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

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arxiv 2001.10293 v4 pith:UDEHKAD6 submitted 2020-01-28 math.AP

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

classification math.AP
keywords probabilisticwell-posednessregularitysobolevspacesuper-criticalapproximatecomplementary
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abstract

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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