Asymptotic behaviors of solutions to viscous conservation laws via \(L^2\)-energy method (Q2767447)

The author considers the Cauchy problem for a scalar viscous conservation law \(u_t+f(u)_x=\mu u_{xx}\), and for \(p\)-systems with viscosity \(v_t-u_x=0\), \(u_t+p(v)_x= \mu(u_x/b)_x\). The corresponding Riemann problems have the nonlinear waves as the weak solutions, i.e. the rarefaction wave, the shock wave or their superpositions, depending on the data. The author shows that the solutions to the Cauchy problems end to the rarefaction wave, viscous shock wave or their superpositions, respectively. Fundamental results are proved via the \(L^2\)-energy method and the recent results are surveyed. Some cases are proposed as open problems, the initial-boundary value problem on the half-line is also discussed.