Raphäel Danchin | Searcharxiv

arXiv2025

Global existence for multi-dimensional partially diffusive systems

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.

math.AP↗

arXiv2024

Local well-posedness in the critical regularity setting for hyperbolic systems with partial diffusion

This paper is dedicated to the local existence theory of the Cauchy problem for a general class of symmetrizable hyperbolic partially diffusive systems (also called hyperbolic-parabolic systems) in the whole space $\mathbb{R}^d$ with $d\ge 1$. We address the question of well-posedness for large data having critical Besov regularity in the spirit of previous works by the second author on the compressible Navier-Stokes equations. Compared to the pioneering of Kawashima (S. Kawashima. Systems of a hyperbolic parabolic type with applications to the equations of magnetohydrodynamics. PhD thesis, Kyoto University, 1983) and to the more recent work by Serre (D. Serre. Local existence for viscous system of conservation laws: $H^s$-data with $s > 1+d/2$. In Nonlinear partial differential equations and hyperbolic wave phenomena, volume 526 of Contemp. Math., pages 339-358. Amer. Math. Soc., Providence, RI, 2010), we take advantage of the partial parabolicity of the system to consider data in functional spaces that need not be embedded in the set of Lipschitz functions. This is in sharp contrast with the classical well-posedness theory of (multi-dimensional) hyperbolic systems where it is mandatory. A leitmotiv of our analysis is to require less regularity for the components experiencing a direct diffusion, than for the hyperbolic components. We then use an energy method that is performed on the system after spectral localization and a suitable Gårding inequality. As an example, we consider the Navier-Stokes-Fourier equations.

math.AP↗