \(\tau \le t\le T\) we have
and therefore
holds. The choice
implies (42) provided \(2\tau \le T\) which is the same as \(d(x)\le \frac{\sqrt{T}}{2C\sqrt{2}}\). On the other hand, \(\Theta \) is exponentially small if t is large enough, so the contribution to the integral in (42) is bounded below by a nonzero constant. This ends the proof of the lemma.
Lemma 2
Let \(0\le \alpha <1\). There exists constant C depending on \(\Omega \) and \(\alpha \) such that
holds for \(0\le t\le T\).
Indeed, the upper bounds (31) and (35) yield
and, in view of the upper bound in (23), \(\frac{1}{d(y)}w_1(y)\le C_0\) and the upper bound in (31), we have
This proves (43). We introduce now a good family of cutoff functions \(\chi \) depending on a length scale \(\ell \).
Lemma 3
Let \(\Omega \) be a bounded domain with \(C^2\) boundary. For \(\ell >0\) small enough (depending on \(\Omega \)) there exist cutoff functions \(\chi \) with the properties: \(0\le \chi \le 1\), \(\chi (y)=0\) if \(d(y)\le \frac{\ell }{4}\), \(\chi (y)= 1\) for \(d(y)\ge \frac{\ell }{2}\), \(|\nabla ^k\chi |\le C\ell ^{-k}\) with C independent of \(\ell \) and
and
hold for \(j>-d\), \(\alpha <d\) and \(d(x)\ge \ell \). We will refer to such \(\chi \) as a “good cutoff”.
Proof
There exists a length \(\ell _0\) such that if P is a point of the boundary \(\partial \Omega \), and if \(|P-y|\le 2\ell _0\), then \(y\in \Omega \) if and only if (after a rotation and a translation) \(y_d>F(y')\), where \(y'=(y_1, \ldots , y_{d-1})\) and F is a \(C^2\) function with \(F(0)=0\), \(\nabla F(0)=0\), \(|\nabla F|\le \frac{1}{10}\). We took thus without loss of generality coordinates such that \(P= (0,0)\) and the normal to \(\partial \Omega \) at P is \((0,\ldots , 0, 1)\). Now if \(\ell <\ell _0\) and \(d(x)\ge \ell \) and \(|y-P|\le \frac{\ell _0}{2}\) satisfies \(d(y)\le \frac{\ell }{2}\), then there exists a point \(Q\in B(P,\ell _0)\) such that
Indeed, if \(|x-P|\ge \ell _0\) we take \(Q=P\) because then \(|x-y| =|x-P+P-y|\ge \ell _0-\frac{\ell _0}{2}\), so \(|x-y|\ge \frac{|y-Q|}{2}\). But also \(|x-y|\ge \frac{d(x)}{2}\) because there exists a point \(P_1 =(p, F(p))\in \partial \Omega \) such that \(|y-P_1| = d(y)\le \frac{\ell }{2}\) while obviously \(|x-P_1|\ge d(x)\ge \ell \). If, on the other hand \(|x-P|< \ell _0\), then x is in the neighborhood of P and we take \(Q=x\). Because \(y-P_1= (y'-p, y_d-F(p))\) we have
for \(y\in B(P,\ell _0)\). We take a partition of unity of the form \(1= \psi _0 +\sum _{j=1}^N\psi _j\) with \(\psi _k\in C_0^{\infty }({\mathbb R}^d)\), subordinated to the cover of the boundary with neighborhoods as above, and with \(\psi _0\) supported in \(d(x)\ge \frac{\ell _0}{4}\), identically 1 for \(d(x)\ge \frac{\ell _0}{2}\), \(\psi _j\) supported near the boundary \(\partial \Omega \) in balls of size \(2\ell _0\) and identically 1 on balls of radius \(\ell _0\). \(\square \)
The cutoff will be taken of the form \(\chi = \alpha _0 +\sum _{j=1}^N \chi _j(\frac{y_d-F(y')}{\ell })\alpha _j(y)\), where of course the meaning of y changes in each neighborhood. The smooth functions \(\chi _j(z)\), are identically zero for \(|z|\le \frac{11}{40}\) and identically 1 for \(|z| \ge \frac{10}{22}\). The integrals in (44) and (45) reduce to integrals of the type
and
This completes the proof.
