Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations

Mascia, Corrado; Porretta, Alessio; Terracina, Andrea

doi:10.1007/s002050200184

Cited by 84 publications

(133 citation statements)

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“…For the homogeneous Dirichlet boundary conditions, this has been achieved by Carrillo in [26]. For non-homogeneous Dirichlet boundary conditions satisfying rather strong regularity assumptions, this was done in [47,48,59] and in [4,2,3]. For the Neumann boundary conditions, a specific procedure was designed in [9].…”

Section: Proof (Sketched)mentioning

confidence: 99%

“…Typical tools are [47, Definition 1.1, Lemma 2.2] and [59, Lemma 1] that are used to "generate" boundary terms from sequences of test functions (ξ h ) h with gradient concentrated at an h-neighbourhood of the boundary. This approach is used in the works Mascia, Porretta, Terracina [47], Michel, Vovelle [48] and Vallet [59], the latter work presenting most general results for hyperbolicparabolic problems with (t, x)-dependent coefficients. The context of these works is much more general than the ours, because it includes hyperbolic degeneracy; yet the application of these arguments to (1) remains lengthy.…”

Section: Proof (Sketched)mentioning

confidence: 99%

“…The first one is to construct ξ h more or less explicitly, using only the geometry of the domain (this is the case in [48,59] and also in [23,46,45] described below). The second one is to construct (ξ h ) h by solving a PDE related to some of the terms in (19) (this was the case in [47]); this is a Holmgren-type approach, and it may lead to finer constructions. 4.1.…”

Section: Kato Inequalities: "Going To the Boundary"mentioning

confidence: 99%

“…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].…”

mentioning

confidence: 99%

“…Yet the Stefan type problems with continuous F serve as a playground for the wide class of practically important hyperbolic-parabolic-elliptic problems (see in particular [26,47,48,59,7]); for these problems, entropy inequalities and the doubling of variables remain the essential technique. It is an open question how to transfer to this context the techniques of [9,12] or those of [11] recalled in this paper; some work in this direction is in progress.…”

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confidence: 99%

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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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Section: Proof (Sketched)mentioning

confidence: 99%

Section: Proof (Sketched)mentioning

confidence: 99%

Section: Kato Inequalities: "Going To the Boundary"mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On uniqueness techniques for degenerate convection-diffusion problems

Andreïanov

Igbida

2012

IJDSDE

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Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

Chen

Torres

Ziemer

2008

Comm Pure Appl Math

123

167

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Abstract. We analyze a class of weakly differentiable vector fields F : R N → R N with the property that F ∈ L ∞ and div F is a Radon measure. These fields are called bounded divergencemeasure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field F over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we first develop an alternative way to establish the Gauss-Green theorem for any smooth bounded set with F ∈ L ∞ . Then we establish a fundamental approximation theorem which states that, given a Radon measure µ that is absolutely continuous with respect to H N−1 on R N , any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure µ . We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter, E, as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for F holds on E.With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an (N − 1)-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measurevalued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.

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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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Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations

Cited by 84 publications

References 10 publications

On uniqueness techniques for degenerate convection-diffusion problems

On uniqueness techniques for degenerate convection-diffusion problems

Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

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

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