Continuity of Some Classes of Functions Related to Degenerate Parabolic Equations and Its Applications

We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in the research of degenerate parabolic equations including porous medium equations. Consequently, we prove that a function in such a class is continuous. As an application, we obtain the estimate for the continuous modulus of the solutions of a few degenerate parabolic equations in divergence form, including the anisotropic porous equations.     

Local Existence And Uniqueness of Solutions of Degenerate Parabolic Equations

We consider a class of degenerate parabolic equaitons on a bounded domain with mixed boundary conditions. These problems arise, for example, in the study of flow through porous media. Under appropriate hypotheses, we establish the existence of a nonegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations. This construction is used establish uniqueness, cinparison, and L¹ continuous dependence theorems, as well some results on blow up of solutions in finite time     

二阶完全非线性蜕化抛物方程

本文研究了一个空间变量的二阶完全非线性蜕化抛物方程ｕｔ＝Ｆ（ｕｘｘ，ｕｘ，ｘ，ｔ）的第一边值问题。在仅要求Ｆ及其一阶导数满足结构条件的情形，给出了蜕化问题连续解的存在唯一性。这个工作将渗流方程的结果推广到非常一般的情形。     

Harnack estimates for quasi-linear degenerate parabolic differential equations

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.     

一类双重退化抛物方程局部解的存在性

This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate p- Laplace equation(p＞2) u_t-div(b(x,t,u)|▽u|~(p-2)▽u)=f(x,u,t). The initial-boundary value problem in a bounded domain of R~N is considered under mixed boundary conditions.The existence of local-in-time weak solutions is obtained.     

Infinite dimensional attractors for porous medium equations in heterogeneous medium

In this paper, we give a detailed study of the global attractors for porous medium equations in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is obtained by showing that their ?‐Kolmogorov entropy behaves as a polynomial of the variable 1 ∕ ? as ? tends to zero, which is not observed for non‐degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ?‐entropy of infinite‐dimensional attractors are also obtained. We believe that the method developed in this paper has a general nature and can be applied to other classes of degenerate evolution equations. Copyright © 2012 John Wiley & Sons, Ltd.     

渗流方程差分解的收敛性

渗流方程是拟线性退化抛物型方程。[1—4]讨论了弱解的存在唯一性问题。由于非线性扩散系数有零点,其解可以不光滑。在[5,6]中研究了渗流方程 u_t=(u~m)_(xx),m>1的差分方法问题,对光滑区和弱间断点给以分别处理。渗流方程的解是连续的,但在有些点上导数不存在。因此,不能用Taylor展开估计截断误差的方法证明差分解的收敛性。     

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.     

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

In this article, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long‐time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all time is established in a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality‐type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates.     

Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes

For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.     

Regularization of outflow problems in unsaturated porous media with dry regions

We study a porous medium with saturated, unsaturated, and dry regions, described by Richards' equation for the saturation s and the pressure p. Due to a degenerate permeability coefficient k(x,s) and a degenerate capillary pressure function pc(x,s), the equations may be of elliptic, parabolic, or of ODE-type. We construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system. An example shows that the convergence fails in general. Our approach provides an existence result for the outflow problem in the case of x-dependent coefficients and a method for a numerical approximation.     

Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

A Harnack Inequality for Degenerate Parabolic Equations

In this paper we apply the Moser iteration method to degenerate parabolic divergence structure equations. Under some conditions we get a Harnack inequality for weak solutions and from it derive Hölder estimates for weak solutions of uniformly degenerate parabolic equations and the continuity of weak solutions of non-uniformly degenerate parabolic equations.     

Local Hölder continuity of nonnegative weak solutions of degenerate parabolic equations

In this paper, we consider nonnegative weak solutions for a class of degenerate parabolic equations. Using Moser’s method, we get the local boundedness of solutions to equations of this class. Then we prove that the solutions are locally Hölder continuous.     

Viscosity solutions of a class of degenerate quasilinear parabolic equations with Dirichlet boundary condition

This paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous Dirichlet boundary condition. The equation under consideration arises from a number of practical model problems including reaction–diffusion processes in a porous medium. The degeneracy of the problem appears on the boundary and possibly in the interior of the domain. The goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them. An application of the above results is given to a porous-medium type of reaction–diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem.     

A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations

Following the lead of [Carrillo, Arch. Ration. Mech. Anal. 147 (1999) 269-361], recently several authors have used Kru?kov's device of “doubling the variables” to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order differential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo's uniqueness result to a class of degenerate parabolic equations with spatially dependent second order differential operator. The class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation-consolidation processes.     

Traveling Waves in Degenerate Diffusion Equations

In this survey, we present a literature review on the study of traveling waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy. The main results on wave existence and stability are presented for the typical degenerate equations, including porous medium equations, flux limited diffusion equations, delayed degenerate diffusion equations, and other strong degenerate diffusion equations.     

On the existence of nonnegative continuous solutions for a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the existence of nonnegative continuous solutions for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations.     

On the existence of nonnegative continuous solutions for a class of fully nonlinear degenerate parabolic equations

This paper is concerned with the existence of nonnegative continuous solutions for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations.     

An improperly posed estimate for the equation |u|α ut =uxx +F(u)(0 <α <1)

The noncharacteristic Cauchy problem of heat equations is not well-posed. But the estimation on the continuous dependence of solutions holds under their prescribed bound and the bound of their Cauchy data. In this paper we show that a similar estimation holds also for some degenerate quasilinear parabolic equation.