We recall from [6] that the Córdoba-Córdoba inequality [14] holds in bounded domains. In fact, more is true: there is a lower bound that provides a strong boundary repulsive term:
Proposition 1
Let \(\Omega \) be a bounded domain with smooth boundary. Let \(0\le s<2\). There exists a constant \(c>0\) depending only on the domain \(\Omega \) and on s, such that, for any \(\Phi \), a \(C^2\) convex function satisfying \(\Phi (0)= 0\), and any \(f\in C_0^{\infty }(\Omega )\), the inequality
holds pointwise in \(\Omega \).
The proof follows in a straightforward manner from the proof of [6] using convexity, approximation, and the lower bound (42). We prove below two nonlinear lower bounds for the case \(\Phi (f)= \frac{f^2}{2}\), one when f is a localized finite difference, and one when f is a localized first derivative. The proof of Proposition 1 can be left as an exercise, following the same pattern as below.
Theorem 3
Let \(f\in L^{\infty }(\Omega )\) be smooth enough (\(C^2\), e.g.) and vanish at the boundary, \(f\in {\mathcal {D}}(\Lambda _D^{s})\) with \(0\le s<2\). Then
holds for all \(x\in \Omega \). Here \(\gamma _0 =\frac{c_{s}}{2}\) with \(c_s\) of (29). Let \(\ell >0\) be a small number and let \(\chi \in C_0^{\infty }(\Omega )\), \(0\le \chi \le 1\) be a good cutoff function, with \(\chi (y)=1\) for \(d(y)\ge \frac{\ell }{2}\), \(\chi (y) =0\) for \(d(y)\le \frac{\ell }{4}\) and with \(|\nabla \chi (y)|\le \frac{C}{\ell }\). There exist constants \(\gamma _1>0\) and \(M>0\) depending on \(\Omega \) such that, if q(x) is a smooth function in \(L^{\infty }(\Omega )\) then if
then
holds pointwise in \(\Omega \) when \(|h|\le \frac{\ell }{16}\), and \(d(x)\ge \ell \) with
Proof
We start by proving (47):
It follows that
where \(\tau >0\) is arbitrary and \(0\le \psi (s)\le 1\) is a smooth function, vanishing identically for \(0\le s\le 1\) and equal identically to 1 for \(s\ge 2\). We restrict to \(t\le T\),
\(\square \)
and open brackets in (51):
with
and
We proceed with a lower bound on I and an upper bound on J. For the lower bound on I we note that, in view of (40) and the fact that
we have
Therefore we have that
with \(c_2 = c_1\int _1^2\psi (u)u^{-1-\frac{s}{2}}du\), a positive constant depending only on \(\Omega \) and s, provided \(\tau \) is small enough,
In order to bound J from above we use (43) with \(\alpha =0\). Now
We have that
Indeed,
so the bound follows from (31) and (45). On the other hand,
and therefore, in view of (43)
and therefore
with
a constant depending only on \(\Omega \) and s. In conclusion
Now, because of the lower bound (52), if we can choose \(\tau \) so that
then it follows that
Because of the bounds (55), (58), if
then a choice
with \(C_9 = c_2 (8C_8)^{-1}\) achieves the desired bound. This concludes the proof.
We are providing now a lower bound for D(f) for a different situation.
Theorem 4
Let \(\ell >0\) be a small number and let \(\chi \in C_0^{\infty }(\Omega )\), \(0\le \chi \le 1\) be a good cutoff function, with \(\chi (y)=1\) for \(d(y)\ge \frac{\ell }{2}\), \(\chi (y) =0\) for \(d(y)\le \frac{\ell }{4}\) and with \(|\nabla \chi (y)|\le \frac{C}{\ell }\). There exist constants \(\gamma _2>0\) and \(M>0\) depending on \(\Omega \) such that, if q(x) is a smooth function in \(C^{\alpha }(\Omega )\) with \(0<\alpha <1\) and
then
holds pointwise in \(\Omega \) when \(d(x)\ge \ell \), with
Proof
We follow exactly the proof of Theorem 3 up to, and including the definition of I(x) given in (53). In particular, the lower bound (55) is still valid, provided \(\tau \) is small enough (56). The term J starts out the same, but is treated slightly differently,
In order to bound J we use (45) and (43).
