Asymptotic Regularity and Attractors for Slightly Compressible Brinkman–Forchheimer Equations

Abstract

Slightly compressible Brinkman–Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified.

Appendix A: An Auxiliary Linear Problem

In this appendix, we study the following linear problem:

$$\begin{aligned} \partial _tp+{\text {div}}(Du)=0,\ \ -\Delta _xu+\nabla _xp=g(t),\ \ p\big |_{t=0}=p_0,\ \ u\big |_{\partial \Omega }=0. \end{aligned}$$

(A.1)

Note that, solving the second equation of (A.1) with respect to u, we get

$$\begin{aligned} u(t)=-(-\Delta _x)^{-1}\nabla _xp+(-\Delta _x)^{-1}g(t), \end{aligned}$$

(A.2)

where the Laplacian is endowed with the homogeneous Dirichlet boundary condition. Inserting this expression to the first equation, we arrive at

$$\begin{aligned} \partial _tp+\mathfrak Ap={\text {div}}(D(-\Delta _x)^{-1}g(t)), \end{aligned}$$

(A.3)

where

$$\begin{aligned} \mathfrak Ap:=-{\text {div}}(D(-\Delta _x)^{-1}\nabla _xp). \end{aligned}$$

(A.4)

Thus, the key question here are the properties of the operator ${\mathfrak {A}}$.

Proposition A.1

The operator ${\mathfrak {A}}\in \mathcal L({\bar{H}}^\delta (\Omega ),\bar{H}^\delta (\Omega ))$ if $\delta >-\frac{1}{2}$. Moreover, this operator is positive definite and self-adjoint in ${\bar{L}}^2(\Omega )$:

$$\begin{aligned} (\mathfrak Ap,p)\ge \alpha \Vert p\Vert ^2_{{\bar{L}}^2}, \ \ p\in \bar{L}^2(\Omega ) \end{aligned}$$

(A.5)

for some $\alpha >0$.

Proof

Indeed, the first statement is an immediate corollary of the classical elliptic regularity estimates for the Laplacian, see e.g., [37], so we only need to check the stated properties for $\delta =0$. The fact that ${\mathfrak {A}}$ is self-adjoint is also straightforward, so we need to verify positiveness. Namely,

$$\begin{aligned} ({\mathfrak {A}} p,p)= & {} -({\text {div}}(Du),p)=(Du,{\text {div}}p)\nonumber \\= & {} -(Du,\Delta _xu)= (D\nabla _xu,\nabla _xu)\ge \alpha _1\Vert \nabla _xu\Vert ^2_{L^2}, \end{aligned}$$

(A.6)

where $-\Delta _xu+\nabla _xp=0$ and $\alpha _1>0$ is the smallest eigenvalue of the matrix D. Using, e.g., the Bogovski operator it is easy to show that $\Vert p\Vert ^2_{{\bar{L}}^2}\le C\Vert \nabla _xu\Vert ^2_{L^2}$ for some positive constant C. Thus, the proposition is proved. $\square$

As an immediate corollary of this proposition, we get the following result.

Corollary A.2

Let $p_0\in {\bar{H}}^\delta (\Omega )$ and $g\in L^1(0,T;H^{\delta -1}(\Omega ))$, $\delta >-\frac{1}{2}$. Then, the solution p(t) of Eq. (A.3) belongs to ${\bar{H}}^\delta$ for all $t\ge 0$ and the following estimate holds:

$$\begin{aligned} \Vert p(t)\Vert _{{\bar{H}}^\delta }\le C_\delta \Vert p(0)\Vert _{\bar{H}^\delta }e^{K_\delta t}+ C_\delta \int _0^te^{K_\delta (t-\tau )}\Vert g(\tau )\Vert _{H^{\delta -1}}\,d\tau , \end{aligned}$$

(A.7)

where the constants $C_\delta$ and $K_\delta$ depend only on $\delta$. In particular, for $\delta =0$, the corresponding exponent $K_0=-\alpha <0$.

