Decay of periodic entropy solutions to degenerate nonlinear parabolic equations☆
Introduction
In the half-space , , we consider the nonlinear parabolic equation where the flux vector is merely continuous: , , and the diffusion matrix is Lebesgue measurable and bounded: , . We also assume that the matrix (nonnegative definite). This matrix may have nontrivial kernel. Hence (1.1) is a degenerate (hyperbolic-parabolic) equation. In particular case it reduces to a first order conservation law Equation (1.1) is endowed with the initial condition
Let be a function of bounded variation on any segment in . We will need the bounded linear operator , where C is the space of constants. This operator is defined up to an additive constant by the relation where is the left limit of g at the point u, and the integral in (1.4) is understood in accordance with the formula where , is the interval if , and , if . Observe that is continuous even in the case of discontinuous . For instance, if then . Notice also that for the operator is uniquely determined by the identity (in ). In the case the function can be defined by the identity . As is easy to see, in this case the correspondence is a linear continuous map from into .
We fix some representation of the diffusion matrix in the form , where is a matrix-valued function with measurable and bounded entries, . We recall the notion of entropy solution of the Cauchy problem (1.1), (1.3) introduced in [7].
Definition 1 A function is called an entropy solution (e.s. for short) of (1.1), (1.3) if the following conditions hold: (i) for each the distributions where vectors , and , ; (ii) for every , (iii) for any convex function (iv) in .
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 . First of all, observe that since then, up to an additive constant, For we consider the set consisting of bounded Borel functions such that and the chain rule (1.6) holds. By (ii) contains all bounded functions such that . Let , , be a sequence which converges pointwise to a function as . It is clear that is a Borel function and . Observe that the functions are bounded and measurable (because are Borel functions), the sequence pointwise as , and . This implies that From the other hand, it follows from (1.8) that converges to as uniformly on Π. Therefore, we can pass to the limit as in relation (1.6) with and derive that (1.6) holds for our limit function . Thus, . We see that is closed under pointwise convergence and by Lebesgue theorem contains all Borel functions g such that . Since is arbitrary, we find that (1.6) holds for all bounded Borel functions. It only remains to notice that the behavior of out of the segment , where , does not matter. Therefore, (1.6) holds for any Borel function bounded on , in particular, for each locally bounded Borel function. Remark also that for correctness of (1.6) we have to choose the Borel representative for (recall that is defined up to equality on a set of full measure and such the representative exists). Notice that 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 was used with . 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 , where is a nondegenerate linear map and . This can be written in the coordinate form as As is easy to verify, the function is an e.s. of (1.1), (1.3) with initial data if and only if is an e.s. of the problem where corresponding to the representation , , of the diffusion matrix . We remark that Recall that , .
We underline that the matrix is not necessarily symmetric and therefore differs from .
Now, we suppose that the initial function is periodic with a lattice of periods L: a.e. in for all . Denote the corresponding torus (which can be identified with a fundamental parallelepiped), dx is the Lebesgue measure on , normalized by the condition , V being volume of a fundamental parallelepiped (observe that V does not depend on its choice). Then there exists a space-periodic e.s. of the problem (1.1), (1.3), a.e. in Π for all . This can be written as . This solution can be constructed as a limit of the sequence of solutions to the regularized problem with smooth flux vectors , and smooth and strictly positive definite diffusion matrices , which approximate and , respectively. The corresponding initial functions are supposed to be smooth approximations of . To justify the convergence of approximations , we can utilize known a priory estimates [13], [24], like in papers [7], [8].
It is known [7, Theorem 1.2] that e.s. satisfies the maximum principle: for a.e. .
Our main result is the long time decay property of entropy solutions. Let be the dual lattice to the lattice of periods L, be the mean value of initial data.
Theorem 1.1 Assume that the following nonlinearity-diffusivity condition holds: for all , there is no vicinity of I, where simultaneously the function is affine and the function almost everywhere. Then
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 . This condition reads: for all , , In the hyperbolic case 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 , a nonzero vector , and a constant such that , on this interval. Then, as is easily verified, the function is an x-periodic (with the lattice of periods L) e.s. of (1.1), (1.3) with the periodic initial data , which has the mean value I. It is clear that does not converge as in , and the decay property fails.
Section snippets
Some properties of periodic entropy solutions
Lemma 2.1 For each convex there exists a locally finite nonnegative x-periodic Borel measure on Π such that This measure can be identified with a finite measure on . Moreover, for almost each (a.e.) , ,
Proof It follows from (1.7) that for every convex the distribution By the Schwartz theorem on the representation of nonnegative distributions we
Reduction of the problem
We will need the following algebraic statement.
Lemma 3.1 Let A be a lattice in , be a subgroup of A, be a linear span of . Assume that . Then any basis of can be completed to a basis of A.
Proof Let be a basis of , that is, any element is uniquely represented as with integer coefficients . It is clear that is a lattice and therefore is a basis of the vector space . We consider the natural projection . Then is an additive subgroup
Measure valued functions and H-measures
Recall (see [4], [22]) that a measure-valued function on a domain is a weakly measurable map of Ω into the space of probability Borel measures with compact support in .
The weak measurability of means that for each continuous function the function is measurable on Ω.
A measure-valued function is said to be bounded if there exists such that for almost all .
Measure-valued functions of the kind , where
The proof of the decay property
We assume that is an x-periodic e.s. of (1.1), (1.3). As was demonstrated in section 3, we may suppose that conditions (R1)–(R4) hold. In view of (1.10) for every the function is an e.s. of the equation where is the vector with components , , , , while the symmetric matrix has the entries , , where for , for .
If is an
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Cited by (5)
On decay of entropy solutions to degenerate nonlinear parabolic equations with perturbed periodic initial data
2022, Journal of Differential EquationsOn some properties of entropy solutions of degenerate non-linear anisotropic parabolic equations
2021, Journal of Differential EquationsCitation Excerpt :More generally we establish below the comparison principle formulated in Theorem 1.3. Theorem 4.1 was established in recent paper [19] and generalizes earlier results [6,17]. The following simple lemmas were proved in [20].
On Decay of Entropy Solutions to Nonlinear Degenerate Parabolic Equation with Almost Periodic Initial Data
2021, Lobachevskii Journal of Mathematics
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This work was supported by the “RUDN University Program 5-100”, the Ministry of Science and Higher Education of the Russian Federation (project no. 1.445.2016/1.4) and by the Russian Foundation for Basic Research (grant 18-01-00258-a).