We have from (31) and (45), as before,
On the other hand,
In view of (43)
and so
with
a constant depending only on \(\Omega \) and s. Regarding \(J_{12}(x)\) we have in view of (35)
because, in view of (23)
on the domain of integration. \(\square \)
In conclusion
The rest is the same as in the proof of Theorem 3: If \(|f(x)|\ge Md(x)^{-1}\Vert q\Vert _{L^{\infty }(\Omega )}\) for suitable M, (\(M= 8C_3c_2^{-1}\)) then we choose \(\tau \) such that
and this yields \(|f(x)| I\ge 4 |J(x)|\), and consequently, in view of (59) which is then valid, the result (61) is proved.
We specialize from now on to \(s=1\).
Bounds for Riesz Transforms
We consider u given in (8),
We are interested in estimates of u in terms of \(\theta \), and in particular estimates of finite differences and the gradient. We fix a length scale \(\ell \) and take a good cutoff function \(\chi \in C_0^{\infty }(\Omega )\) which satisfies \(\chi (x) =1 \) if \(d(x)\ge \frac{\ell }{2}\), \(\chi (x) = 0\) if \(d(x)\le \frac{\ell }{4}\), \(|\nabla \chi (x)|\le C\ell ^{-1}\), (44) and (45). We take \(|h|\le \frac{\ell }{14}\). In view of the representation
we have on the support of \(\chi \)
We split
with
and \(\rho =\rho (x,h)>0\) a length scale to be chosen later but it will be smaller than the distance from x to the boundary of \(\Omega \):
We represent
where
and where
with
and
We used here the fact that \((\chi \theta )(\cdot )\) and \((\chi \theta )(\cdot + h)\) are compactly supported in \(\Omega \) and hence
The following elementary lemma will be used in several instances:
Lemma 4
Let \(\rho >0\), \(p>0\). Then
if \(m\ge 0\), \(j\ge 0\), \(m+j>0\), and
if \(m=0\) and \(j=0\), with constants \(C_{K,m,j}\) and \(C_K\) independent of \(\rho \) and p. Note that when \(m+j>0\), \(\rho =\infty \) is allowed.
We start estimating the terms in (73). For \(e_1\) we use the inequality (36), and it then follows from Lemma 4 with \(m= {d} +1\) that
and therefore we have from (44) that
holds for \(d(x)\ge \ell \). Concerning \(e_3\) we use Lemma (4) with \(m= d\) and \(j=0,1\) in conjunction with (32) and obtain
and therefore we obtain from (45)
holds for \(d(x)\ge \ell \). Regarding \(e_4\) we can split it into
with
and
Now \(e_6\) is bounded using the Lemma (4) with \(m= d\) and \(j=0,1\) in conjunction with (32) and (44) and obtain
for \(d(x)\ge \ell \), with a constant independent of \(\ell \). For \(e_5\) we use the fact that \(\chi \) is a fixed smooth function which vanishes at the boundary.
In order to bound the terms \(e_2\) and \(e_5\) we need to use additional information, namely the inequalities (37) and (38). For \(e_5\) we write
and using (37) and Lemma 4 with \(m=0\), \(j=0\) and (45) we obtain the bound
and therefore, in view of (45) and \(\rho \le d(x)\) we have
for \(d(x)\ge \ell \), with C depending on \(\Omega \) but not on \(\ell \). Consequently, we have
for \(d(x)\le \ell \), with a constant C depending on \(\Omega \) only. In order to estimate \(e_2\) we write
and use (38) and Lemma 4 with \(m=1\), \(j=0\) to obtain
and thus
holds for \(d(x)\ge \ell \). Summarizing, we have that
for \(d(x)\ge \ell \). We now estimate \(u_h\) using (32) and a Schwartz inequality
We have therefore
where \(f=\chi \delta _h\theta \) and D(f) is given in Theorem 3. Regarding \(\delta _h u^{out}\) we have
in view of (36). Putting together the estimates (87), (88) and (89) we have
Proposition 2
Let \(\chi \) be a good cutoff, and let u be defined by (8). Then
holds for \(d(x)\ge \ell \), \(\rho \le cd(x)\), \(f=\chi \delta _h \theta \) and with C a constant depending on \(\Omega \).
