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

## TL;DR

This paper demonstrates that partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition can still have globally existing smooth solutions if the source term is nonresonant, using a novel coupling of Green function estimates and space-time resonance analysis.

## Contribution

It introduces a new approach to ensure global existence of solutions for nonresonant, partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition.

## Key findings

- Global smooth solutions exist despite lack of dissipation in some directions.
- Nonresonant source terms prevent singularity formation.
- Coupling Green function estimates with space-time resonance analysis is effective.

## Abstract

In the context of hyperbolic systems of balance laws, the Shizuta-Kawashima coupling condition guarantees that all the variables of the system are dissipative even though the system is not totally dissipative. Hence it plays a crucial role in terms of sufficient conditions for the global in time existence of classical solutions. However, it is easy to find physically based models that do not satisfy this condition, especially in several space dimensions. In this paper, we consider two simple examples of partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition ([SK]) in 3D, such that some eigendirections do not exhibit dissipation at all. We prove that, if the source term is non resonant (in a suitable sense) in the direction where dissipation does not play any role, then the formation of singularities is prevented, despite the lack of dissipation, and the smooth solutions exist globally in time. The main idea of the proof is to couple Green function estimates for weakly dissipative hyperbolic systems with the space-time resonance analysis for dispersive equations introduced by Germain, Masmoudi and Shatah. More precisely, the partially dissipative hyperbolic systems violating [SK] are endowed, in the non-dissipative directions, with a special structure of the nonlinearity, the so-called Nonresonant Bilinear Form for the wave equation (see Pusateri and Shatah, CPAM 2013).