A class of degenerate diffusion equations with mixed boundary conditions

This paper is concerned with a class of degenerate diffusion equations subject to mixed boundary conditions. Under some structure conditions, we discuss the blow-up property of local solutions and estimate the bounds of “blow-up time.”     

Asymptotic finite solutions of degenerate quasilinear parabolic equations with small diffusion

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 77–88, August, 1991.     

Asymptotic behavior of the unbounded solutions to some degenerate boundary layer equations revisited

We reconsider the boundary-layer flow of a non-Newtonian fluid corresponding to the classical Ostwald de Waele power-law model. The physical problem can be described in terms of solutions of the degenerate differential equation

Asymptotic expansions for degenerate parabolic equations

We prove asymptotic convergence results for some analytical expansions of solutions to degenerate PDEs with applications to financial mathematics. In particular, we combine short-time and global-in-space error estimates, previously obtained in the uniformly parabolic case, with some a priori bounds on “short cylinders”, and we achieve short-time asymptotic convergence of the approximate solution in the degenerate parabolic case.     

Asymptotic behavior of degenerate linear transport equations

We study in this paper a few simple examples of hypocoercive systems in which the coercive part is degenerate. We prove that the (completely explicit) speed of convergence is at least of inverse power type (the power depending on the features of the considered system).     

A free boundary problem for quasilinear degenerate parabolic equations with general degeneracy

In this paper, we study the free boundary problem for degenerate parabolic equations (1.1)–(1.4). The existence of generalized solutions inBV ₁, 1/2 is obtained by the means of parabolic regularization under certain restrictions. The uniqueness and regularity of generalized solutions are also discussed. In addition, a C¹⁺ smoothness for the free boundary is obtained in the parabolic case.     

Asymptotic behavior for doubly degenerate parabolic equations

We use mass transportation inequalities to study the asymptotic behavior for a class of doubly degenerate parabolic equations of the form

(1)$\text{?ρ}\text{?t}\text{=}\text{div}\text{ρ?c}{}^1\text{?}\text{F′(ρ)+V}\text{in}\text{(0,∞)×Ω,}\text{and}\text{ρ(t=0)=ρ}{}_0\text{in}\text{{0}×Ω,}$

where $\text{Ω}$ is $\text{R}{}^{\mathrm{n}}$, or a bounded domain of $\text{R}{}^{\mathrm{n}}$ in which case $\text{ρ?c}{}^1\text{[?(F′(ρ)+V)]·ν=0}$ on $\text{(0,∞)×?Ω}$. We investigate the case where the potential V is uniformly c-convex, and the degenerate case where V=0. In both cases, we establish an exponential decay in relative entropy and in the c-Wasserstein distance of solutions – or self-similar solutions – of (1) to equilibrium, and we give the explicit rates of convergence. In particular, we generalize to all p>1, the HWI inequalities obtained by Otto and Villani (J. Funct. Anal. 173 (2) (2000) 361–400) when p=2. This class of PDEs includes the Fokker–Planck, the porous medium, fast diffusion and the parabolic p-Laplacian equations. To cite this article: M. Agueh, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Self-similar solutions of the Euler equations with spherical symmetry

We consider self-similar flows arising from the uniform expansion of a spherical piston and preceded by a shock wave front. With appropriate boundary conditions imposed on the piston surface and the spherical shock, the isentropic compressible Euler system is transformed into a nonlinear ODE system. We formulate the problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions.     

Asymptotic solution of the unsteady boundary layer equations

Summary An extension of the Meksyn asymptotic method to unsteady boundary layers in laminar, incompressible flow is investigated. The results indicate that unsteady boundary layers can be calculated by the Meksyn asymptotic method with comparable accuracy to that obtained for steady flows. Several differences from the well developed steady-flow application exist and require further work before general problems can be treated. The calculation technique is more straight-forward for cases involving acceleration because three or four terms in the expansions may then yield sufficient accuracy. The form of the governing equation required by the Meksyn method indicates that it is most useful for unsteady stagnation boundary layers since some basic unsteady flows are not directly accessible in their simplest form from that equation. The effect of unsteadiness on the rate of asymptotic convergence is assessed by detailed comparison of a similar solution for unsteady, stagnation flow with analogous results from the Falkner-Skan equation and of reliable numerical results for both cases.

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Résumé On étudie une extension de la méthode asymptotique de Meksyn aux couches limites instables des écoulements laminaires de fluides incompressibles. Les résultats montrent que les couches limites instables peuvent être calculées à l'aide de la méthode asymptotique de Meksyn avec une précision comparable à celle obtenue pour les écoulements stables. Plusieurs différences existent par rapport à l'application, bien mise au point, aux écoulements stables; elles demandent encore du travail avant que les problèmes généraux puissent être traités. La méthode de calcul est plus directe dans les cas impliquant une accélération, car 3 ou 4 termes dans les développements assurent alors une précision suffisante. La forme de l'équation principale nécessaire à la méthode de Meksyn indique qu'elle est très utile pour les couches limites instables au repos; en effet, certains écoulements instables de base ne peuvent être atteints directement dans leur forme la plus simple à partir de cette équation. L'effet de l'instabilité sur la vitesse de convergence asymptotique est établi grâce à une comparaison détaillée d'une solution analogue pour un écoulement instable stagnant avec les résultats semblables obtenus par l'équation Falkner-Skan, et des résultats numériques sûrs obtenus dans les deux cas.

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This work was supported by the Energy Research and Development Administration.     

Source-type solutions of degenerate diffusion equations with absorption

The object of this paper is to study the existence of a solution of the Cauchy problemu _t=Δu^m−u^p, u(x,0)=δ(x) and when a solution exists, to study its behaviour ast→0.     

Nonlinear degenerate parabolic equations with Neumann boundary condition

We study Neumann problem for a class of nonlinear degenerate parabolic PDE. A typical nonlinearity we have in mind is, for instance, β(u)=−1/u(u>0). We establish a necessary and sufficient condition on given data for existence of solution.     

Inverse problem for the diffusion equation in the case of spherical symmetry

The initial boundary value problem for the diffusion equation is considered in the case of spherical symmetry and an unknown initial condition. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplace operator applied to the solution of the initial boundary value problem. The uniqueness of the solution of the inverse problem is studied depending on the parameters entering into the boundary conditions. It is shown that the solution of the inverse problem is either unique or not unique up to a one-dimensional linear subspace.     

Asymptotic symmetry of singular solutions of semilinear elliptic equations

For dimensions 3?n?6, we derive lower bound for positive solution of

     

Asymptotic symmetry of solutions of nonlinear partial differential equations

For a large class of partial differential equations on exterior domains or on ?^N we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi equations.