Now we will obtain similar estimates for \(\nabla u\). We start with the representation
where
and \(\rho = \rho (x) \le c d(x)\). In view of (36) we have
We split now
where
and with
and
and
Now
holds for \(d(x)\ge \ell \) because of (36), time integration using Lemma 4 and then use of (44). For \(g_2(x)\) we use (38) and then Lemma 4 to obtain
for \(d(x)\ge \ell \). Now
holds because of (32), Lemma 4 and then use of (45). Regarding \(g_4\), in view of
we have
and, we thus obtain from (37) and from Lemma 4 with \(m=j=0\)
because \(\rho \le cd(x)\).
Finally we have using a Schwartz inequality like for (88)
Gathering the bounds we have proved
Proposition 3
Let \(\chi \) be a good cutoff with scale \(\ell \) and let u be given by (8). Then
holds for \(d(x)\ge \ell \), \(\rho \le cd(x)\) and \(f=\chi \nabla \theta \) with a constant C depending on \(\Omega \).
Commutators
We consider the finite difference
with \(d(x)\ge \ell \) and \(|h|\le \frac{\ell }{16}\). We use a good cutoff \(\chi \) again and denote
We start by computing
Lemma 5
There exists a constant \(\Gamma _0\) such that the commutator
obeys
for \(d(x)\ge \ell \), \(|h|\le \frac{\ell }{16}\), \(f=\chi \delta _h\theta \) and \(\theta \in H_0^1(\Omega )\cap L^{\infty }(\Omega )\).
Proof
We use (110). For \(E_1(x)\) we use a similar argument as for \(e_1\) leading to (80), namely the inequality (31) and Lemma 4 with \(m=d+2\), \(j=0\), and (44) to obtain
For \(E_2\) we proceed in a manner analogous to the one leading to the bound (86), by using (85), (37), Lemma 4 with \(m=d+2\), \(j=0\), and (45) to obtain
For \(E_3\) we use
and using Lemma 4 with \(m=d+1\), \(j=0\) and (45) we obtain
concluding the proof. \(\square \)
We consider now the commutator \([\nabla , \Lambda _D]\).
Lemma 6
There exists a constant \(\Gamma _3\) depending on \(\Omega \) such that for any smooth function f vanishing at \(\partial \Omega \) and any \(x\in \Omega \) we have
If \(\chi \) is a good cutoff with scale \(\ell \) and if \(\theta \) is a smooth bounded function in \({\mathcal {D}}\left( \Lambda _D\right) \), then
obeys
for \(d(x)\ge \ell \), with a constant \(\Gamma _3\) independent of \(\ell \).
Proof
We note that
and therefore
The inequality (113) follows from (37) and Lemma 4. For the inequality (115) we need also to estimate
by the right hand side of (115), and this follows from (45) in view of (31). \(\square \)
SQG: Hölder Bounds
We consider the equation (7) with u given by (8) and with smooth initial data \(\theta _0\) compactly supported in \(\Omega \). We note that by the Córdoba-Córdoba inequality we have
We prove the following uniform interior Hölder bound:
Theorem 5
Let \(\theta (x,t)\) be a smooth solution of (7) in the smooth bounded domain \(\Omega \). There exists a constant \(0<\alpha <1\) depending only on \(\Vert \theta _0\Vert _{L^{\infty }(\Omega )}\), and a constant \(\Gamma >0\) depending on the domain \(\Omega \) such that, for any \(\ell >0\) sufficiently small
holds.
