Development of singularities for a single quasi-linear equation and uniqueness of its generalized solutions

Abstract

Physically, the solution of the quasi-linear equation $\{u_t+f(u)_x=0$, x∈R, t>0 $u(x,0)=u_0(x)$, x∈R$, has discontinuities after a finite time. We find the maximum time T for which its solution is smooth in R x (0,T) when the initial data are smooth and then show that the solution must have discontinuities after the time T. Next we compute, by using a numerical scheme, the development of its discontinuities. Moreover, we show the uniqueness of its generalized solution when the flux f depends not only on u but on x and t.