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Global existence for multi-dimensional partially diffusive systems

Jean-Paul Adogbo, Raphäel Danchin

TL;DR

This work develops a global existence theory for multi-dimensional partially diffusive hyperbolic systems in critical Besov spaces by combining a hybrid low/high-frequency framework with a Beauchard–Zuazua–style Lyapunov functional. A parabolic mode is introduced to reveal smoothing in the diffusive components, while a transfer mechanism links diffusion-dominated low frequencies to hyperbolic components, yielding decay under the Shizuta–Kawashima condition and Kalman-type rank criteria. The authors prove global-in-time existence and decay for small data in the critical Besov setting, extend results to an L2-based critical regime under structural Assumption E, and apply the theory to magnetohydrodynamics and Navier–Stokes–Fourier equations. The MHD application demonstrates global well-posedness near equilibrium with explicit decay, highlighting the framework’s relevance to complex coupled hyperbolic–parabolic PDEs arising in fluid and plasma physics. Overall, the paper advances global well-posedness results for partially dissipative systems by exploiting frequency splitting, parabolic smoothing, and structural hypotheses to obtain sharp existence, uniqueness, and decay results.

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.

Global existence for multi-dimensional partially diffusive systems

TL;DR

This work develops a global existence theory for multi-dimensional partially diffusive hyperbolic systems in critical Besov spaces by combining a hybrid low/high-frequency framework with a Beauchard–Zuazua–style Lyapunov functional. A parabolic mode is introduced to reveal smoothing in the diffusive components, while a transfer mechanism links diffusion-dominated low frequencies to hyperbolic components, yielding decay under the Shizuta–Kawashima condition and Kalman-type rank criteria. The authors prove global-in-time existence and decay for small data in the critical Besov setting, extend results to an L2-based critical regime under structural Assumption E, and apply the theory to magnetohydrodynamics and Navier–Stokes–Fourier equations. The MHD application demonstrates global well-posedness near equilibrium with explicit decay, highlighting the framework’s relevance to complex coupled hyperbolic–parabolic PDEs arising in fluid and plasma physics. Overall, the paper advances global well-posedness results for partially dissipative systems by exploiting frequency splitting, parabolic smoothing, and structural hypotheses to obtain sharp existence, uniqueness, and decay results.

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.
Paper Structure (26 sections, 22 theorems, 248 equations)

This paper contains 26 sections, 22 theorems, 248 equations.

Table of Contents

  1. Assumptions and approaches
  2. Normal form
  3. Assumption D
  4. The Shizuta-Kawashima and Kalman rank conditions
  5. Parabolic mode
  6. Transfer of integrability from the parabolic to the hyperbolic mode
  7. Main results
  8. Assumption E
  9. Proof of Theorem \ref{['thm:glob:cri']}
  10. Establishing the a priori estimates
  11. $\bigstar$ Estimates for $\partial_t V$:
  12. $\bigstar$ Low frequencies analysis:
  13. $\bigstar$ High frequencies analysis:
  14. Proving the existence and uniqueness parts of Theorem \ref{['thm:glob:cri']}
  15. Proof of Theorem \ref{['thm:decay:sucri']}
  16. ...and 11 more sections

Key Result

Lemma 1.1

Let $\mathcal{N}_\omega= S^{-1}\mathcal{A}_\omega$ and $\mathcal{M}_\omega=S^{-1} \mathcal{B}_\omega$ for all $\omega\in\mathbb S^{d-1}.$ There exist positive parameters $\varepsilon_0,\cdots\varepsilon_{n-1}$ (that are defined inductively and can be taken arbitrarily small) and a Lyapunov functiona such that the following inequalities hold for some positive $c$ and $C,$ and all $\rho>0$ and $\ome

Theorems & Definitions (37)

  • Remark 1.1
  • Lemma 1.1
  • Definition 1.1
  • Lemma 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • ...and 27 more