Stability of Entropy Solutions to the Cauchy Problem for a Class of Nonlinear Hyperbolic-Parabolic Equations

Chen, Gui–Qiang; DiBenedetto, Emmanuele

doi:10.1137/s0036141001363597

Cited by 49 publications

(35 citation statements)

References 8 publications

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“…The well-posedness of multi-dimensional Dirichlet initial-boundary value problems for strongly degenerate parabolic equations is shown in [40]. Further recent contributions to the analysis of strongly degenerate parabolic equations include [41,42,43,44].…”

Section: Multiresolution Schemesmentioning

confidence: 99%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Bürger

Ruiz

Schneider³

et al. 2007

J Eng Math

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Abstract. A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the approximation of Engquist-Osher for the flux and explicit time stepping. An adaptive multiresolution scheme with cell averages is then used to speed up CPU time and memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier-thickener model illustrate the efficiency of this method.

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Section: Multiresolution Schemesmentioning

confidence: 99%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Bürger

Ruiz

Schneider³

et al. 2007

J Eng Math

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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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“…Extensions of this uniqueness result to the initial value problem can be found in [61,62] for bounded entropy solutions (of more general equations). Uniqueness for unbounded entropy solutions and kinetic solutions is studied in [30] and [31], respectively. The inhomogeneous Dirichlet boundary value problem is treated in [72].…”

Section: Strongly Degenerate Parabolic Problemsmentioning

confidence: 99%

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Tory¹,

Karlsen²,

Bürger³

et al. 2003

SIAM J. Appl. Math.

140

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Abstract. We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression" or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convectiondiffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.Key words. polydisperse suspensions, sedimentation, systems of conservation laws, strongly degenerate parabolic-hyperbolic systems, central difference approximation AMS subject classifications. 35K65, 35L40, 35L65, 65M06, 76T05 DOI. 10.1137/S00361399024081631. Introduction. Mathematical models for the (controlled) sedimentation of polydisperse suspensions of small particles, which belong to a finite number of species differing in size or density and are suspended in a viscous fluid, are important to many applications such as the chemical engineering, ceramic, pulp and paper, and food industries, mineral processing, wastewater treatment, and medicine [3,50,88,89,101,122]. The characteristic behavior of such mixtures is differential sedimentation, which leads to areas of different composition if an initially homogeneous suspension is allowed to settle. In this paper, we consider the additional property that the solid particles possibly form a compressible sediment layer. A mathematical model for polydisperse suspensions forming compressible sediments is developed, analyzed, and simulated, focusing on three different aspects.First, we show how two existing sedimentation models-one for monodisperse flocculated suspensions, which are described by scalar strongly degenerate parabolichyperbolic equations, and one for polydisperse suspensions of rigid spheres differing

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Stability of Entropy Solutions to the Cauchy Problem for a Class of Nonlinear Hyperbolic-Parabolic Equations

Cited by 49 publications

References 8 publications

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

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

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

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