High Frequency Waves and the Maximal Smoothing Effect for Nonlinear Scalar Conservation Laws

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 new 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. Key-words: multidimensional conservation laws, nonlinear smooth flux, geometric optics, Sobolev spaces, smoothing effect. Mathematics Subject Classification: Primary: 35L65, 35B65; Secondary: 35B10, 35B40, 35C20.