Capillary problem for singular degenerate parabolic equations on a half space
type: Item
instance of: scholarly article
title: Capillary problem for singular degenerate parabolic equations on a half space
zbMATH Open document ID: 0879.35011
published in: Differential and Integral Equations
publication date: 1997-12-10T00:00:00Z
review text: The paper is devoted to the study of solutions to a geometric evolution of an interface \(\{\Gamma_t\}_{t \geq 0}\) in a fixed domain \(\Omega\) under the condition that the geometric boundary \(b\Gamma_t = \partial \Omega \cap \bar {\Gamma_t}\) of \(\Gamma_t\) touches the boundary \(\partial \Omega\) with a given angle. Intrinsically, \(\Gamma_t\) means a surface dividing two regions \(D^{+}_{t}, D^{-}_{t}\) in \(\Omega\) occupied in the time \(t\) by two phases of the studied materials. In the paper, the interface \(\Gamma_t\) is looked for as the zero level set of a function \(u\) which is positive on \(D^{+}\) and negative on \(D^{-}\). If \(u \in C^2\) and \(\nabla u \neq 0\) near \(\Gamma_t\), the problem can be formulated as follows: \[ \partial_t u + F(\nabla u, \nabla^2 u) = 0\;\;\text{on} \;\Omega \cap \Gamma_t,\qquad\langle \nu (x), \nabla u\rangle - k |\nabla u |= 0\;\;\text{on} b\Gamma_t \] with \(F\) given, \(k = \cos \theta\) (\(\theta\) being the angle between \(\Gamma_t\) and \(\partial \Omega \)) and \(\nu(x)\) the outer unit normal of \(\partial \Omega\) at a point \(x\). The crucial step in the proof is a comparison principle for viscosity sub- and super-solutions. It allows to construct a unique viscosity solution for any continuous initial value \(a\) (constant outside a compact subset of \(\Omega \)). For simplicity reasons, only the case of \(\Omega\) equal to a halfspace is considered here. The general case is treated in a forthcoming paper.
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 1095808
zbMATH Keywords: capillary problem
zbMATH Keywords: viscosity solutions
zbMATH Keywords: level-set approach
zbMATH Keywords: comparison principle
author: Moto-Hiko Sato
MaRDI profile type: Publication
Recommended article: C-singular solutions of the capillary problem
Recommended article: Global existence and uniqueness results for singular solutions of the capillarity equation
Recommended article: Q3766171
Recommended article: Capacitary estimates of solutions of semilinear parabolic equations
Recommended article: Capillary surfaces arising in singular perturbation problems
Recommended article: Semilinear degenerate parabolic systems and distributed capacitance models
Recommended article: Singular degenerate parabolic equations with applications
Recommended article: Non-negative global weak solutions for a degenerate parabolic system modelling thin films driven by capillarity
Recommended article: On a problem for the degenerating parabolic-hyperbolic equation involving Caputo derivative of fractional order and non-linear terms
Recommended article: On a degenerate parabolic equation with singular convection
description: scientific article; zbMATH DE number 1095808