概率方法解拟线性偏微分方程

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。

^StochastictMethod for Quasilinear Partial Differential Equations

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.