A Local Energy Identity for Parabolic Equations with Divergence-Free Drift

TL;DR

This paper establishes a local energy identity for certain solutions of parabolic equations with divergence-free drift, advancing understanding of their energy behavior in a distributional framework.

Contribution

It introduces a local energy identity for distributional solutions of parabolic equations with divergence-free drift, a novel theoretical result.

Findings

Proves a local energy identity for solutions in $L_{2, abla} imes W^{1,0}_2$.

Provides a new tool for analyzing parabolic equations with divergence-free drift.

Enhances theoretical understanding of energy distribution in such PDEs.

Abstract

We prove a local energy identity for a class of distributional solutions, in $L_{2,\infty} \cap W^{1,0}_2$, of parabolic equations with divergence-free drift.

Equations

$${\partial }_{t}u-\text{mbox}\mathrm{div}(a\nabla u)+b.\nabla u=0,$$

$${\partial }_{t}u-\text{mbox}\mathrm{div}(A\nabla u)=0,$$

$$\sup \left\{\frac{1}{|B(0,r)|}{\int }_{B(x_0,r)}|d-[d{]}_{x_0,r}|dx:B(x_0,r)\subset \subset \Omega \right\},$$

$$u_{\varphi }\in L_1({\mathbb{R}}^n),$$

$$\Vert u{,}_{j}v{,}_i-v{,}_{j}u{,}_i{\Vert }_{{\mathcal{H}}^1}\le C\Vert \nabla u{\Vert }_{L_p}\Vert \nabla v{\Vert }_{L_q},\forall i,j=1,\dots ,n.$$

$$\Vert f{\Vert }_{H^{-1}(\Omega )}^2=\sum _{k=1}^{\infty }{f}_k^2/{\lambda }_k,$$

$$L_{2,\infty }(Q)\cap W_2^{1,0}(Q),$$

$${\int }_{Q}u{\partial }_t\varphi dz={\int }_Q(A\nabla u).\nabla \varphi dz\forall \varphi \in C_0^{\infty }(Q),$$

$$\nu {\mathbb{I}}_n\le a\le {\nu }^{-1}{\mathbb{I}}_n$$

$$\begin{gathered} \frac{1}{2}{\int }_B\varphi (x,t_0)|u(x,t_0){|}^{2}dx+{\int }_{-1}^{t_0}{\int }_B\varphi \nabla u.a\nabla udz=\frac{1}{2}{\int }_{-1}^{t_0}{\int }_B|u{|}^2{\partial }_t\varphi dz \\ -{\int }_{-1}^{t_0}{\int }_B(A\nabla u).\nabla \varphi udz. \end{gathered}$$

$${\partial }_{t}u\in L_2(-1,0;H^{-1}(B))$$

$$g(x,t)=A(x,t)\nabla u(x,t),$$

$$\begin{cases}-\Delta U(\cdot,t)=\operatorname*{\mbox{div}}g(\cdot,t)\quad\mbox{for a.e $t\in(-1,0)$}\\ U(\cdot,t)|_{\partial B}=0.\end{cases}$$

$${\int }_B\text{mbox}\mathrm{div}g(\cdot ,t)vdx$$

$$=-{\int }_B(a\nabla u).\nabla vdx-{\int }_B(d\nabla u).\nabla vdx$$

$$=:A_1+A_2.$$

$$|A_{1}|\leq\|a\|_{L_{\infty}(Q)}\|\nabla u(\cdot,t)\|_{L_{2}(B)}\|\nabla v\|_{L_{2}(B)}\quad\mbox{for a.e $t\in(-1,0)$}.$$

$$-A_2=\frac{1}{2}\sum _{i,j=1}^n{\int }_B{d}_{ij}(u{,}_{j}v{,}_i-v{,}_{j}u{,}_i)dx.$$

$$\|\bar{u}(\cdot,t)\|_{W^{1}_{2}(\mathbb{R}^{n})}\leq c\|u(\cdot,t)\|_{W^{1}_{2}(B)}\quad\mbox{for a.e $t\in(-1,0)$},$$

