L1-Contraction and Uniqueness for Quasilinear Elliptic–Parabolic Equations

Otto, Félix

doi:10.1006/jdeq.1996.0155

Cited by 269 publications

(252 citation statements)

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“…It can be obtained either by the technique of Otto [70] (doubling the time variable) or using the theory of integral solutions and nonlinear semigroup methods, consult for example [30]. Besides, u ρ verifies the entropy formulation of Definition 2.2 with fluxes f ρ , A ρ , where η ± c can be replaced by regular "boundary" entropies η ± c,ε , whenever we prefer to do so.…”

Section: Definition 22 (Entropy Solution) An Entropy Solution Of Thmentioning

confidence: 99%

“…The product between ∂ t u ρ and A ρ (u ρ ) is handled using the usual chain rule argument (see, e.g., [4,70,30]), where the relevant duality is between the space E :…”

Section: Definition 22 (Entropy Solution) An Entropy Solution Of Thmentioning

confidence: 99%

“…We refer to [4,20,70,30,6] and the references cited therein for more information on elliptic-parabolic equations.…”

Section: Introductionmentioning

confidence: 99%

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

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

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Section: Definition 22 (Entropy Solution) An Entropy Solution Of Thmentioning

confidence: 99%

“…The product between ∂ t u ρ and A ρ (u ρ ) is handled using the usual chain rule argument (see, e.g., [4,70,30]), where the relevant duality is between the space E :…”

Section: Definition 22 (Entropy Solution) An Entropy Solution Of Thmentioning

confidence: 99%

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

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

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“…The lack of regularity of F is the only reason why the doubling of variables in space can be needed for the Stefan-type problems (1) (the doubling of variables in time, see Otto [51], Blanchard and Porretta [24], does not interfere with different boundary conditions; moreover, it can be avoided thanks to the nonlinear semigroup techniques, see Bénilan and Wittbold [21] and Section 5). Let us also mention that for diffusion-convection operators under the general form −div a(t, x; w,∇w), the explicit dependence in x is a major obstacle to apply the doubling of variables technique (except for the case treated by Vallet in [59]); some results for this case were obtained by Blanchard and Porretta in [24] and by Zimmermann [60] under regularity assumptions on F .…”

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confidence: 99%

On uniqueness techniques for degenerate convection-diffusion problems

Andreïanov

Igbida

2012

IJDSDE

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Abstract. We survey recent developments and give some new results concerning uniqueness of weak and renormalized solutions for degenerate parabolic problems of the form ut − div (a 0 (∇w) + F (w)) = f , u ∈ β(w) for a maximal monotone graph β, a Leray-Lions type nonlinearity a 0 , a continuous convection flux F , and an initial condition u| t=0 = u 0 . The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space.We avoid the degeneracy that could make the problem hyperbolic in some regions; yet our starting point is the notion of entropy solution, notion that underlies the theory of general hyperbolic-parabolic-elliptic problems. Thus, we focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the "parabolic-elliptic aspects". We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of "going to the boundary" in the Kato inequality for comparison of two solutions; uniqueness for renormalized solutions obtained via reduction to weak solutions. On several occasions, the results are achieved thanks to the notion of integral solution coming from the nonlinear semigroup theory.1. Introduction 1.1. A survey of literature. 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].The notion of entropy solution (or, as in the present paper, entropy solutions techniques used on weak solutions) was retained by most of the authors; yet, let us mention the version of Bendahmane and Karlsen [18,19] We thank Safimba Soma for fruitful discussions on the subject of this note, and the anonymous referee for a careful reading and very pertinent remarks.

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“…One of them, introduced by Kruzhkov in [28] to prove an L 1 contraction property of entropy solutions of hyperbolic equations, is based in doubling the time variable and performing a passing to the limit in which these variables collapse. This technique has been applied to parabolic scalar equations, see, e.g., [29], [9], [18], [19], [35], and also to certain systems of parabolic equations coupled through reaction terms, but not through transport terms, see [36]. Notice that when applying succesfully this technique, uniqueness is always obtained as a by-product of a comparison principle.…”

mentioning

confidence: 99%

On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions

Díaz

Galiano

Jüngel

2001

Nonlinear Analysis: Real World Applications

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Abstract. A drift-diffusion model for semiconductors with nonlinear diffusion is considered. The model consists of two quasilinear degenerate parabolic equations for carrier densities and the Poisson equation for electric potential. We assume Lipschitz continuous nonlinearities in the drift and generation-recombination terms.Existence of weak solutions is proven by using a regularization technique. Uniqueness of solutions is proven when either the diffusion term ϕ is strictly increasing and solutions have spatial derivatives in L 1 (Q T ) or when ϕ is non-decreasing and a suitable entropy condition is fullfilled by the electric potential.

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L1-Contraction and Uniqueness for Quasilinear Elliptic–Parabolic Equations

Cited by 269 publications

References 5 publications

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

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

On uniqueness techniques for degenerate convection-diffusion problems

On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions

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