Global existence for multi-dimensional partially diffusive systems
Résumé
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 Navier-Stokes-Fourier equations. def = d α,β=1 Y αβ (U )ω α ω β is a 2010 Mathematics Subject Classification. 35M11, 35Q30. 76N10.
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