Generic IllPosedness for Wave Equation of Power Type on Three-Dimensional Torus

Abstract

In this article, we prove that the equationThis work also indicates that, only properly regularizing the initial data can we smoothly approximate the solutions constructed in [2] and [12].

On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs

“…When s ≥ 1 2 , local well-posedness of (3.1) in H s (T 3 ) follows from a standard fixed point argument with the (deterministic) Strichartz estimates. Moreover, the equation (3.1) is known to be ill-posed in H s (T 3 ) for s < 1 2 [26,21,90]. In the following, we take initial data (u 0 ,…”

“…(v) Unlike the usual deterministic theory, the approximation property of the random solution u ω constructed in Theorem 3.1 by smooth solutions crucially depends on a method of approximation. On the one hand, Xia [90] showed that the solution map: (u 0 , u 1 ) → u for (3.1) is discontinuous everywhere in H s (T 3 ) when s < 1 2 . This in particular shows that the solution map for (3.1), a priori defined on smooth functions, does not extend continuously to rough functions, including the case of the random initial data (u ω 0 , u ω 1 ) considered in Theorem 3.1.…”

“…Moreover, the limit is independent of the mollification kernel η. See [88,90,64]. (vi) In [63], Oh-Okamoto-Pocovnicu proved almost sure local well-posedness of the following NLS without gauge invariance:…”

“…On the other hand, it is known that (1.1) and (1.2) are ill-posed in the supercritical regime: s < s crit . See [26,21,48,76,90,62,24,64]. Regardless of these ill-posedness results, by considering random initial data (see Section 2), one may still prove almost sure local well-posedness 3 in the supercritical regime.…”

Almost sure global well posedness for the BBM equation with infinite <inline-formula><tex-math id="M1">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula> initial data

“…Norm inflation based at general initial data has been studied for NLW [50,47] and NLS [38] in negative Sobolev spaces. We establish norm inflation at any u 0 ∈ H s (M), with s < 0, for the BBM equation 1.…”