An Oleinik-type estimate for a convection–diffusion equation and convergence to N-waves

In this article we propose an Oleinik-type estimate for sign-changing solutions to a convection–diffusion equationut U1+(|u|^(gamma)/(gamma))x=(mu)ux(gamma), u(x,0)=u0(x), u,x(set membership)R, 1<(gamma)<2, [(mu),t]>0. Since the Oleinik entropy inequality holds for nonnegative solutions or inviscid case ((mu)=0) only, the theoretical progress for the case was limited.