For fractional Laplacian order 1 < alpha < 2, solutions of the scaled fractal Burgers equation converge in L^2 to the entropy shock, with a shift and a convergence rate that tends to zero.
$L^2$-type contraction for shocks of scalar viscous conservation laws with strictly convex flux
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abstract
We study the $L^2$-type contraction property of large perturbations around shock waves of scalar viscous conservation laws with strictly convex fluxes in one space dimension. The contraction holds up to a shift, and it is measured by a weighted related entropy, for which we choose an appropriate entropy associated with the strictly convex flux. In particular, we handle shocks with small amplitude. This result improves the recent article [18] of the author and Vasseur on $L^2$-contraction property of shocks to scalar viscous conservation laws with a special flux, that is almost the Burgers flux.
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Inviscid limit to the shock waves for the fractal Burgers equation
For fractional Laplacian order 1 < alpha < 2, solutions of the scaled fractal Burgers equation converge in L^2 to the entropy shock, with a shift and a convergence rate that tends to zero.