$L^2$-contraction for shock waves of scalar viscous conservation laws

We consider the $L^2$-contraction up to a shift for viscous shocks of scalar viscous conservation laws with strictly convex fluxes in one space dimension. In the case of a flux which is a small perturbation of the quadratic burgers flux, we show that any viscous shock induces a contraction in $L^2$, up to a shift. That is, the $L^2$ norm of the difference of any solution of the viscous conservation law, with an appropriate shift of the shock wave, does not increase in time. If, in addition, the difference between the initial value of the solution and the shock wave is also bounded in $L^1$, the $L^2$ norm of the difference converges at the optimal rate $t^{-1/4}$. Both results do not involve any smallness condition on the initial value, nor on the size of the shock. In this context of small perturbations of the quadratic Burgers flux, the result improves the Choi and Vasseur's result in [7]. However, we show that the $L^2$-contraction up to a shift does not hold for every convex flux. We construct a smooth strictly convex flux, for which the $L^2$-contraction does not hold any more even along any Lipschitz shift.