Occurrence and non-appearance of shocks in fractal Burgers equations

We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently "large" initial conditions, by giving a result which states that, for smooth "small" initial data, the solution remains at least Lipschitz continuous.