On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits

We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (u _t+uu_x)_x=1/2u _x ² with the simplest initial data such that u _x blows up in finite time. We prove that the zero-viscosity limit selects a unique global weak solution of the partial differential equation without viscosity. We also present numerical experiments which indicate that the zero-dispersion limit selects a different global weak solution of the same initial-value problem.