On the existence, uniqueness and stability of entropy solutions to scalar conservation laws

We consider one-dimensional scalar conservation laws with and without viscosity where the flux function F(x,t,u) is only assumed to be absolutely continuous in x, locally integrable in t and continuous in u. The existence and uniqueness of entropy solutions for the associated initial-value problem are obtained through the vanishing viscosity method and the doubling variables technique. We also prove the stability of entropy solutions in and in C([0,T];L1(R)) with respect to both initial data and flux functions.