$$\|\bar{d}(\cdot,t)\|_{BMO(\mathbb{R}^{n};\mathbb{R}^{n\times n})}\leq c\|d(\cdot,t)\|_{BMO(B;\mathbb{R}^{n\times n})}\quad\mbox{for a.e $t\in(-1,0)$},$$

$$-A_2=\frac{1}{2}\sum _{i,j=1}^n{\int }_{{\mathbb{R}}^n}{\bar{d}}_{ij}(\bar{u}{,}_{j}v{,}_i-v{,}_j\bar{u}{,}_i)dx.$$

$$\Vert \bar{u}{,}_{j}v{,}_i-v{,}_j\bar{u}{,}_i{\Vert }_{{\mathcal{H}}^1({\mathbb{R}}^n)}\le C\Vert \nabla \bar{u}{\Vert }_{L_2({\mathbb{R}}^n)}\Vert \nabla v{\Vert }_{L_2({\mathbb{R}}^n)},$$

$$|A_2|\le C\Vert \bar{d}{\Vert }_{L_{\infty }(-1,0;BMO({\mathbb{R}}^n;{\mathbb{R}}^{n\times n}))}\Vert \nabla \bar{u}{\Vert }_{L_2({\mathbb{R}}^n)}\Vert \nabla v{\Vert }_{L_2({\mathbb{R}}^n)},$$

$$|A_2|\le C\Vert d{\Vert }_{L_{\infty }(-1,0;BMO(B;{\mathbb{R}}^{n\times n}))}\Vert \nabla u{\Vert }_{L_2(B)}\Vert \nabla v{\Vert }_{L_2(B)},$$

$$\|\operatorname*{\mbox{div}}g(\cdot,t)\|_{H^{-1}(B)}\leq C(n,a,d)\|\nabla u(\cdot,t)\|_{L_{2}(B)}\quad\mbox{for a.e $t\in(-1,0)$}.$$

$$\|\nabla U(\cdot,t)\|_{L_{2}(B)}\leq C(n,a,d)\|\nabla u(\cdot,t)\|_{L_{2}(B)}\quad\mbox{for a.e $t\in(-1,0)$}.$$

$$\Vert \nabla U{\Vert }_{L_2(Q)}\le C(n,a,d)\Vert \nabla u{\Vert }_{L_2(Q)}.$$

$${\int }_{Q}u{\partial }_t\varphi dz={\int }_Q\nabla U.\nabla \varphi dz\forall \varphi \in C_0^{\infty }(Q).$$

$$u(x,t)=\sum _{k=1}^{\infty }{d}_k(t){\varphi }_k(x),$$

$$U(x,t)=\sum _{k=1}^{\infty }{b}_k(t){\varphi }_k(x),$$

$$\Vert \nabla U{\Vert }_{L_2(Q)}^2={\int }_{-1}^0\sum _{k=1}^{\infty }{b}_k^2(t){\lambda }_{k}dt\le C(n,a,d)\Vert \nabla u{\Vert }_{L_2(Q)}^2<\infty .$$

Full text

1 Introduction

We are considering the parabolic equations of the type

[TABLE]

where $a$ is a bounded, symmetric and uniformly elliptic matrix and $b$ a divergence-free vector field belonging to $L_{\infty}(BMO^{-1})$. We say that a divergence-free vector field $b$ belongs to the space $BMO^{-1}$ if there exits a skew symmetric matrix $d$ belonging to $BMO$ such that $b=\operatorname*{\mbox{div}}(d)$. Therefore, the above equation can be rewritten as follows:

[TABLE]

where $A=a+d$, with $a$ as before and $d\in L_{\infty}(BMO)$ a skew symmetric matrix.

G. Seregin and co-authors introduced, in their paper [1], the notion of suitable weak solutions to equation (1), which are distributional solutions that belong to the energy class $L_{2,\infty}\cap W^{1,0}_{2}$ and that satisfy a particular local energy inequality. In this paper, we establish a local energy identity for distributional solutions of (1) which belong to the energy class $L_{2,\infty}\cap W^{1,0}_{2}$, and therefore, we prove at the same time that the local energy inequality required in the definition of suitable weak solutions, introduced in [1], is a direct consequence of being a distributional solution in the above energy class.

