The Cauchy Problem for a Strongly Degenerate Quasilinear Equation

Abstract

Abstract. We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation u t = div a(u, Du), where a(z, ξ ) = ∇ ξ f (z, ξ ), and f is a convex function of ξ with linear growth as ξ → ∞, satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.

Flux limited generalized porous media diffusion equations

“…Recently, following the strategy suggested by Brenier [25], McCann and Puel [43] have constructed solutions of the Neumann problem associated with equation (1.2) for bounded initial data assuming that they are also bounded from below. For that, they considered (1.2) as the gradient flow of the Boltzmann entropy for the Wasserstein metric corresponding to the cost function Our main purpose here is to prove existence and uniqueness results for (1.1) to cover the case where u 0 ∈ L 1 (R N ) ∩ L ∞ (R N ), u 0 ≥ 0, thus, extending the results in [5], [6]. We consider here that Φ : [0, ∞) → [0, ∞) is a strictly increasing function such that Φ(0) = 0, Φ, Φ −1 ∈ W 1,∞ ([a, b]) for any 0 < a < b.…”

“…Let us briefly review them. Using the CrandallLiggett's iterative scheme [35] and the notion of entropy solutions, a general existence and uniqueness theory for (1.1) when Φ(r) = r has been developed in [5], [6] when the initial condition 0 ≤ u 0 ∈ L 1 (R N ) ∩ L ∞ (R N ) and later extended to initial conditions u 0 ∈ BV (R N ) for (1.2) in [29]. The case of the Neumann problem in a bounded domain was previously considered in [3], [4] when the initial condition u 0 ≥ 0 was bounded and bounded away from 0 or the Lagrangian was coercive.…”

“…The notion of entropy solution permits to prove a uniqueness result based on Kružkov's technique of doubling variables [42] (see [28] for an extension to the second order case) suitably adapted to work with functions whose truncatures are of bounded variation [5], [6]. This last condition is necessary since, in general we are only able to prove that min(max(u, a), b) − a ∈ BV (R N ), 0 < a < b.…”

“…As a technical tool to prove these results we need some lower semi-continuity results for energy functionals whose density is a function g(x, u, Du) convex in Du with a linear growth rate as |Du| → ∞ which were proved in [36] and [39]. Since the proofs of these results follow closely the techniques given in [6], [29] we shall not give them in detail in the parabolic case.…”

“…In the parabolic case, the proofs will not be given in detail and can be reconstructed using the methods in [6], [29] together with the necessary adaptations that are used in Subsection 3.4 for the stationary case. Finally, in Section 6 we apply our previous results to the class of equations (1.14) under suitable assumptions on Λ, Φ.…”