Proof
We take a good cutoff \(\chi \) used above, \(|h|\le \frac{\ell }{16}\) and observe that, from the SQG equation we obtain the equation
where \(C_h(\theta )\) is the commutator given above in (111). Denoting (as before in (109)) \(f=\chi \delta _h\theta \) we have after multiplying by \(\delta _h\theta \) and using the fact that \(\chi (x)=1\) for \(d(x)\ge \ell \),
where
and D(f) is given in Theorem 3. \(\square \)
Multiplying by \(|h|^{-2\alpha }\) where \(\alpha >0\) will be chosen small to be small enough we obtain
The factor \(2\alpha \) comes from the differentiation \(\delta _h u\cdot \nabla _h (|h|^{-2\alpha })\) and its smallness will be crucial below. Let us record here the inequality (47) in the present case:
valid pointwise, when \(|h|\le \frac{\ell }{16}\) and \(d(x)\ge \ell \), where
and \(|(\delta _h\theta )_d|=0\) otherwise.
We use now the estimates (90), (112) and a Young inequality for the term involving \(\sqrt{\rho D(f)}\) to obtain
for \(d(x)\ge \ell \), \(|h|\le \frac{\ell }{16}\). Let us choose \(\rho \) now. We set
where we put
where M is the constant from Theorem 3, \(\Gamma _0\) is the constant from (112) and \(\gamma _1\) is the constant from (124). This choice was made in order to use the lower bound on D(f) to estimate the contribution due to the inner piece \(u_h\) (see (72)) of \(\delta _h u\) and the contribution from the commutator \(C_h(\theta )\). We distinguish two cases. The first case is when \(|\delta _h\theta (x)|\ge M_1\Vert \theta \Vert _{L^{\infty }}\frac{|h|}{d(x)}\). Then we have
The choice of \(M_1\) was such that, in this case
We choose now \(\alpha \) by requiring
to satisfy
and obtain from (128)
for \(d(x)\ge \ell \), \(|h|\le \frac{\ell }{16}\), in the case \(|f|\ge M_1\Vert \theta \Vert _{L^{\infty }}\frac{|h|}{d(x)}\).
The second case is when the opposite inequality holds, i.e, when \(|\delta _h\theta (x)|\le M_1\Vert \theta \Vert _{L^{\infty }}\frac{|h|}{d(x)}\). Then, using \(\rho = d(x)\) we obtain from (125)
Summarizing, in view of the inequalities (131) and (132), the damping term \(\frac{\gamma _1}{d(x)}|\delta _h\theta (x)|^2\) in (124) and the choice of small \(\epsilon \) in (130), we have that
holds for \(d(x)\ge \ell \) and \(|h|\le \frac{\ell }{16}\) where
with \(\Gamma _1\) depending on \(\Omega \). Without loss of generality we may take \(\Gamma _1> 4(16)^{2\alpha }\) so that
when \(|h|\ge \frac{\ell }{16}\). We note that
holds for any t, \(x\in \Omega \) with \(d(x)\ge \ell \) and \(|h|\le \frac{\ell }{16}\).
We take \(\delta >0\), \(T>0\). We claim that, for any \(\delta >0\) and any \(T>0\)
holds.
The rest of the proof is done by contradiction. Indeed, assume by contradiction that there exists \(\tilde{t}\le T\), \(\tilde{x}\) and \(\tilde{h}\) with \(d({\tilde{x}})\ge \ell \) and \(|{\tilde{h}}|\le {\frac{\ell }{16}}\) such that
holds. Because the solution is smooth, we have
for a short time \(0\le t\le t_1\). (Note that this is not a statement about well-posedness in this norm: \(t_1\) may depend on higher norms.) Also, because the solution is smooth, it is bounded in \(C^1\), and
on the time interval [0, T]. It follows that there exists \(\delta _1>0\) such that
In view of these considerations, we must have \(\tilde{t} >t_1\), \(|\tilde{h}|\ge \delta _1\). Moreover, the supremum is attained: there exists \(\bar{x}\in \Omega \) with \(d(\bar{x})\ge \ell \) and \(\bar{h} \ne 0\) such that \(\delta _1\le |\bar{h}|\le \frac{\ell }{16}\) such that
Because of (135) we have that
and therefore there exists \(t'<\tilde{t}\) such that \(s(t')>s(\tilde{t})\). This implies that \(\inf \{t>t_1\left| \right. s(t)>R\} = t_1\) which is absurd because we made sure that \(s(t_1)< R\). Now \(\delta \) and T are arbitrary, so we have proved
which finishes the proof of the theorem.
