Uniqueness for nonlinear degenerate problems

Abstract

We prove existence and uniqueness results for weak solutions of nonlinear degenerate problems arising in various physical models. The main novelty in the article concerns the uniqueness, which employs a technique based in showing that weak solutions are also entropy solutions, for which uniqueness follows from a straightforward adaptation of known results. We treat equations with lower order terms that have a particular structure and show with a counterexample that for general lower order terms the uniqueness does not hold.2000 Mathematics Subject Classification: 35D99, 35K65, 35R35.

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

“…Since then, many authors extended the Carrillo results in various directions (see e.g. [30,59,66,73,25,46,60,62,67,49,5,13]). Some additional techniques are required for anisotropic diffusion problems, where a kinetic approach (see Chen, Perthame [35]) and an accurate entropic approach (see Bendahmane, Karlsen [17,18]) were developed in the few last years; see also Souganidis, Perthame [75] and and Chen, Karlsen [34].…”

On uniqueness techniques for degenerate convection-diffusion problems

“…This leads to inequalities (16) with k = min{ϕ −1 (κ)} and with k = max{ϕ −1 (κ)}; then a "passage inside the flat regions" is needed in order to recover (16) with k in the interior of the interval ϕ −1 (κ). This technique of [26] was further developed in [38]; a non-restrictive in practice technical assumption on ϕ was required.…”

“…The fourth term is treated in the same way as the third one; here the y-dependent term G(ŵ) + a 0 (∇ŵ) plays the role of S(w) in the calculation (39), and the integration by parts is in x. Finally, the last term in (38) gives rise to the following contribution:…”

“…Study of degenerate parabolic problems has undergone a considerable progress in the last ten years, thanks to the fundamental paper of J. Carrillo [26] in which the Kruzhkov device of doubling of variables was extended to hyperbolic-parabolic-elliptic problems of the form j(v)−div(f (v)+∇ϕ(v)) = 0, and a technique for treating the homogeneous Dirichlet boundary conditions was put forward. In [26], the appropriate notion of entropy solution was established, and this definition (or, sometimes, parts of the uniqueness techniques of [26]) led to many developments; among them, let us mention [2,3,5,4,6,8,9,10,11,12,13,18,19,24,25,27,29,30,34,37,38,39,41,42,45,46,47,48,52,53,57,58,59]. Also numerical aspects of the problem were investigated; see, e.g., [7,32,33,35,40,49].…”

Renormalized solution for Stefan type problems: existence and uniqueness

“…In [15], Carrillo proves that problems of type E(u 0 , g) are well posed using the concept of "entropy solutions", which are weak solutions that satisfy some additional conditions called entropy conditions. However, under the additional structure condition (H 2 ), it is well known by now (see [4,15,21,22]) that Problem E(u 0 , g) is expected to admit at most one weak solution which, by definition, is a function u ∈ L 1 (Q) such that w ∈ L 2 (0, T ; H 1 0 (Ω)) and satisfies the equation in D (Q). As to the existence of a weak solution, this requires additional assumptions on the data u 0 and g, for instance u 0 ∈ L ∞ (Ω) and g ∈ L ∞ (Q).…”