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Pré-Publication, Document De Travail Année : 2024

Sharp blow-up stability for self-similar solutions of the modified Korteweg-de Vries equation

Simão Correia
  • Fonction : Auteur
  • PersonId : 1045272
Raphaël Côte
  • Fonction : Auteur
  • PersonId : 181956
  • IdHAL : cote
  • IdRef : 119749351

Résumé

We consider the modified Korteweg-de Vries equation. Given a self-similar solution, and a subcritical perturbation of any size, we prove that there exists a unique solution to the equation which behaves at blow-up time as the self-similar solution plus the perturbation. To this end, we develop the first robust analysis in spaces of functions with bounded Fourier transforms. To begin, we prove the local well-posedness in subcritical spaces through an appropriate restriction norm method. As this method is not sufficient to capture the critical self-similar dynamics, we develop an infinite normal form reduction (INFR) to derive time-dependent a priori $L^\infty$ bounds in frequency variables. Both approaches rely on frequency-restricted estimates, which are specific positive multiplier estimates capable of capturing the oscillatory nature of the equation. As a consequence of our analysis, we also prove local well-posedness for small subcritical perturbations of self-similar solutions at positive time.
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Dates et versions

hal-04475652 , version 1 (23-02-2024)

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  • HAL Id : hal-04475652 , version 1

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Simão Correia, Raphaël Côte. Sharp blow-up stability for self-similar solutions of the modified Korteweg-de Vries equation. 2024. ⟨hal-04475652⟩
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