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Infiltration Equation with Degeneracy on the Boundary

Infiltration Equation with Degeneracy on the Boundary This paper is mainly about the infiltration equation u t = div ( a ( x ) | u | α | ∇ u | p − 2 ∇ u ) , ( x , t ) ∈ Ω × ( 0 , T ) , $$ {u_{t}}= \operatorname{div} \bigl(a(x)|u|^{\alpha }{ \vert { \nabla u} \vert ^{p-2}}\nabla u\bigr),\quad (x,t) \in \Omega \times (0,T), $$ where p > 1 $p>1$ , α > 0 $\alpha >0$ , a ( x ) ∈ C 1 ( Ω ‾ ) $a(x)\in C^{1}(\overline{\Omega })$ , a ( x ) ≥ 0 $a(x)\geq 0$ with a ( x ) | x ∈ ∂ Ω = 0 $a(x)|_{x\in \partial \Omega }=0$ . If there is a constant β $\beta $ such that ∫ Ω a − β ( x ) d x ≤ c $\int_{\Omega }a^{-\beta }(x)dx\leq c$ , p > 1 + 1 β $p>1+\frac{1}{\beta }$ , then the weak solution is smooth enough to define the trace on the boundary, the stability of the weak solutions can be proved as usual. Meanwhile, if for any β > 1 p − 1 $\beta >\frac{1}{p-1}$ , ∫ Ω a − β ( x ) d x d t = ∞ $\int_{\Omega }a^{-\beta }(x)dxdt=\infty $ , then the weak solution lacks the regularity to define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Infiltration Equation with Degeneracy on the Boundary

Zhan, Huashui
Acta Applicandae Mathematicae , Volume 153 (1) – Aug 24, 2017
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References (22)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer Science+Business Media B.V.
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-017-0124-3
Publisher site
See Article on Publisher Site

Abstract

This paper is mainly about the infiltration equation u t = div ( a ( x ) | u | α | ∇ u | p − 2 ∇ u ) , ( x , t ) ∈ Ω × ( 0 , T ) , $$ {u_{t}}= \operatorname{div} \bigl(a(x)|u|^{\alpha }{ \vert { \nabla u} \vert ^{p-2}}\nabla u\bigr),\quad (x,t) \in \Omega \times (0,T), $$ where p > 1 $p>1$ , α > 0 $\alpha >0$ , a ( x ) ∈ C 1 ( Ω ‾ ) $a(x)\in C^{1}(\overline{\Omega })$ , a ( x ) ≥ 0 $a(x)\geq 0$ with a ( x ) | x ∈ ∂ Ω = 0 $a(x)|_{x\in \partial \Omega }=0$ . If there is a constant β $\beta $ such that ∫ Ω a − β ( x ) d x ≤ c $\int_{\Omega }a^{-\beta }(x)dx\leq c$ , p > 1 + 1 β $p>1+\frac{1}{\beta }$ , then the weak solution is smooth enough to define the trace on the boundary, the stability of the weak solutions can be proved as usual. Meanwhile, if for any β > 1 p − 1 $\beta >\frac{1}{p-1}$ , ∫ Ω a − β ( x ) d x d t = ∞ $\int_{\Omega }a^{-\beta }(x)dxdt=\infty $ , then the weak solution lacks the regularity to define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition.

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Aug 24, 2017

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