Renormalized Entropy Solutions for Quasi-linear Anisotropic Degenerate Parabolic Equations

Bendahmane, Mostafa; Karlsen, Kenneth H.

doi:10.1137/s0036141003428937

Cited by 105 publications

(103 citation statements)

References 17 publications

(39 reference statements)

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“…considered by Bendahmane, Karlsen in [17,18], the discrete entropy inequalities on Cartesian meshes are as easy to obtain as for the isotropic case a(ξ) = k(ξ)ξ considered in the present paper.…”

Section: On the Choice Of Fv Scheme And Various Generalizationsmentioning

confidence: 53%

“…• Convection-diffusion problems with anisotropic linear and nonlinear diffusion were considered in [35,34] and in [17,18]. General DDFV schemes do not seem easy to adapt to the nonlinear anisotropic framework, because of the presence of "privileged" directions of diffusion.…”

Section: On the Choice Of Fv Scheme And Various Generalizationsmentioning

confidence: 99%

“…[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]. In this paper, we use a variant of the Carrillo entropic approach.…”

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: On the Choice Of Fv Scheme And Various Generalizationsmentioning

confidence: 53%

Section: On the Choice Of Fv Scheme And Various Generalizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

View full text Add to dashboard Cite

show abstract

“…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.…”

mentioning

confidence: 97%

“…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. …”

mentioning

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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Anisotropic Doubly Nonlinear Degenerate Parabolic Equations

Bendahmane

Karlsen

Numerical Mathematics and Advanced Applications

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Renormalized Entropy Solutions for Quasi-linear Anisotropic Degenerate Parabolic Equations

Cited by 105 publications

References 17 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

Anisotropic Doubly Nonlinear Degenerate Parabolic Equations

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