2 Preliminaries

In what follows, we will use the following abbreviated notations: $B:=B(0,1)$ (the unit ball of $\mathbb{R}^{n}$), $Q:=B\times(-1,0)$, as well as $z:=(x,t)$.

We recall that a function $d$ is in the space $BMO(\Omega;\mathbb{R}^{n\times n})$ if the following quantity

[TABLE]

with $[d]_{x_{0},r}$ the average of $d$ over $B(x_{0},r)$, is bounded; and a function $u$ belongs to the Hardy space $\mathcal{H}^{1}(\mathbb{R}^{n})$ if there exists a function $\phi\in C^{\infty}_{0}(B)$ such that

[TABLE]

where $u_{\phi}(x):=\sup_{t>0}|(\phi_{t}\star u)(x)|$, and $\phi_{t}(x):=t^{-n}\phi(x/t)$.

For simplicity we adopt the following notation convention $\partial_{i}f={f,}_{i}$. We have the following classical div-curl type lemma for Hardy spaces, which is a direct consequence of Theorem II.1 in [5].

Lemma 1**.**

Let $u\in W^{1}_{p}(\mathbb{R}^{n})$ and $v\in W^{1}_{q}(\mathbb{R}^{n})$, with $1<p<\infty$ and $1/p+1/q=1$. Then $u,_{j}v,_{i}-v,_{j}u,_{i}\in\mathcal{H}^{1}(\mathbb{R}^{n})$ for all $i,j=1,\ldots,n$ and we have

[TABLE]

We recall also some basic facts related to the spectral decomposition of the Laplace operator on a bounded domain $\Omega$ of $R^{n}$, with smooth boundary. The Laplacian viewed as an unbounded operator from $L_{2}(\Omega)$ into itself has a discrete spectrum; we denote by $0<\lambda_{1}<\lambda_{2}<\ldots<\lambda_{n}<\ldots$ (with $\lambda_{n}\to\infty$), its eigenvalues and $\{\phi_{k}\}_{k=1}^{\infty}$ the corresponding eigenvectors which form a Hilbert basis of $L_{2}(\Omega)$. Setting $\mathring{L}^{1}_{2}(\Omega)$ to be the completion of $C^{\infty}_{0}(\Omega)$ with respect to the Dirichlet semi-norm $\|u\|_{L^{1}_{2}}^{2}:=\displaystyle\int|\nabla u|^{2}$, and $H^{-1}(\Omega)$ to be the dual of $\mathring{L}^{1}_{2}(\Omega)$, we have the following classical lemma, which gives us a Hilbert basis of $\mathring{L}^{1}_{2}(\Omega)$ and a representation of the norm of $H^{-1}$ by means of the eigenvectors and eigenvalues of the Laplace operator.

Lemma 2**.**

$(\phi_{k}/\sqrt{\lambda_{k}})_{k=1}^{\infty}$* is a Hilbert basis of $\mathring{L}^{1}_{2}(\Omega)$ and as a direct consequence, we have that, if $f\in H^{-1}(\Omega)$, then*

[TABLE]

where $f_{k}=\langle f,\phi_{k}\rangle$.

Proof.

The proof of this result is quiet classical, therefore we skip it. ∎

3 Main Theorem

We now state the main result of this paper.

Theorem 3.1**.**

Let $u$ belonging to the energy class

[TABLE]

such that

[TABLE]

where $A=a+d$, with $a\in L_{\infty}(Q;\mathbb{R}^{n\times n})$ a symmetric matrix satisfying

[TABLE]

and $d\in L_{\infty}(-1,0;BMO(B;\mathbb{R}^{n\times n}))$ a skew symmetric matrix. Then the following energy identity holds for all $t_{0}\in(-1,0)$ and for all test functions $\phi\in C^{\infty}_{0}(B\times(-1,1))$:

[TABLE]

4 Proof of Theorem 3.1

The method we use for this proof are due to Seregin, in his lecture notes:"Parabolic Equations".

We start by proving a simple regularity result for the time derivative of $u$ defined as in Theorem 3.1.

Proposition 4.1**.**

Let $u$ defined as in Theorem 3.1. Then

[TABLE]

Proof.

