The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations

Abstract

Le problème de Cauchy pour leséquations d'ondeà coefficients non Lipschtziens; application au prolongement de solutions d'équations d'ondes non linéaires. AbstractIn this paper we study the Cauchy problem for second order strictly hyperbolic operators of the formwhen the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant version of a previous paper of the first author with N.Lerner [6]. We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients above depend on u: the smooth solution can be extended as long as it remains Log-Lipschitz.

A Note on Complete Hyperbolic Operators with log-Zygmund Coefficients

“…These logarithmic Sobolev spaces were introduced in [11]. Due to the low regularity of the coefficients of L, this kind of spaces will come into play in our computations.…”

“…(here we set y = (t, x) ∈ R t × R N x ), was considered by Colombini and Métivier in [11]. They assumed the same isotropic log-Lipschitz condition of [10] on the coefficients of the second order part of L, while b j and c j were supposed to be α-Hölder continuous (for some α ∈ ]1/2, 1[ ) and d to be only bounded.…”

“…This means that we will focus on log-Lipschitz continuous a jk 's and Hölder continuous b j 's, while the 0-th order coefficient c can be taken just bounded (see also [10], [11] and [9] about these hypothesis).…”

“…Let us note that if a = a(x) ∈ L ∞ and if we take the cut-off function ψ 1 , then T a is actually the usual paraproduct operator. If we take ψ γ as defined in (3.1), instead, we get a paraproduct operator which starts from high enough frequencies, which will be indicated with T µ a (see [11,Par. 3.3]).…”

“…In particular, according with the results of [10], [11] and [9], we will assume the first order coefficients to be Hölder continuous in the space variable, and the zero order coefficient to be just bounded. From an energy estimate analogous to (1.6), we will infer, for smooth in space coefficients, the H ∞ well-posedness of the related Cauchy problem with a finite loss of derivatives.…”