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

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, 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.1. Introduction. This paper deals with supercritical geometric optics to highlight the maximal regularizing effect for nonlinear multidimensional scalar conservation laws. This effect is studied in Sobolev spaces by Lions, Perthame, and Tadmor in [23]. They obtain a uniform fractional Sobolev bounds for any ball of L ∞ initial data under a nonlinearity condition on the flux quantified by a parameter α ∈ ]0, 1]. They also conjectured the better Sobolev exponent s = α for entropy solutions.For the first time, the multidimensional case is investigated to bound this maximal smoothing effect. Furthermore, all smooth nonlinear fluxes are considered in this paper. We build sequences of solutions uniformly bounded in W s,1 loc with the conjectured maximal exponent s = α and with no possible improvement of the Sobolev exponent.High frequency periodic solutions are used for this purpose. Near a constant state and for L ∞ data, geometric optics expansions with various frequencies and various phases are validated in the framework of weak entropy solutions and of L 1

Fractional BV spaces and applications to scalar conservation laws

“…Since b ∈ C s , we have |b(z)| = |b(z) − b(c)| ≤ D s |z − c| s . The Lax-Oleinik formula (14) and Inequality (18) conclude the proof:…”

“…It suffices to study the case u < v by symmetry: with v = u + h, h > 0, We say that the flux is nonlinear if p is finite. This general example has been studied recently for the multidimensional case in [1,14]. These examples allow to compute the parameter of degeneracy of any smooth flux given in the paper of P.-L. Lions, B. Perthame and E. Tadmor [18].…”

“…The best parameter s quantifying the smoothing effect is not known in the multidimensional case. It is improved in [25] and bounded in [9,14]. For the one dimensional case, the best smoothing effect in W s,1 conjectured in [18] was first proved in [13].…”

On the maximal smoothing effect for multidimensional scalar conservation laws

“…In the one-dimensional case, De Lellis and Westdickenberg showed in 2003 that s ≤ α for power-law convex uxes ( [11]) and Jabin showed in 2010 that s = α for C 2 uxes under a generalized Oleinik condition ( [13]). For a nonlinear multidimensional smooth ux the parameter α is determined explicitly in [16] with an equivalent denition of nonlinearity recalled in Section 2 below. In particular the parameter α depends on the space dimension n and satises: α ≤ 1 n .…”

“…In particular the parameter α depends on the space dimension n and satises: α ≤ 1 n . Moreover, Denition 4 naturally yields the construction of a supercritical family of oscillating smooth solutions -on a bounded time before shocks-exactly uniformly bounded in the optimal Sobolev space conjectured ( [16]). In this paper:…”