On the uniqueness of the solution of the evolution dam problem

Carrillo, José

doi:10.1016/0362-546x(94)90084-1

Cited by 54 publications

(35 citation statements)

References 8 publications

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“…Due to this loss of regularity, it is necessary to work with weak solutions; moreover, to single out a physically relevant and unique weak solution, we need to impose additional "entropy inequalities", in the spirit of Kruzhkov [63]. Early results on hyperbolic-parabolic equations were obtained by Volpert, Hudjaev [78]; see also [80,82,81], [74], [28], [19] and references cited therein, and [76], [37], [32]. L 1 entropy techniques for degenerate convection-diffusion equations like (3), which take into account both hyperbolic and parabolic features, were developed by Carrillo [29] for the homogeneous Dirichlet problem in bounded domains.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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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“…water and air) must be modelled, we are led to the closely related two-phase flow equations, see e. g. [1,8,9,10,11,17,26]. Instead, if the medium is partially saturated and the unsaturated phase is not modelled, we are led to the dam problem [6], where the outflow condition is of particular importance. A system with similar double degeneracy appears in a model for the evolution of bacterium species in [5].…”

Section: Pop and Schweizermentioning

confidence: 99%

Regularization Schemes for Degenerate Richards Equations and Outflow Conditions

Pop

Schweizer

2011

Math. Models Methods Appl. Sci.

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We analyze regularization schemes for the Richards equation and a time discrete numerical approximation. The original equations can be doubly degenerate, therefore they may exhibit fast and slow diffusion. Additionally, we treat outflow conditions that model an interface separating the porous medium from a free flow domain. In both situations we provide a regularization with a non-degenerate equation and standard boundary conditions, and discuss the convergence rates of the approximations.

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“…However, when the diusion of heat is no longer linear, and specially when it is degenerate, the problem turns to be more involved: natural estimates for the heat equation are obtained in L 1 but proving the well posedness of Navier-Stokes equation in this space seems to be a dicult task. This fact makes dicult to use the L 1 techniques developed in the last years, and still in progress, that have been successfully used to prove uniqueness for degenerate scalar equations as well as for certain systems of equations where comparison principles still hold (see [20], [5], [13], [8], [27]). In this paper we shall approach the problem from a duality technique, i.e., from the search of suitable test functions (perturbations of the sign function) that allows to conclude the uniqueness property.…”

Section: Uniqueness Of Solutionsmentioning

confidence: 99%