On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory

• 作者: OUYANG Miao
• 来源期刊: 偏微分方程（英文版）
• 学科: 数学
• 关键词: Prandtl boundary layer theory entropy solution Kru(z)kov's bi-variables method partial boundary value condition stability
• 年，卷（期）: 2020,（2）
• 页码范围: 119-142
• 页数: 24页
• 分类号: O175.29
• 语种: 英文
• DOI: 10.4208/jpde.v33.n2.3

摘要

The equation arising from Prandtl boundary layer theory (e)u/(e)t-(e)/(e)x1(a(u,x,t)(e)u/(e)xi)-fi(x)Diu+c(x,t)u=g(x,t)is considered.The existence of the entropy solution can be proved by BV estimate method.The interesting problem is that,since a(·,x,t) may be degenerate on the boundary,the usual boundary value condition may be overdetermined.Accordingly,only dependent on a partial boundary value condition,the stability of solutions can be expected.This expectation is turned to reality by Kru(z)kov's bi-variables method,a reasonable partial boundary value condition matching up with the equation is found first time.Moreover,if axi(·,x,t)|x∈(e)Ω=a(·,x,t)|x∈(e)Ω=0 and fi(x)|x∈(e)Ω=0,the stability can be proved even without any boundary value condition.