On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory

The equation arising from Prandtl boundary layer theory partial derivative u/partial derivative t - partial derivative/x(i) (a(u,x,t) partial derivative u/partial derivative x(i)) - f(i)(x)D(i)u + 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 Kruzkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if ax(i)(., x, t)vertical bar(x is an element of partial derivative Omega) = a(., x, t)vertical bar(x is an element of partial derivative Omega)=0 and f(i)(x)vertical bar(x is an element of partial derivative Omega) = 0, the stability can be proved even without any boundary value condition.