Global existence for multi-dimensional partially diffusive systems

doi:10.1016/j.jde.2025.113596

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Global existence for multi-dimensional partially diffusive systems

2025-11-01

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Jean-Paul Adogbo

Raphaël Danchin

DOI：10.1016/j.jde.2025.113596

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Abstract

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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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Keywords:

Hyperbolic-parabolic systems

Partial diffusion

Critical regularity

Global well-posedness