Global existence for multi-dimensional partially diffusive systems

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 [1], and second, to refine and enhance the analysis of Kawashima [15].

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 [3] and recent contributions [8], [7], [6]. 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 barotropic compressible Navier-Stokes equations.