Large-Time Behavior of Periodic Entropy Solutions to Anisotropic Degenerate Parabolic-Hyperbolic Equations

We are interested in the large-time behavior of periodic entropy solutions in $L^\infty$ to anisotropic degenerate parabolic-hyperbolic equations of second-order. Unlike the pure hyperbolic case, the nonlinear equation is no longer self-similar invariant and the diffusion term in the equation significantly affects the large-time behavior of solutions; thus the approach developed earlier based on the self-similar scaling does not directly apply. In this paper, we develop another approach for establishing the decay of periodic solutions for anisotropic degenerate parabolic-hyperbolic equations. The proof is based on the kinetic formulation of entropy solutions. It involves time translations and a monotonicity-in-time property of entropy solutions, and employs the advantages of the precise kinetic equation for the solutions in order to recognize the role of nonlinearity-diffusivity of the equation.


Introduction
We study the large-time behavior of periodic solutions in L ∞ to nonlinear anisotropic degenerate parabolic-hyperbolic equations of second-order. Consider the Cauchy problem for the second-order equations: where P = (P 1 , ..., P d ) is the period with P i > 0, a(·) := f ′ (·) ∈ L ∞ loc (R; R d ), and the d × d matrix A(u) = (a ij (u)) is symmetric, nonnegative, and locally bounded so that it can be always written under the form with (σ ik (u)) the square root matrix of A(u).
Equation (1.1) and its variants model degenerate diffusion-convection motions of ideal fluids and arise in a wide variety of important applications (cf. [1,2,4,12,20] and the references cited therein), for which a deep understanding of solutions to (1.1) is in great demand, at both the theoretical and numerical level.
In [7], a well-posedness theory has been established for L 1 solutions of the Cauchy problem (1.1) and (1.2) of anisotropic degenerate parabolichyperbolic equations of second-order. It extends the isotropic theory for degenerate parabolic-hyperbolic equations and several latter studies, see for instance [3,5,16,18] and the references therein. A notion of kinetic solutions, a new concept in this context, and a corresponding kinetic formulation have been extended. In particular, it has been also proved that, when u 0 ∈ L ∞ , the kinetic solution is equivalent to the entropy solution, or to the dissipative solution [23], which is unique; this provides a new path to study the behavior of entropy solutions in L ∞ through the corresponding kinetic equations for anisotropic degenerate parabolic-hyperbolic equations. In this paper, we employ the advantages of this path to develop a new approach for establishing the decay of periodic entropy solutions in L ∞ to (1.1) and (1.2) as t → ∞. The main theorem of this paper is the following.
be the unique periodic entropy solution to (1.1) and (1.2). Assume that the flux function and the diffusion matrix A(u) satisfy the nonlinearity-diffusivity condition: For any δ > 0, Then we have The nonlinearity-diffusivity condition (1.5) for equation (1.1) is developed from [13,17] and is reminiscent from the theory of velocity averaging lemmas [13,21,22,25]. For smooth coefficients, condition (1.5) is equivalent to the simpler and more standard setting: For any (τ, κ) ∈ R d+1 with τ 2 + |κ| 2 = 1, we have Several explicit examples are given in [17,25]. It implies that there is no interval of ξ on which both the flux function f (ξ) is affine and the diffusion matrix A(ξ) is degenerate, and thus also makes the relation with the applications of the theory of compensated compactness [19,26] to one-dimensional hyperbolic conservation laws. Such a nonlinearity-diffusivity condition is necessary for the decay of periodic solutions and the compactness of solution operators. See also the books [8,24]. Theorem 1.1 also extends naturally the non-degeneracy condition for the purely hyperbolic case in [11] where the first long-time convergence result of periodic solutions was obtained in one or two dimensions for BV initial data and higher order local non-degeneracy conditions that replace (1.5).
Unlike the pure hyperbolic case, equation (1.1) is no longer self-similar invariant and the diffusion term in the equation significantly affects the largetime behavior of solutions; thus the approach in [6] based on the self-similar scaling for the pure hyperbolic case does not apply and we have to change the strategy of proof. The approach developed in this paper is based on the kinetic formulation of entropy solutions in [7], involves time translations and a monotonicity-in-time of entropy solutions, and employs the advantages of the kinetic equations of the solutions, in order to recognize the role of the nonlinearity-diffusivity of the equation.
The rest of this paper is organized as follows. We first recall the notion of entropy solutions and their precise kinetic formulation, and then analyze some basic properties of entropy solutions. Finally, we develop a new approach to give a rigorous proof for the long-time asymptotic result.