Proof of Theorem 1
The proof follows from (136) because \(\Gamma _1\) does not depend on \(\ell \). For any fixed \(x\in \Omega \) we may take \(\ell \) such that \(\ell \le d(x)\le 2 \ell \). Then (136) implies
Gradient Bounds
We take the gradient of (7). We obtain
where \((\nabla u)^*\) is the transposed matrix. Let us take a good cutoff \(\chi \). Then \(g=\nabla \theta \) obeys everywhere
with \(C_{\chi }\) given in (114). We multiply by g and, using the fact that \(\chi (x)=1 \) when \(d(x)\ge \ell \) we obtain
when \(d(x)\ge \ell \), where \(L_{\chi }\) is similar to the one defined in (122):
and \(f=\chi g\). Recall that \(D(f) = f\Lambda _Df-\Lambda _D\left( \frac{f^2}{2}\right) \). Then, using (115) and (107) we deduce
for \(d(x)\ge \ell \). Using a Young inequality we deduce
for \(d(x)\ge \ell \). Now \(|g| = |f|\) when \(d(x)\ge \ell \). If \(|g(x)|\ge M\Vert \theta \Vert _{L^{\infty }(\Omega )}d(x)^{-1}\) then, in view of (61)
which is a super-cubic lower bound. We choose in this case
and the right hand side of (142) becomes at most cubic in g:
In view of (143) we see that
holds for \(d(x)\ge \ell \), if \(|g|\ge M\Vert \theta \Vert _{L^{\infty }}d(x)^{-1}\). In the opposite case, \(|g(x)|\le M\Vert \theta \Vert _{L^{\infty }}d(x)^{-1}\) we choose
and obtain from (142)
and using the convex damping inequality (61)
we obtain in this case
Putting together (146) and (149) and 119 we obtain
Theorem 6
Let \(\theta \) be a smooth solution of (7). Then
where \(P(\Vert \theta \Vert _{L^{\infty }(\Omega )})\) is a polynome of degree four.
Proof of Theorem 2
The proof follows by choosing \(\ell \) depending on x, because the constants in (150) do not depend on \(\ell \).
Example: Half Space
The case of the half space is interesting because global smooth solutions of (7) are easily obtained by reflection: If the initial data \(\theta _0\) is smooth and compactly supported in \(\Omega = {\mathbb R}^d_+\) and if we consider its odd reflection
then the solution of the critical SQG equation in the whole space, with intitial data \(\widetilde{\theta _0}\) is globally smooth and its restriction to \(\Omega \) solves (7) there. This follows because of reflection properties of the heat kernel and of the Dirichlet Laplacian.
The heat kernel with Dirichlet boundary conditions in \(\Omega ={\mathbb R}^d_{+}\) is
where \(\widetilde{y} = (y_1,\ldots , y_{d-1}, -y_d)\). More precisely,
with \(x'= (x_1,\ldots , x_{d-1})\),
and
Let us note that
We check that (32) is obeyed. Indeed, because \(1-e^{-p}\ge \frac{p}{2}\) when \(0\le p\le 1\) it follows that
if \(\frac{x_dy_d}{t}\le 1\), and if \(p=\frac{x_dy_d}{t} \ge 1\) then
In this case, if \(\frac{x_d}{\sqrt{t}} \ge 1\) then \(\frac{y_d}{t}\le t^{-\frac{1}{2}}p\) and \(pe^{-p}\) is bounded; if \(\frac{x_d}{\sqrt{t}}\le 1\) we write \(\frac{y_d}{t} = t^{-\frac{1}{2}}(\frac{y_d-x_d}{\sqrt{t}} + \frac{x_d}{\sqrt{t}})\) and thus we obtain:
We check (37): First we have
and then
Indeed, the only nonzero component occurs when the differentiation is with respect to the normal direction, and then
Therefore
We check (38): first
Consequently
and (38) follows:
We compute \(\Theta \) and \(\Lambda _D1\):
and therefore
Remark 3
We note here that \(\Lambda _D^{s} 1 = C_sy_d^{-s}\) is calculated by duality:
where we used the symmetry of the kernel H and (163).
