Concerning the pathological set in the context of probabilistic well-posedness

Sun, Chenmin; Tzvetkov, Nikolay

Comptes Rendus. Mathématique, Tome 358 (2020) no. 9-10, pp. 989-999.

On démontre un résultat complémentaire à ceux manifestant le caractère bien posé probabiliste de l’équation des ondes avec des données initiales de régularité de Sobolev super critique par rapport au changement d’échelle laissant invariant l’équation.

We prove a complementary result to the probabilistic well-posedness for the nonlinear wave equation. More precisely, we show that there is a dense set $S$ of the Sobolev space of super-critical regularity such that (in sharp contrast with the probabilistic well-posedness results) the family of global smooth solutions, generated by the convolution with some approximate identity of the elements of $S$, does not converge in the space of super-critical Sobolev regularity.

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