Entropy Solutions and Kinetic Formulation
In this section, we first recall the notion of entropy solutions and their precise kinetic formulation, which requires some care to define appropriately the various terms of the equation. Then we analyze some basic properties of entropy solutions, which will be used in the proof of Theorem 1.1.
(ii) For any function ψ ∈ C 0 (R) with ψ(u) ≥ 0 and any k = 1, · · · , d, the chain rule holds: (iii) For any smooth function S(u), there exists an entropy dissipation mea- for n(t, x, ξ) the parabolic defect measure of u(t, x) defined as: We point out that the chain rule in (ii) has to be assumed in the anisotropic case, and this makes the main difference with the isotropic case in [3] where this property follows from an argument reminiscent to the theory of Sobolev spaces. The requirement (iii) has been made with notations that are adapted to the kinetic formulation we introduce now. The L 2 -condition in (i) is required to define the parabolic defect measure n(t, x, ξ) in (2.4) (also see [5]).
To do so, we may factor out an S ′ (u) in equation (2.3) and obtain a more handful kinetic formulation of nonlinear degenerate parabolic-hyperbolic equations of second-order with form (1.1). The new ingredient of this formulation is the identification of the kinetic defect measure m(t, x, ξ) and the degenerate parabolic defect measure n(t, x, ξ) in a precise manner, even in the region where u(t, x) is discontinuous. Compare with the classical kinetic formulation for scalar hyperbolic conservation laws in [17] (see also [21]).
We introduce the kinetic function χ on R 2 : We notice that, if u is in L ∞ [0, ∞) × R d and periodic in x with period P , then χ(ξ; u) is in L ∞ [0, ∞) × R d+1 and periodic in x with period P .

ANISOTROPIC DEGENERATE PARABOLIC-HYPERBOLIC EQUATIONS 5
The simple representation formula S(u) = R S ′ (ξ) χ(ξ; u) dξ leads to the following kinetic equation, which is equivalent to the entropy identity (2.3): where n(t, x, ξ) is still defined through (2.4). In [7], it has been proved that the entropy solutions in L ∞ are equivalent to the kinetic solutions determined by the kinetic formulation (2.5)-(2.7). Furthermore, we have for any constant v. The results (i)-(ii) in Theorem 2.1 are direct corollaries of the wellposedness results and the arguments in [7] (i.e. standard entropy inequalities for |u − v| and u 2 ) as in the hyperbolic case [8,24]; while (iii) is a direct corollary of the kinetic averaging compactness result of [17]; also see more recent results in [15,22,25].

Decay of Periodic Entropy Solutions: Proof of Theorem 1.1
In this section, we develop a new approach to give a rigorous proof for the decay property of periodic solutions, which takes the advantage of the precise kinetic formulation (2.5)-(2.7). Without loss of generality, we first set T P u(t, x)dx = 0; otherwise, we may replace u(t, x) by u(t, x) −ū, f (u) by f (u +ū), and A(u) by A(u +ū) in (1.1), so that all the arguments below remain unchanged. Then, we divide the proof into four steps.
1. Limit. Theorem 2.1 indicates that the periodic solution u(t, x) belongs to L ∞ , bounded by u 0 L ∞ , and is compact as the solution operator. Also the function is a non-increasing, bounded function, which implies that the following limit exists: Then we find that, for t ≥ −k, Correspondingly, we have Furthermore, multiplying both sides of (3.2) by ξ and then integrating over (t, x, ξ) ∈ (−T, T ) × T P × R for any T > 0, we obtain (3.3) This implies that the nonnegative measure sequence (m k + n k )(t, x, ξ) is uniformly bounded in k over (−T, T ) × T P × R, and hence there exists a subsequence k j and a measure M (t, x, ξ) such that On the other hand, since I(t) converges, we also have Then we conclude from (3.3) that M ((−∞, ∞)× T P × R) = 0, which implies In particular, multiplying (3.6) by ξ and then integrating dxdξ, we have where I ∞ = I(∞) is a constant, independent of t, determined in (3.1). The rest of the proof consists in showing that such a function χ is very particular and is in fact constant (see also this type of "rigidity" results in [14,10,9]).
This leads toĝ Integrating in ξ and using the Cauchy-Schwarz inequality, we find Notice that the frequencies κ are discrete and may include κ = 0. In particular, when κ = 0, then there exists δ 0 > 0 such that |κ| ≥ δ 0 . On the other hand, since v(t, x) has mean zero in x over T P , we have φv(τ, 0) = 0.
On the contrary, if I ∞ > 0, then we can choose λ small enough so that Cω δ (λ)/I 1/2 ∞ ≤ 1 2 and find from (3.9) that It remains to choose a sequence of functions φ B (t) = 1 for |t| ≤ B, with B a given large number and φ ′ B (t) = 2B−|t| B for B ≤ |t| ≤ 2B, and φ B (t) = 0 for |t| ≥ 2B. In the above inequality, we find where C > 0 is a constant independent of B and λ. When B tends to ∞, this implies that I ∞ must vanish, which is a contradiction. Therefore, (3.10) holds.
Therefore, for any T > k j + 1 > k j + s for s ∈ (0, 1), we employ (2.8) for the monotonicity-in-time of solution to obtain We conclude that This completes the proof.