Boundary Problem For Parabolic Equation Research Articles

Boundary Problems for Parabolic Equations with Degenerate Boundary Condition of the Third Kind

Boundary Problems for Parabolic Equations with Degenerate Boundary Condition of the Third Kind

The existence of solutions for drying with coupled phase change in a porous medium

This paper deals with a theoretical mathematical analysis of freezing (desublimation) of moisture in a finite porous medium with a heat flux condition at the boundary. The goal is to generalize [E.A. Santillan Marcus, D.A. Tarzia, Exact solutions for drying with coupled phase-change in a porous medium with a heat flux condition on the surface, Comput. Appl. Math. 22 (2003) 293–311], proving the local existence and uniqueness in time of the solution of this problem. We give the model equations as a free boundary problem, and we prove that the problem is equivalent to a system of Volterra integral equations following the Friedman–Rubinstein’s method given in [A. Friedman, Free boundary problems for parabolic equations, I. Melting of solids, J. Math. Mech. 8 (1959) 499–517]. Then, we prove that the problem has a unique local solution in time by using the Banach contraction theorem.

Regularity of the Solutions for Parabolic Two-Phase Free Boundary Problems

In this paper we start to develop the regularity theory of general two-phase free boundary problems for parabolic equations. In particular we consider uniformly parabolic operators in nondivergence form and we are mainly concerned with the optimal regularity of the viscosity solutions. We prove that under suitable nondegenerate conditions the solution is Lipschitz across the free boundary.

Exact solutions of one-dimensional two-phase free boundary problems for parabolic equations

We construct automodel solutions for the one-dimensional two-phase Stefan, Florin, and Verigin free boundary problems for parabolic equations in the case where the initial and boundary data are not adjusted. It is shown that in the Stefan problem with “supercooling,” the liquid temperature may be less than the temperature of the phase transition, i.e., the liquid may be “supercooled” while the solid may be “superheated.” Bibliography: 8 titles.

On a parabolic free boundary problem arising from a Bingham-like flow model with a visco-elastic core

In this paper we study a two-phase one-dimensional free boundary problem for parabolic equation, arising from a mathematical model for Bingham-like fluids with visco-elastic core presented in [L. Fusi, A. Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004) 826–847]. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Local existence is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a non-standard technique based on a weak formulation of the problem.

Estimates for Solutions to Model Problems Connected with Quasistationary Approximation for the Stefan Problem

Model problems for elliptic equations are considered. The time-derivative of a solution can be contained in the boundary condition or in the conjugation condition. Suich problems appear, for example, in the study of free boundary problems for elliptic equations that can be considered as quasistationary approximations to free boundary problems for parabolic equations. Estimates for solutions to the model problems are obtained. Bibliography: 11 titles.

Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain

We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial differential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature flow equation and apply to more general Neumann-type boundary problems for parabolic equations in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and curvature.

Two-phase nonlinear free boundary problems for parabolic equations

On etablit un theoreme sur le caractere bien pose local d'un probleme parabolique non lineaire a une dimension d'espace avec des conditions aux limites a 2 phases libres

Free boundary problems for parabolic equations

Free boundary problems for parabolic equations

Second-Order Correct Boundary Conditions for the Numerical Solution of the Mixed Boundary Problem for Parabolic Equations

Second-Order Correct Boundary Conditions for the Numerical Solution of the Mixed Boundary Problem for Parabolic Equations

On the existence of a solution of Verigin's problem

On the existence of a solution of Verigin's problem

Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations

Received June 21, 1962. This research was supported by the Air Force Office of Scientific Research. In May, 1961 the author submitted a thesis containing the principle part of this paper to the Rice University in partial fulfillment of the requirements for a degree of Master of Arts. * ,p(x, t) E Ca'd(R) if and only if sp is continuously differentiable a times with respect to x and a times with respect to t in the region R. 405