Decay of periodic entropy solutions to degenerate nonlinear parabolic equations

Abstract

Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation.

Introduction

In the half-space $\mathrm{\Pi }={\mathbb{R}}_{+}\times {\mathbb{R}}^n$, ${\mathbb{R}}_{+}=(0,+\infty )$, we consider the nonlinear parabolic equation

$$u_t+{\mathrm{div}}_x(\phi (u)-a(u){\mathrm{\nabla }}_{x}u)=0,$$

where the flux vector $\phi (u)=({\phi }_1(u),\dots ,{\phi }_n(u))$ is merely continuous: ${\phi }_i(u)\in C(\mathbb{R})$, $i=1,\dots ,n$, and the diffusion matrix $a(u)={(a_{ij}(u))}_{i,j=1}^n$ is Lebesgue measurable and bounded: $a_{ij}(u)\in L^{\infty }(\mathbb{R})$, $i,j=1,\dots ,n$. We also assume that the matrix $a(u)\ge 0$ (nonnegative definite). This matrix may have nontrivial kernel. Hence (1.1) is a degenerate (hyperbolic-parabolic) equation. In particular case $a\equiv 0$ it reduces to a first order conservation law

$$u_t+{\mathrm{div}}_x\phi (u)=0.$$

Equation (1.1) is endowed with the initial condition

$$u(0,x)=u_0(x).$$

Let $g(u)\in B{V}_{loc}(\mathbb{R})$ be a function of bounded variation on any segment in $\mathbb{R}$. We will need the bounded linear operator $T_g:C(\mathbb{R})/C\to C(\mathbb{R})/C$, where C is the space of constants. This operator is defined up to an additive constant by the relation

$$T_g(f)(u)=g(u-)f(u)-\underset{0}{\overset{u}{\int }}f(s)dg(s),$$

where $g(u-)=\underset{v\to u-}{\lim }g(v)$ is the left limit of g at the point u, and the integral in (1.4) is understood in accordance with the formula

$$\underset{0}{\overset{u}{\int }}f(s)dg(s)=\mathrm{sign}u\underset{J(u)}{\int }f(s)dg(s),$$

where $\mathrm{sign}u=1$, $J(u)$ is the interval $[0,u)$ if $u>0$, and $\mathrm{sign}u=-1$, $J(u)=[u,0)$ if $u\le 0$. Observe that $T_g(f)(u)$ is continuous even in the case of discontinuous $g(u)$. For instance, if $g(u)=\mathrm{sign}(u-k)$ then $T_g(f)(u)=\mathrm{sign}(u-k)(f(u)-f(k))$. Notice also that for $f\in C^1(\mathbb{R})$ the operator $T_g$ is uniquely determined by the identity $T_g{(f)}^{\prime }(u)=g(u)f^{\prime }(u)$ (in ${\mathcal{D}}^{\prime }(\mathbb{R})$). In the case $f^{\prime }(u),g(u)\in L_{loc}^2(\mathbb{R})$ the function $T_g(f)\in C(\mathbb{R})/C$ can be defined by the identity $T_g{(f)}^{\prime }(u)=g(u)f^{\prime }(u)\in L_{loc}^1(\mathbb{R})$. As is easy to see, in this case the correspondence $g\to T_g(f)$ is a linear continuous map from $L_{loc}^2(\mathbb{R})$ into $C(\mathbb{R})/C$.

We fix some representation of the diffusion matrix $a(u)$ in the form $a(u)=b^{\top }(u)b(u)$, where $b(u)={(b_{ij}(u))}_{i,j=1}^n$ is a matrix-valued function with measurable and bounded entries, $b_{ij}(u)\in L^{\infty }(\mathbb{R})$. We recall the notion of entropy solution of the Cauchy problem (1.1), (1.3) introduced in [7].

