Nonlinear Diffusion Equations

Wu, Zhong; Yin, Jingxue; Li, Huilai; Zhao, Junning

doi:10.1142/4782

Cited by 259 publications

(222 citation statements)

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“…The first term in (81) converges to the first term in the right-hand side of (77). This argument is standard; one can use, e.g., the properties of the Lebesgue points of L 1 functions and the upper-semicontinuity of the…”

Section: On the Choice Of Fv Scheme And Various Generalizationsmentioning

confidence: 95%

“…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%

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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: 95%

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

View full text Add to dashboard Cite

show abstract

“…comes from a host of applied fields such as the theory of non-Newtonian fluid, the study of water infiltration through porous media, and combustion theory; one can refer to [1][2][3][4] and the references therein. Here > 1, = / , ∈ (Ω), Ω ⊂ R is a bounded domain with the appropriately smooth boundary Ω.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

A New Kind of Weak Solution of Non-Newtonian Fluid Equation

Zhan

2017

Journal of Function Spaces

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“…In particular, equation (1.1) may be used to describe the non-stationary flows in a porous medium of fluids under polytropic conditions. In this case, equation (1.1) is called the non-Newtonian polytropic filtration equations (see [5], [8], [11] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, a lot of papers are devoted to discuss the following degenerate equation 2) with source and without boundary flux and to reveal the phenomena described as [4], [9]. Following from [2], [8], [11], we can get the local existence of weak solutions in the sense of the following definition.…”

Section: Introductionmentioning

confidence: 99%

Critical Blow-Up and Extinction Exponents for Non-Newton Polytropic Filtration Equation With Source

Zhou¹,

Mu²

2009

Bulletin of the Korean Mathematical Society

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Abstract. This paper deals with the critical blow-up and extinction exponents for the non-Newton polytropic filtration equation. We reveals a fact that the equation admits two critical exponents q 1 , q 2 ∈ (0, +∞) with q 1 < q 2 . In other words, when q belongs to different intervals (0, q 1 ), (q 1 , q 2 ), (q 2 , +∞), the solution possesses complete different properties. More precisely speaking, as far as the blow-up exponent is concerned, the global existence case consists of the interval (0, q 2 ]. However, when q ∈ (q 2 , +∞), there exist both global solutions and blow-up solutions. As for the extinction exponent, the extinction case happens to the interval (q 1 , +∞), while for q ∈ (0, q 1 ), there exists a non-extinction bounded solution for any nonnegative initial datum. Moreover, when the critical case q = q 1 is concerned, the other parameter λ will play an important role. In other words, when λ belongs to different interval (0, λ 1 ) or (λ 1 , +∞), where λ 1 is the first eigenvalue of p-Laplacian equation with zero boundary value condition, the solution has completely different properties.

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Nonlinear Diffusion Equations

Cited by 259 publications

References 0 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

A New Kind of Weak Solution of Non-Newtonian Fluid Equation

Critical Blow-Up and Extinction Exponents for Non-Newton Polytropic Filtration Equation With Source

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