On the maximal smoothing effect for multidimensional scalar conservation laws

In 1994, Lions, Perthame and Tadmor conjectured an optimal smoothing effect for entropy solutions of multidimensional scalar conservation laws. This effect estimated in fractional Sobolev spaces is linked to the flux nonlinearity. We use a new definition of a nonlinear smooth flux in order to show that the conjectured smoothing effect cannot be exceeded. First one-dimensional solutions are considered in fractional BV spaces. Then the multidimensional case is handled with a monophase solution.