Step 1. Let us set

[TABLE]

and consider the problem

[TABLE]

Let $v\in C^{\infty}_{0}(B)$, we have:

[TABLE]

We have by a straightforward computation that

[TABLE]

On the other hand, we have thanks to the skew symmetry of $d$, that $A_{2}$ can be rewritten as follows

[TABLE]

Denote by $\bar{u}$ the extension of $u$ from $B$ to $\mathbb{R}^{n}$ such that

[TABLE]

where $c$ depends only on $n$. Similarly, let us denote by $\bar{d}$ the extension of $d$ from $B$ to $\mathbb{R}^{n}$ such that

[TABLE]

where, again, $c$ depends only on $n$. In the later case, to construct such an extension, one can use a reflection on the boundary (See, e.g., Theorem 2 in [2], where this is done for very general domains $\Omega\subset\mathbb{R}^{n}$). Therefore, because $v$ is compactly supported in $B$, we have that

[TABLE]

We have from Lemma 1 that $\bar{u},_{j}v,_{i}-v,_{j}\bar{u},_{i}\in\mathcal{H}^{1}(\mathbb{R}^{n})$ with

[TABLE]

and since $BMO(\mathbb{R}^{n})$ is the dual of the Hardy space $\mathcal{H}^{1}(\mathbb{R}^{n})$, we derive that

[TABLE]

and a fortiori

[TABLE]

(with $C$ depending only on $n$). Hence, we have that $\operatorname*{\mbox{div}}g(\cdot,t)\in H^{-1}(B)$, with

[TABLE]

Therefore, there exists a unique $U(\cdot,t)\in\mathring{L}^{1}_{2}(B)$ which solves (3) and such that

[TABLE]

We also deduce that $\nabla U\in L_{2}(Q)$ and

[TABLE]

Step 2. Now, we can rewrite (2) as follows

[TABLE]

By a density arguments, we can test (4) with functions $\phi(x,t)=\chi(t)\phi_{k}(x)$, where $\chi\in C^{\infty}_{0}(-1,0)$ and $\phi_{k}$ is an eigenfunction (introduced in the second part of the preliminaries section; here we choose $\Omega=B$). Since $(\phi_{k})_{k=1}^{\infty}$ is a Hilbert basis of $L_{2}(B)$, we can write $u$ as follows

[TABLE]

where $d_{k}(t)=\int_{B}u(\cdot,t)\phi_{k}dx$; we also have

[TABLE]

where $b_{k}(t)=\int_{B}U(\cdot,t)\phi_{k}dx$. So we have, thanks to Lemma 2, that

[TABLE]

We have now

[TABLE]

So, $d_{k}^{\prime}(t)=-b_{k}(t)\lambda_{k}$ and we derive that

[TABLE]

where the convergence of this sum occurs in the space of distributions; thus we have, for every $w\in C^{\infty}_{0}(B)$ and $\chi\in C^{\infty}_{0}(-1,0)$, that

[TABLE]

and the statement follows. ∎

Remark 1*.*

Let $\varphi\in C^{\infty}_{0}(B)$. Then, we readily deduce from the above proposition that

[TABLE]

Since obviously $u\varphi\in L_{2}(-1,0;\mathring{L}^{1}_{2}(B))$, we conclude that

[TABLE]

Proof of Theorem 3.1.

Step 1. Consider the following auxiliary equation

[TABLE]

where $F=u\partial_{t}\varphi-(A\nabla u).\nabla\varphi$ and $G=u(A\nabla\varphi)$. We have that the distribution $F-\operatorname*{\mbox{div}}G$ belongs to $L_{2}(-1,0;H^{-1}(B))$. This is a direct consequence of the fact that $u\varphi$ is distributional solution of (5) together with Remark 1 and Step 1 of the proof of Proposition 4.1. But, for our purpose we are interested, more precisely, in the bounds of the terms which appear in the definition of $F-\operatorname*{\mbox{div}}G$ and which belong to $L_{2}(-1,0;H^{-1}(B))$. Therefore let us consider a function $w\in C^{\infty}_{0}(B)$; then for a.e $t\in(-1,,0)$

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

We rewrite the term $J_{3}$ as follows

[TABLE]

But again, thanks to the skew symmetry of $d$, we have that

[TABLE]

and making the same computations as for $A_{2}$ in Step 1 of the proof of Proposition 4.1, with the only difference being that we keep $p$ arbitrary (instead of choosing $p=2$ as in the proof of Proposition 4.1), we obtain

[TABLE]

(for $1<p<\infty$ to be suitably chosen in function of $n$) which implies that

[TABLE]

If $\mathbf{n\geq 3}$, we steadily have for

[TABLE]

that (here $2^{*}=\frac{2n}{n-2}$)

[TABLE]

where Sobolev embedding and Poincaré’s inequality are used in the last estimate.

