摘要
本文用随机分析方法证明了拟线性抛物型方程ut+f(u)ux、uxx=0,u(0,x)=u0(x)在u0有界可测,f连续且f>0条件下,其解当→0时收敛于拟线性方程ut+f(u)ux=0,u(0,x)=u0(x)的熵解,即论证了“沾性消失法”解此方程的正确性,1957年Oleinik曾用差分方法解决了此问题。这里用概率方法重新获得此结果。
lt is proved by stochastic analysis method that if u0 is bounded and mea- surable and f> 0 then the solution of Cauchy problem of the quasilinear parabolic partial differential equation u0+f(u)ux+=0 , u(0,x) =u0 (x) xonverges to the entropy solution of the quasilinear partial differential equation u0 +f (u)ux=0 , u(0,x) = u0(x), as - 0, that is the validity of ' vanishing viscosity' method for the equation. This is the well known result due to Oleinik by using difference method in 1975. The result of this paper suggests the importance of probabilistic method for solving nonlinear partial differeatial equations.
出处
《数学学报(中文版)》
SCIE
CSCD
北大核心
1997年第3期333-344,共12页
Acta Mathematica Sinica:Chinese Series
关键词
拟线性
偏微分方程
解
概率方法
有界变差函数
Quasilinear partial differential equation , Eatropy condition , Stochastic partial differential equation, lto formula, Bounded variation function