We observe that if we consider horizontal finite differences, i.e. \(h_d= 0\) then \(C_h(\theta )\) vanishes, and we deduce that
with \(C_{1,\alpha }\) the partial \(C^{\alpha }\) norm of the initial data. This inequality can be used to prove that \(u_2\) is bounded when \(d=2\). Indeed
and the bound is obtained using the partial Hölder bound on \(\theta \) (164) and the uniform bounds \(\Vert \theta \Vert _{L^p}\) for \(p=1, \infty \). The outline of the proof is as follows: we split the integral
with
and
where in (167) we used the fact that the kernel is odd in the first variable. Then, for \(u^{in}\) we use the bound (164) to derive
and for \(u^{out}\), if we have no other information on \(\theta \) we just bound
with some \(L\ge \delta \). Both \(\delta \) and L are arbitrary.
Finally, let us note that even if \(\theta \in C_0^{\infty }(\Omega )\), the tangential component of the velocity need not vanish at the boundary because it is given by the integral
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Acknowledgements
The work of PC was partially supported by NSF Grant DMS-1209394.
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Appendices
Appendix 1
We sketch here the proofs of (36) (37) and (38). We take a point \(\bar{x}\in \Omega \), a point \(y\in \Omega \) and distinguish between two cases, if \(d(\bar{x})< \frac{|\bar{x}-y|}{4}\) and if \(d(\bar{x})\ge \frac{|\bar{x}-y|}{4}\). In the first case we take a ball B of radius \(\delta =\frac{d(\bar{x})}{8}\) centered at \(\bar{x}\) and in the second case we take also a ball B centered at \(\bar{x}\) but with radius \(\delta =\frac{d(\bar{x})}{2}\). We note that in both cases the radius \(\delta \) is proportional to \(d(\bar{x})\). We take \(x\in B(\bar{x}, \frac{\delta }{2})\), we fix \(y\in \Omega \), take the function \(h(z,t) = H_D(z,y,t)\), and apply Green’s identity in the domain \(U = B\times (0,t)\). We obtain
and thus
We note that the x dependence is only via G, and \(x-z\) is bounded away from zero. We differentiate twice under the integral sign, and use the upper bounds (31), (32). We have
where \(p_k(\xi )\) are polynomials of degree k. The integrals are not singular. In both cases \(|x-z|\ge \frac{\delta }{2}\), and any negative power \((t-s)^{-\frac{k}{2}}\) can be absorbed by \(e^{-\frac{|x-z|^2}{8(t-s)}}\) at the price \(|x-z|^{-k}\le C\delta ^{-k}\), still leaving \(e^{-\frac{|x-z|^2}{8(t-s)}}\) available. Similarly, in the first case \(|y-z|\ge |\bar{x}-y| -\delta \ge \delta \) and in the second case \(|y-z|\ge |\bar{x}-z| - |\bar{x}-y| \ge \frac{\delta }{2}\). Any power \(s^{-\frac{k}{2}}\) can be absorbed by \(e^{-\frac{|y-z|^2}{2Ks}}\) at the price \(|y-z|^{-k}\le C\delta ^{-k}\) still leaving \(e^{-\frac{|y-z|^2}{2Ks}}\) available. We note that if \(d(y)<d(x)\) so that \(d(y)^2<t\) is possible, then, in view of (23) we have \(\frac{w_1(y)}{|y-z|d(y)}\le C\delta ^{-1}\). We also note that view of the fact that
we have a bound
with \(\tilde{K} = 16 + 4K\). Pulling this exponential out and estimating all the rest in terms of \(\delta \) we obtain, in both cases, all the integrals bounded by \(Ct\delta ^{-d-4}\) and therefore we have, in both cases,
because \(t\le c\delta ^2\). This proves (36).