Remark A.3

The result of Corollary A.2 gives the dissipative estimate for $\delta =0$ only. For other values of $\delta$, the constant $K_\delta$ a priori may be positive, then the obtained estimate will be not dissipative. This is related with the fact that we do not know a priori that the spectrum of operator ${\mathfrak {A}}$ is the same in all Sobolev spaces ${\bar{H}}^\delta (\Omega )$, so if it depends on $\delta$, then it may happen that Eq. (A.3) may become unstable for some values of $\delta$. We expect that, in a fact, the spectrum of ${\mathfrak {A}}$ is independent of $\delta$, but failed to find the proper reference. So, in order to avoid the technicalities, we restrict ourselves to the most important for our purposes case $\delta =1$ and verify that the corresponding $K_1$ is also negative.

Proposition A.4

Let $p_0\in {\bar{H}}^1(\Omega )$ and $g\in L^1(0,T;L^2(\Omega ))$. Then, the solution p(t) of the truncated problem (A.3) satisfied the following estimate:

$$\begin{aligned} \Vert p(t)\Vert _{{\bar{H}}^1}\le C\Vert p(0)\Vert _{{\bar{H}}^1}e^{-\alpha t}+C\int _0^t\Vert g(s)\Vert _{L^2}\,ds, \end{aligned}$$

(A.8)

where the positive constants C and $\alpha$ are independent of t and p.

Proof

In the case of periodic boundary conditions, the desired estimate can be obtained just by multiplying Eq. (A.3) by $\Delta _xp$. However, this does not work in the case of Dirichlet boundary conditions because of the presence of extra boundary integrals arising after integration by parts. So, in this case we will use the localization technique instead. Note also that we only need to verify (A.8) for $g=0$. The general case will follow then form the Duhamehl formula.

Step 1. Interior estimates. Let us fix a non-negative cut-off function $\phi (x)\in C^1_0({\mathbb {R}})$ such that $\phi (x)=0$ if x is in the $\mu /2$-neighbourhood of the boundary $\partial \Omega$ and $\phi \equiv 1$ if $x\in \Omega$ and is outside the $\mu$-neighbourhood of $\Omega$. In addition, we require that

$$\begin{aligned} |\nabla _x\phi (x)|\le C\phi (x)^{1/2},\ \ x\in \Omega . \end{aligned}$$

It is not difficult to see that such a function exists for all $\mu >0$ small enough.

We write Eq. (A.3) as a system (A.1) with $g=0$ (in order to avoid the inverse Laplacian) and multiply the first equation by $-{\text {div}}(\phi \nabla _xp)$. Then, after integration by x, we get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}(\phi ,|\nabla _xp|^2)= & {} ({\text {div}}(Du),{\text {div}}(\phi \nabla _xp))\nonumber \\&\quad = \sum _{i=1}^3({\text {div}}(Du),\partial _{x_i}(\phi \partial _{x_i}p))= -\sum _{i=1}^3(Du,\partial _{x_i}(\phi \partial _{x_i}\nabla _xp))\nonumber \\&\quad - \sum _{i=1}^3(Du,\partial _{x_i}(\nabla _x\phi \partial _{x_i}p)= -\sum _{i=1}^3(\partial _{x_i}(D\partial _{x_i}u),\nabla _xp)\nonumber \\&\quad + ({\text {div}}(Du),\nabla _x\phi \cdot \nabla _xp)= -\sum _{i=1}^3(\partial _{x_i}(D\phi \partial _{x_i}u),\Delta _xu)\nonumber \\&\quad + ({\text {div}}(Du),\nabla _x\phi \cdot \nabla _xp)= -(\phi D\Delta _xu,\Delta _xu)-(D\Delta _xu\cdot \nabla _x\phi ,{\text {div}}(u))\nonumber \\&\quad +({\text {div}}(Du),\nabla _x\phi \cdot \nabla _xp)\le -\alpha _1(\phi ,|\Delta _xu|^2)+\frac{\alpha _1}{4}(\phi ,|\Delta _xu|^2)\nonumber \\&\quad +\frac{\alpha _1}{4}(\phi ,|\nabla _xp|^2)+ C\Vert \nabla _xu\Vert ^2_{L^2}\le -\frac{\alpha _1}{2}(\phi ,|\nabla _xp|^2)+C\Vert p\Vert ^2_{{\bar{L}}^2}. \end{aligned}$$

(A.9)

Since we have already known from Corollary A.2 that

$$\begin{aligned} \Vert p(t)\Vert _{{\bar{L}}^2}+\Vert \nabla _xu(t)\Vert ^2_{L^2}\le Ce^{-\alpha t}\Vert p(0)\Vert _{L^2}, \end{aligned}$$

(A.10)

then applying the Gronvall inequality to the obtained relation, we get the desired interior dissipative estimate:

$$\begin{aligned} (\phi ,|\nabla _xp(t)|^2)\le C\left( (\phi ,|\nabla _xp(0)|^2)+\Vert p(0)\Vert ^2_{\bar{L}^2}\right) e^{-\beta t}, \end{aligned}$$

(A.11)

where C and $\beta$ are some positive constants.

