Random data Cauchy theory for supercritical wave equations

In the study of nonlinear evolutionary PDE’s, one often encounters the presence of a critical threshold for the well-posedness theory. A typical situation is to have a method showing well-posedness in Sobolev spaces Hs where s is greater than a critical index scr. This index is often related to a scale invariance (leading to solutions concentrating at a point of the space-time) of the considered equation. In some cases (but not all), a good local well-posedness theory is valid all the way down to the scaling regularity. On the other hand, at least in the context of nonlinear dispersive equations, no reasonable local well-posedness theory is known for any supercritical equation, i.e. for data having less regularity than the scaling one. In fact, recently, several methods to show ill-posedness, or high frequency instabilities, for s < scr emerged (see the works by Burq, Gérard and Tzvetkov [4, 3], Lebeau [6] and Christ Colliander and Tao [5]). The goal of this paper is to give a class of equations for which, using probabilistic arguments, one can still obtain a suitable well-posedness theory below the critical threshold. Our model will be the cubic nonlinear wave equation posed on a compact manifold. Let (M, g) be a three dimensional compact smooth riemannian manifold (without boundary) and let ∆ be the Laplace-Beltrami operator associated to the smooth met-ric g. For s ∈ R, we denote by Hs(M) the classical Sobolev space equipped with the norm ‖u‖Hs(M) = ‖(1−∆)s/2u‖L2(M). Consider the following cubic wave equation (0.1) (∂2t −∆)u+ u3 = 0, (u, ∂tu)|t=0 = (f1, f2) with real valued initial data (f1, f2) ≡ f ∈ Hs(M)×Hs−1(M) ≡ Hs(M). Using Strichartz estimates for the free evolution (see Section 2) one can show that for s> 1/2 the Cauchy problem (0.1) is locally well-posed for data in Hs(M). This means that for every f ∈ Hs(M) there exists T> 0 and a unique solution u of (0.1), in a suitable class, such that (u, ut) ∈ C([0, T];Hs(M)), i.e. the solution u represents a continuous curve in Hs(M) (we call such a solution strong solution since the classical construction of weak solutions does not yield the continuity in time). Moreover, we can show that the time existence T may be chosen the same for all f belonging to a fixed bounded set B of Hs(M) and the map f 7 → (u, ut) is continuous (and even Lipschitz continuous) from B t