The case $\mathbf{n=2}$ is a straightforward adaptation of the previous (since $H^{1}(B)$ embeds continuously in every $L_{s}(B)$, $1\leq s<\infty$), whereas for the case $\mathbf{n=1}$, we take $p=2$, use the fact that $H^{1}(B)$ is continuously embedded in $L_{\infty}(B)$ and Poincaré’s inequality for the term in $w$.

Next, we have the following easy bound for the terms $J_{1}$ and $J_{2}$:

[TABLE]

So in conclusion, we have, for a.e $t\in(-1,0)$ that

[TABLE]

and we get a fortiori

[TABLE]

Step 2. Let us now tackle the question of well-posedness of (5). Consider the time-indexed family of bilinear forms

[TABLE]

Let us first notice that the map $t\in(-1,0)\mapsto\delta_{t}(w,v)$ is measurable for every $w,v\in\mathring{L}^{1}_{2}(B)$. Furthermore, we have by similar computations as those made in Step 1 in the proof of Proposition 4.1, that there exists a constant $C=C(n,a,d)>0$ independent of $t$ such that

[TABLE]

for all $w,v\in\mathring{L}^{1}_{2}(B)$ i.e $\delta_{t}$ is a bounded bilinear operator on $\mathring{L}^{1}_{2}(B)$. We have, additionally, the following coercivity estimate

[TABLE]

In view of these previous estimates and the regularity proved for the right-hand side of (5) and considering the evolution triple $\mathring{L}^{1}_{2}(B)\subset L_{2}(B)\subset H^{-1}$, we have by applying J-L. Lions abstract theorem for well-posedness of evolution equations (see e.g.,[4], Theorem 4.1, Chapter 3, section 4) that there exists a unique solution

[TABLE]

with

[TABLE]

such that

[TABLE]

for any $v\in C^{\infty}_{0}(Q)$. Let us notice that, from Remark 1, the fact that $u\varphi$ is a distributional solution of (5) and by the above uniqueness result:

[TABLE]

On another hand, by the regularity obtained for $w$, we can extend identity (7) to functions $v$ in $L_{2}(-1,0;\mathring{L}^{1}_{2}(B))$ and therefore, test (7) with $w$ itself. Thus, we get

[TABLE]

Denote by $L$, the left-hand side of the above identity. By the skew symmetry of $d$, we obtain that

[TABLE]

therefore coming back to $u$, we get

[TABLE]

Now, denote by $R$ the right hand side of (8); we easily obtain that

[TABLE]

Therefore, (8) implies that

[TABLE]

for all $\varphi\in C^{\infty}_{0}(Q)$. Let us notice that all the integrals in the above identity are finite, especially the last one of the right hand side of (9). To see this we rewrite

[TABLE]

and use the same method as in the estimation of $J_{3}$ in the previous step.

Step 3. Now, we choose $\varphi(x,t)=\chi_{\epsilon}(t)\phi(x,t)$, where $\phi\in C^{\infty}_{0}(B\times(-1,1))$ and $\chi_{\epsilon}(t)=1$ if $t\leq t_{0}-\varepsilon$, $\chi_{\epsilon}(t)=(t_{0}+\epsilon-t)/(2\epsilon)$, if $t_{0}-\epsilon<t<t_{0}+\epsilon$, and $\chi_{\epsilon}(t)=0$ when $t\geq t_{0}+\epsilon$, with $t_{0}\in(-1,0)$. Therefore passing to the limit $\epsilon\to 0$ in (9), we have that Theorem 3.1 is proved.

∎

Acknowledgement

This work was supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]. The author would like to thank Siran Li and Ghozlane Yahiaoui for their careful reading and insightful discussion.