Definition 1

A function $u=u(t,x)\in L^{\infty }(\mathrm{\Pi })$ is called an entropy solution (e.s. for short) of (1.1), (1.3) if the following conditions hold:

(i) for each $r=1,\dots ,n$ the distributions

$${\mathrm{div}}_x{B}_r(u(t,x))\in L_{loc}^2(\mathrm{\Pi }),$$

where vectors $B_r(u)=(B_{r1}(u),\dots ,B_{rn}(u))\in C(\mathbb{R},{\mathbb{R}}^n)$, and $B_{ri}^{\prime }(u)=b_{ri}(u)$, $r,i=1,\dots ,n$;

(ii) for every $g(u)\in C^1(\mathbb{R})$, $r=1,\dots ,n$

$${\mathrm{div}}_x{T}_g(B_r)(u(t,x))=g(u(t,x)){\mathrm{div}}_x{B}_r(u(t,x))\text{ in }{\mathcal{D}}^{\prime }(\mathrm{\Pi });$$

(iii) for any convex function $\eta (u)\in C^2(\mathbb{R})$

$$\eta {(u)}_t+{\mathrm{div}}_x(T_{{\eta }^{\prime }}(\phi )(u))-D^2\cdot T_{{\eta }^{\prime }}(A(u))+{\eta }^{″}(u)\sum _{r=1}^n{({\mathrm{div}}_x{B}_r(u))}^2\le 0\text{ in }{\mathcal{D}}^{\prime }(\mathrm{\Pi });$$

(iv) $\underset{t\to 0}{\mathrm{ess}\lim }u(t,\cdot )=u_0$ in $L_{loc}^1({\mathbb{R}}^n)$.

In (1.7) the matrix $A(u)$ is a primitive of the matrix $a(u)$, $A^{\prime }(u)=a(u)$, the operator

$$D^2\cdot T_{{\eta }^{\prime }}(A(u))\doteq \sum _{i,j=1}^n\frac{{\partial }^2}{\partial x_i\partial x_j}{T}_{{\eta }^{\prime }}(A_{ij}(u)).$$

Relation (1.7) means that for any non-negative test function $f=f(t,x)\in C_0^{\infty }(\mathrm{\Pi })$

$$\underset{\mathrm{\Pi }}{\int }[\eta (u)f_t+T_{{\eta }^{\prime }}(\phi )(u)\cdot {\mathrm{\nabla }}_{x}f+T_{{\eta }^{\prime }}(A(u))\cdot D_x^{2}f-f{\eta }^{″}(u)\sum _{r=1}^n{({\mathrm{div}}_x{B}_r(u))}^2]dtdx\ge 0,$$

where $D_x^{2}f$ is the symmetric matrix of second order derivatives of f, and “⋅” denotes the standard scalar multiplications of vectors or matrices.

In the case of conservation laws (1.2) Definition 1 reduces to the known definition of entropy solutions in the sense of S.N. Kruzhkov [12].

Remark 1.1

The chain rule postulated in (ii) actually holds for arbitrary locally bounded Borel function $g(u)\in L_{loc}^{\infty }(\mathbb{R})$. First of all, observe that since $B_r^{\prime }(u)\in L^{\infty }(\mathbb{R},{\mathbb{R}}^n)$ then, up to an additive constant,

$$T_g(B_r)(u)=\underset{0}{\overset{u}{\int }}g(v)B_r^{\prime }(v)dv.$$

For $R>0$ we consider the set $F_R$ consisting of bounded Borel functions $g(u)$ such that $\sup |g(u)|\le R$ and the chain rule (1.6) holds. By (ii) $F_R$ contains all bounded functions $g(u)\in C^1(\mathbb{R})$ such that $\sup |g(u)|\le R$. Let $g_l(u)\in F_R$, $l\in \mathbb{N}$, be a sequence which converges pointwise to a function $g(u)$ as $l\to \infty$. It is clear that $g(u)$ is a Borel function and $\sup |g(u)|\le R$. Observe that the functions $g_l(u(t,x))$ are bounded and measurable (because $g_l$ are Borel functions), the sequence $g_l(u(t,x))\to g(u(t,x))$ pointwise as $l\to \infty$, and ${\Vert g_l(u(t,x))\Vert }_{\infty }\le R$. This implies that

$$g_l(u(t,x)){\mathrm{div}}_x{B}_r(u(t,x))\underset{l\to \infty }{\to }g(u(t,x)){\mathrm{div}}_x{B}_r(u(t,x))\text{ in }{L}_{loc}^2(\mathrm{\Pi }).$$

