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Lions, Perthame, Tadmor conjectured in 1994 an optimal smoothing effect forentropy solutions of nonlinear scalar conservations laws . In this short paperwe will restrict our attention to the simpler one-dimensional case. First,supercritical geometric optics lead to sequences of $C^\infty$ solutionsuniformly bounded in the Sobolev space conjectured. Second we give continuoussolutions which belong exactly to the suitable Sobolev space. In order to do sowe give two new definitions of nonlinear flux and we introduce fractional $BV$spaces.

Oscillating waves and optimal smoothing effect for one-dimensional nonlinear scalar conservation laws