Step 2. Boundary estimates: tangential directions. Let us introduce in a small neighbourhood of the boundary three smooth orthonormal vector fields

$$\begin{aligned} \tau _3(x):=n=(n^1(x),n^2(x),n^3(x)), \ \ \tau _1(x):=(\tau _1^1(x),\tau _1^2(x),\tau _1^3(x)) \end{aligned}$$

and $\tau _2(x):=(\tau _2^1(x),\tau _2^2(x),\tau _2^3(x))$ such that n(x) coincides with the outer normal vector when $x\in \partial \Omega$ and $\tau _1(x),\tau _2(x)$ give the complement pair of tangential vectors. This triple of vector field may not exist globally near the boundary, but only locally, so being pedantic we need to use the partition of unity near the boundary to localize them, but we ignore this standard procedure in order to avoid technicalities (this localization can be done exactly in the way how we get interior estimates). After defining the triple of vector fields near the boundary, we use the proper scalar cut-off function in order to extend these fields to the whole domain ${\bar{\Omega }}$.

Let us define the corresponding differentiation operators along these vector fields:

$$\begin{aligned} \partial _{\tau _i}u:=\sum _{j=1}^3\tau ^j_i(x)\partial _{x_j}u \end{aligned}$$

In contrast to the differentiation with respect to coordinate directions, these operators do not commute in general, but their commutator is a lower order operator (again first order differential operator):

$$\begin{aligned}{}[\partial _{\tau _i},\partial _{\tau _j}]=\partial _{\{\tau _i,\tau _j\}}, \end{aligned}$$

where $\{\tau _i,\tau _j\}$ is a Lie bracket of vector fields $\tau _i$ and $\tau _j$. This commutation up to lower order terms is important for our method. One more crucial fact for us is that the condition $u\big |_{\partial \Omega }=0$ implies that $\partial _{\tau _i}u\big |_{\partial \Omega }=0$, $i=1,2$ so differentiation with respect to tangential derivatives preserve the Dirichlet boundary conditions.

We are now ready to get the desired estimates for tangential derivatives. To this end, we denote $q:=\partial _{\tau _i}p$ and $v=\partial _{\tau _i}u$. Then, differentiating Eqs. (A.1) in the direction $\tau _i$, we arrive at

$$\begin{aligned} \partial _tq+{\text {div}}(Dv)+M(x)\nabla _xu=0,\ \ -\Delta _xv+\nabla _xq=N(x)\nabla _xp+Ru, \end{aligned}$$

(A.12)

where the matrices M and N are smooth and R is a linear second order differential operator with smooth coefficients. Multiplying the first and second equations of (A.12) by q and Dv respectively, we arrive at

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert q\Vert ^2_{L^2}= & {} (Dv,\nabla _xq)-(M\nabla _xu,q)-(D\nabla _xv,\nabla _xv)\nonumber \\&-(\nabla _xq, Dv)+(N\nabla _xp,Dv)+(Ru,Dv)\nonumber \\\le & {} -\alpha _1\Vert \nabla _xv\Vert ^2_{L^2} +\varepsilon \Vert \nabla _xv\Vert ^2_{L^2}+\varepsilon \Vert q\Vert ^2_{L^2}+C_\varepsilon (\Vert \nabla _xu\Vert ^2_{L^2}+\Vert p\Vert ^2_{L^2}),\nonumber \\ \end{aligned}$$

(A.13)

where $\varepsilon >0$ can be arbitrarily small. Moreover, multiplying the second equation by $\mathfrak Bq$ after the standard estimates, we get

$$\begin{aligned} \Vert q\Vert _{L^2}^2\le C\Vert \nabla _xv\Vert _{L^2}+C\Vert \nabla _xu\Vert ^2_{L^2}+C\Vert p\Vert _{L^2}^2. \end{aligned}$$

