Global existence for multi-dimensional partially diffusive systems

TL;DR

This paper establishes the global existence of strong solutions for multi-dimensional partially diffusive hyperbolic systems using advanced Besov space techniques and Lyapunov functionals, extending previous local results and refining analysis methods.

Contribution

It introduces a hybrid Besov norm approach and a parabolic mode to analyze interactions between low and high frequency regimes, advancing the understanding of partially diffusive hyperbolic systems.

Findings

Proves global existence of solutions in critical Besov spaces.

Develops a hybrid norm method for frequency regime analysis.

Introduces a parabolic mode for high-frequency smoothing.

Abstract

In this work, we explore the global existence of strong solutions for a class of partially diffusive hyperbolic systems within the framework of critical homogeneous Besov spaces. Our objective is twofold: first, to extend our recent findings on the local existence presented in J.-P. Adogbo and R. Danchin. Local well-posedness in the critical regularity setting for hyperbolic systems with partial diffusion. arXiv:2307.05981, 2024, and second, to refine and enhance the analysis of Kawashima (S. Kawashima. Systems of a hyperbolic parabolic type with applications to the equations of magnetohydrodynamics. PhD thesis, Kyoto University, 1983). To address the distinct behaviors of low and high frequency regimes, we employ a hybrid Besov norm approach that incorporates different regularity exponents for each regime. This allows us to meticulously analyze the interactions between these regimes, which exhibit fundamentally different dynamics. A significant part of our methodology is based on the study of a Lyapunov functional, inspired by the work of Beauchard and Zuazua (K. Beauchard and E. Zuazua. Large time asymptotics for partially dissipative hyperbolic system. Arch. Rational Mech. Anal, 199:177-227, 2011.) and recent contributions (T. Crin-Barat and R. Danchin. Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case. J. Math. Pures Appl. (9), 165:1-41, 2022). To effectively handle the high-frequency components, we introduce a parabolic mode with better smoothing properties, which plays a central role in our analysis. Our results are particularly relevant for important physical systems, such as the magnetohydrodynamics (MHD) system and the Navier-Stokes-Fourier equations.