Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition

We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-Kawashima condition, i.e. such that some eigendirections do not exhibit dissipation at all. In the space-time res- onances framework introduced by Germain, Masmoudi and Shatah, we prove that, when the source term has a Nonresonant Bilinear Form, as proposed by Pusateri and Shatah CPAM 2013, the formation of singular- ities is prevented, despite the lack of dissipation. This allows us to show that smooth solutions to this preliminary case-study model exist globally in time.

Roberta Bianchini (1) , Roberto Natalini (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"

We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-Kawashima condition, i.e. such that some eigendirections do not exhibit dissipation at all. In the space-time resonances framework introduced by Germain, Masmoudi and Shatah, we prove that, when the source term has a Nonresonant Bilinear Form, as proposed by Pusateri and Shatah CPAM 2013, the formation of singular-ities is prevented, despite the lack of dissipation. This allows us to show that smooth solutions to this preliminary case-study model exist globally in time.

Mots clés: Partially dissipative hyperbolic systems; Shizuta-Kawashima condition; space-time resonances; null condition

Domaines: Equations aux dérivées partielles [math.AP]

Roberta Bianchini, Roberto Natalini. Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition. [Research Report] UMPA. 2019. ⟨hal-02150020⟩