For (37) and (38) we start by noticing that it is enough to prove the estimates
and
for \(t<cd^2(x)\). Indeed, if \(|x-y|\ge \frac{d(x)}{14}\), individual Gaussian upper bounds for up to two derivatives of \(H_D\) suffice (there is no need for cancellations). In order to prove (171) and (172) we use a good cutoff \(\chi \) with a scale \(\ell = \frac{d(x)}{100}\). We take \(y\in B(x, \frac{d(x)}{14})\). Both x and y are fixed for now. We note that the function
solves
vanishes for \(z\in \partial \Omega \), and has initial datum \(h_0= \chi (z)\delta (z-y)\), so, by Duhamel
which, in view of \((e^{t\Delta }f)(z) = \int _{\Omega }H_D(z,w,t)f(w)dw\) yields
for all z, and recalling that \(\chi (x)=\chi (y) =1\), and reading at \(z=x\) we have
The right hand side integral is not singular and can be differentiated because the support of \(\nabla \chi \) is far from the ball \(B(x,\frac{d(x)}{14})\). Differentiation \(\nabla _x+\nabla _y\) cancels the Gaussian \(G_t(x-y)\). The estimates of the right hand side
and
for \(t<cd^2(x)\) follow from Gaussian upper bounds. Integration dy on the ball \(B(\frac{d(x)}{14})\) picks up the volume of the ball, and thus (171) and (172) are verified.
Appendix 2
We sketch here the proof of local wellposedness of the equation (7). We start by defining a Galerkin approximation. We consider the projectors \(P_n\)
with \(f_j= \int _{\Omega }f(x)w_j(x)dx\). We consider for fixed n the approximate system
where
with
and with initial data \(\theta _n(0) = P_n\theta _0\) where \(\theta _0\) is a fixed smooth function belonging to \(H_0^1(\Omega )\cap H^2(\Omega )\). Although it was written as a PDE, the system (175) is a system of ODEs for the coefficients \(\theta _{n,j}(t)= \int _{\Omega }\theta _nw_jdx\). Let us note that \(P_n\) does not commute with \(\nabla \) but does commute with \(-\Delta \) and functions of it. The function \(u_n\) is divergence-free and it is a finite sum of divergence-free functions,
Note however that \(u_n\notin P_nL^{2}(\Omega )\). The normal component of \(u_n\) vanishes at the boundary because \(\nabla ^{\perp }w_j\cdot \nu _{\left| \right. \Omega } =0\). Moreover, because
it follows that \(\Vert \theta _n(t)\Vert _{L^2(\Omega )}\) is bounded in time and therefore the solution exists for all time. The following upper bound for higher norms is uniform only for short time, and it is the bound that is used for local existence of smooth solutions. We apply \(\Lambda _D^2=-\Delta \) to (175) and use the fact that it is a local operator, it commutes with \(P_n\) and with derivatives:
We take the scalar product with \(\Lambda _D^2\theta _n\). Because this is finite linear combinations of eigenfunctions, it vansihes at \(\partial \Omega \) and integration by parts is allowed. We obtain
We note now that
Now \(R_D\) is bounded in \(L^2(\Omega )\) (It is in fact an isometry on components; this follows from (27)), therefore
The fact that \(R_D\) is bounded in \(L^4(\Omega )\) is also true [19]. Then
Moreover, it is known (see for instance [1]) that in \(d=2\) we have
and therefore
and
Now we use the Sobolev embedding
and deduce, using also a Poincaré inequality
Thus, after a Young inequality we deduce that
holds for T depending only on \(\Vert \Lambda _D^2\theta _0\Vert _{L^2(\Omega )}\), with a constant independent of n. The following result can now be obtained by assing to the limit in a subsequence and using a Aubin-Lions lemma [21]:
Proposition 4
Let \(\theta _0\in H_0^1(\Omega )\cap H^{2}(\Omega )\) in \(d=2\). There exists \(T>0\) a unique solution of (7) with initial datum \(\theta _0\) satisfying
Higher regularity can be obtained as well. Because the proof uses \(L^2\)- based Sobolev spaces and Sobolev embedding, it is dimension dependent. A proof in higher dimensions is also possible, but it requires using higher powers of \(\Delta \), and will not be pursued here.
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Constantin, P., Ignatova, M. Critical SQG in Bounded Domains. Ann. PDE 2, 8 (2016). https://doi.org/10.1007/s40818-016-0017-1
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DOI: https://doi.org/10.1007/s40818-016-0017-1