(A.14)

Inserting this estimate in (A.13) and fixing $\varepsilon >0$ to be small enough, we finally arrive at

$$\begin{aligned} \frac{d}{dt}\Vert q\Vert _{L^2}^2+{\bar{\alpha }}\Vert q\Vert ^2_{L^2}\le C\Vert \nabla _xu\Vert ^2_{L^2}+\Vert p\Vert ^2_{L^2} \end{aligned}$$

(A.15)

for some ${\bar{\alpha }}>0$. Applying the Gronwall inequality to this relation and using (A.10), we have

$$\begin{aligned} \Vert \partial _{\tau _1}p(t)\Vert ^2_{L^2}+\Vert \partial _{\tau _2} p(t)\Vert _{L^2}^2\le C e^{-\alpha t}\Vert \nabla _xp(0)\Vert ^2_{L^2}, \end{aligned}$$

(A.16)

where $\alpha >0$ and C are independent of p and t. Thus, the desired estimates for tangential derivatives are obtained.

Step 3. Boundary estimates: normal direction. We now want to estimate the normal derivative $\partial _n p$ using Eq. (A.1) and the already obtained estimates for the tangential derivatives. To this end, we need some preparations. Let us write the vector u in the form

$$\begin{aligned} u=u_n n+u_{\tau _1}\tau _1+u_{\tau _2}\tau _2,\ \ u_n:=u.n,\ \ u_{\tau _i}=u.\tau _i. \end{aligned}$$

Then, multiplying the second equation of (A.1) by $\tau _i$, $i=1,2$ and using the fact that the $L^2$-norm of $\partial _{\tau _i}p$ as well as $H^1$-norm of u are already estimated, we get

$$\begin{aligned} \Vert u_{\tau _1}\Vert _{H^2}+\Vert u_{\tau _2}\Vert _{H^2}\le Ce^{-\alpha t}\Vert p(0)\Vert _{H^1}. \end{aligned}$$

(A.17)

Moreover, multiplying the second equation of (A.1) by n and using that the $H^1$-norms of tangential derivatives of u are already under the control, we arrive at

$$\begin{aligned} \Vert \partial ^2_nu_n(t)-\partial _n p(t)\Vert _{L^2}\le Ce^{-\alpha t}\Vert p(0)\Vert _{H^1}. \end{aligned}$$

(A.18)

We now return to the first equation of (A.1) (the equation for pressure). Taking the normal derivative from both sides of this equation and using (A.17) and the fact that the $H^1$-norm of $\partial _{\tau _j}u_{n}$, $j=1,2$ are also under the control, we arrive at

$$\begin{aligned} \partial _t\partial _n p+(Dn.n)\partial _n^2u_n=h(t),\ \ \Vert h(t)\Vert _{L^2}\le Ce^{-\alpha t}\Vert p(0)\Vert _{H^1}. \end{aligned}$$

Multiplying the obtained equation by $\partial _n p$, integrating over x and using (A.18) together with positivity of the matrix D, we finally get

$$\begin{aligned} \frac{d}{dt}\Vert \partial _n(t)\Vert ^2_{L^2}+\alpha _2\Vert \partial _n p(t)\Vert ^2_{L^2}\le Ce^{-2\alpha t}\Vert p(0)\Vert ^2_{H^1} \end{aligned}$$

and applying the Gronwall inequality, we get the desired estimate for the normal derivative:

$$\begin{aligned} \Vert \partial _n p(t)\Vert ^2_{L^2}\le Ce^{-\alpha t}\Vert p(0)\Vert _{H^1}^2 \end{aligned}$$

where $\alpha >0$ and C are independent of t and u.

Combining together the obtained interior, tangential and normal estimates, we derive that

$$\begin{aligned} \Vert p(t)\Vert _{{\bar{H}}^1}^2\le Ce^{-\alpha t}\Vert p(0)\Vert ^2_{{\bar{H}}^1} \end{aligned}$$

and finish the proof of the proposition. $\square$

Corollary A.5

Let the assumptions of Corollary A.2 hold and let $\delta \in [0,1]$. Then, the corresponding estimate (A.7) holds with $K_\delta \le -\alpha <0$.

Indeed, we have verified this property for $\delta =0$ and $\delta =1$. For fractional values $0<\delta <1$, the result follows by the interpolation.