Preprint 2015-012
High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws
Stephane Junca
Abstract: The article first studies the propagation of well prepared high frequency waves with small amplitude ε near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, ([23]), about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a definition of smooth nonlinear flux is stated and compared to classical definitions. Then it is proved that the uniform smoothness expected by [23] in Sobolev spaces cannot be exceeded for all smooth nonlinear fluxes.