From the other hand, it follows from (1.8) that $T_{g_l}(B_r)(u(t,x))$ converges to $T_g(B_r)(u(t,x))$ as $l\to \infty$ uniformly on Π. Therefore, we can pass to the limit as $l\to \infty$ in relation (1.6) with $g=g_l$ and derive that (1.6) holds for our limit function $g(u)$. Thus, $g(u)\in F_R$. We see that $F_R$ is closed under pointwise convergence and by Lebesgue theorem $F_R$ contains all Borel functions g such that $\sup |g|\le R$. Since $R>0$ is arbitrary, we find that (1.6) holds for all bounded Borel functions. It only remains to notice that the behavior of $g(u)$ out of the segment $[-M,M]$, where $M={\Vert u\Vert }_{\infty }$, does not matter. Therefore, (1.6) holds for any Borel function bounded on $[-M,M]$, in particular, for each locally bounded Borel function.

Remark also that for correctness of (1.6) we have to choose the Borel representative for $g(u)$ (recall that $g(u)$ is defined up to equality on a set of full measure and such the representative exists). Notice that $T_g(B_r)(u)$ does not depend on the choice of a Borel representative of g. Hence, the right-hand side of (1.6) does not depend on this choice either.

To be precise, in [7] the representation $a=b^{\top }b$ was used with $b=a{(u)}^{1/2}$. In order to make the definition invariant under linear changes of the independent variables, we have to extend the class of admissible representations. For instance, let us introduce the change $y=y(t,x)=qx-tc$, where $q:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a nondegenerate linear map and $c=(c_1,\dots ,c_n)\in {\mathbb{R}}^n$. This can be written in the coordinate form as

$$y_i=\sum _{j=1}^n{q}_{ij}{x}_j-c_{i}t,i=1,\dots ,n.$$

As is easy to verify, the function $u=v(t,y(t,x))$ is an e.s. of (1.1), (1.3) with initial data $u_0=v_0(y(0,x))$ if and only if $v(t,y)$ is an e.s. of the problem

$$v_t+{\mathrm{div}}_y(\tilde{\phi }(v)-\tilde{a}(v){\mathrm{\nabla }}_{y}v)=0,v(0,y)=v_0(y),$$

where

$$\tilde{\phi }(v)=q\phi (v)-vc,\tilde{a}(v)=qa(v)q^{\top },$$

corresponding to the representation $\tilde{a}(v)={(b(v)q^{\top })}^{\top }(b(v)q^{\top })$, $b(v)q^{\top }\in L^{\infty }(\mathbb{R},{\mathbb{R}}^{n\times n})$, of the diffusion matrix $\tilde{a}(v)$. We remark that

$${\mathrm{div}}_y{(B(v)q^{\top })}_r{|}_{y=y(t,x)}=\sum _{l,j=1}^n{\partial }_{y_l}{B}_{rj}(v)q_{lj}=\sum _{j=1}^n{\partial }_{x_j}{B}_{rj}(u)={\mathrm{div}}_x{B}_r(u).$$

Recall that $B_r^{\prime }(u)=b_r(u)=(b_{r1}(u),\dots ,b_{rn}(u))$, $r=1,\dots ,n$.

We underline that the matrix $b{q}^{\top }$ is not necessarily symmetric and therefore differs from $\tilde{a}{(v)}^{1/2}$.

