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, and Tadmor [J. Amer. Math. Soc., 7 (1994), pp. 169--192] 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 these authors in Sobolev spaces cannot be exceeded for all smooth nonlinear fluxes.