Now, we suppose that the initial function $u_0$ is periodic with a lattice of periods L: $u_0(x+e)=u_0(x)$ a.e. in ${\mathbb{R}}^n$ for all $e\in L$. Denote ${\mathbb{T}}^n={\mathbb{R}}^n/L$ the corresponding torus (which can be identified with a fundamental parallelepiped), dx is the Lebesgue measure on ${\mathbb{T}}^n$, normalized by the condition $dx({\mathbb{T}}^n)=V$, V being volume of a fundamental parallelepiped (observe that V does not depend on its choice). Then there exists a space-periodic e.s. $u=u(t,x)$ of the problem (1.1), (1.3), $u(t,x+e)=u(t,x)$ a.e. in Π for all $e\in L$. This can be written as $u\in L^{\infty }((0,+\infty )\times {\mathbb{T}}^n)$. This solution can be constructed as a limit of the sequence $u_k$ of solutions to the regularized problem

$$u_t+{\mathrm{div}}_x({\phi }_k(u)-a_k(u){\mathrm{\nabla }}_{x}u)=0,$$

with smooth flux vectors ${\phi }_k(u)$, and smooth and strictly positive definite diffusion matrices $a_k(u)$, which approximate $\phi (u)$ and $a(u)$, respectively. The corresponding initial functions $u_k(0,x)$ are supposed to be smooth approximations of $u_0(x)$. To justify the convergence of approximations $u_k$, we can utilize known a priory estimates [13], [24], like in papers [7], [8].

It is known [7, Theorem 1.2] that e.s. $u(t,x)$ satisfies the maximum principle: $|u(t,x)|\le {\Vert u_0\Vert }_{\infty }$ for a.e. $(t,x)\in \mathrm{\Pi }$.

Our main result is the long time decay property of entropy solutions. Let

$$L^{\prime }=\{\xi \in {\mathbb{R}}^n|\xi \cdot e\in \mathbb{Z}\forall e\in L\}$$

be the dual lattice to the lattice of periods L,

$$I=\frac{1}{V}\underset{{\mathbb{T}}^n}{\int }{u}_0(x)dx$$

be the mean value of initial data.

Theorem 1.1

Assume that the following nonlinearity-diffusivity condition holds: for all $\xi \in L^{\prime }$, $\xi \ne 0$ there is no vicinity of I, where simultaneously the function $\xi \cdot \phi (u)$ is affine and the function $a(u)\xi \cdot \xi =0$ almost everywhere. Then

$$\underset{t\to +\infty }{\mathrm{ess}\lim }u(t,x)=I\mathit{\text{in}}{L}^1({\mathbb{T}}^n).$$

The decay property was established in [9] under the more restrictive version of the nonlinearity-diffusivity condition, in the case of locally Lipschitz flux vector when ${\phi }^{\prime }(u)\in L_{loc}^{\infty }(\mathbb{R},{\mathbb{R}}^n)$. This condition reads: for all $\tau \in \mathbb{R}$, $\xi \in L^{\prime }$, $\xi \ne 0$

$$\mathrm{meas}\{u\in \mathbb{R},|u|\le {\Vert u_0\Vert }_{\infty }|\tau u+\xi \cdot {\phi }^{\prime }(u)=a(u)\xi \cdot \xi =0\}=0.$$

In the hyperbolic case $a\equiv 0$ decay property (1.11) was proved in [20], see also the previous papers [6], [10], [19].

Let us show that our nonlinearity-diffusivity condition is exact. In fact, if it fails, we can find an interval $(I-\delta ,I+\delta )$, a nonzero vector $\xi \in L^{\prime }$, and a constant $c\in \mathbb{R}$ such that $\xi \cdot \phi (u)-cu\equiv \mathrm{const}$, $a(u)\xi \cdot \xi \equiv 0$ on this interval. Then, as is easily verified, the function $u(t,x)=I+\delta \sin (2\pi (\xi \cdot x-ct))$ is an x-periodic (with the lattice of periods L) e.s. of (1.1), (1.3) with the periodic initial data $u_0(x)=I+\delta \sin (2\pi (\xi \cdot x))$, which has the mean value I. It is clear that $u(t,x)$ does not converge as $t\to +\infty$ in $L^1({\mathbb{T}}^n)$, and the decay property fails.