Timothée Crin-Barat

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• PhD
• Maître de Conférences at Toulouse Mathematics Institute

We study two relaxation problems in the class of partially dissipative hyperbolic systems: the compressible Euler system with damping and the compressible Euler-Maxwell system. In classical Sobolev spaces, we derive a global convergence rate of $\mathcal{O}(\varepsilon)$ between strong solutions of the relaxed Euler system and the porous medium equ...

We investigate the time-asymptotic stability of the Jin-Xin model and its diffusive relaxation limit toward viscous conservation laws in ℝd for d ⩾ 1. First, we establish a priori estimates that are uniform with respect to both the time and the relaxation parameter ε > 0, for initial data in hybrid Besov spaces based on Lp-norms. This uniformity en...

We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fu...

We study the time-asymptotic behavior of linear hyperbolic systems subject to partial dissipation that is localized in suitable subsets of the domain. Specifically, we recover the classical decay rates of partially dissipative systems that satisfy the stability condition (SK), with a time-delay that depends only on the velocity of each component an...

We characterize the Leray--Hopf solutions of the 2D inhomogeneous Navier--Stokes system that become strong for positive times. This characterization relies on the strong energy inequality and the regularity properties of the pressure. As an application, we establish a weak-strong uniqueness result and provide a unified framework for several recent...

We present a weak–strong uniqueness result for the inhomogeneous Navier–Stokes equations in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document...

We investigate the three-dimensional compressible Euler-Maxwell system, a model for simulating the transport of electrons interacting with propagating electromagnetic waves in semiconductor devices. First, we show the global well-posedness of classical solutions being a sharp small perturbation of constant equilibrium in a critical regularity setti...

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta–Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for...

We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer–Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction f...

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in \(\dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)\) with \(s > 3\) and for any \(0< \tau <1\). Such a result improves upon the existing literature, where the asymptotic stability is...

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for...

We study the diffusive relaxation limit of the Jin-Xin system toward viscous conservation laws in the multi-dimensional setting. For initial data being small perturbations of a constant state in suitable homogeneous Besov norms, we prove the global well-posedness of strong solutions satisfying uniform estimates with respect to the relaxation parame...

In this paper, we study a singular limit problem for a compressible one-velocity bifluid system. More precisely, we show that solutions of the Kapila system generated by initial data close to equilibrium are obtained in the pressure-relaxation limit from solutions of the Baer–Nunziato (BN) system. The convergence rate of this process is a consequen...

We study the diffusive relaxation limit of the Jin-Xin system toward viscous conservation laws in the multi-dimensional setting. For initial data being small perturbations of a constant state in suitable homogeneous Besov norms, we prove the global well-posedness of strong solutions satisfying uniform estimates with respect to the relaxation parame...

In this paper we investigate two types of relaxation processes quantitatively in the context of small data global-in-time solutions for compressible one-velocity multi-fluid models. First, we justify the pressure-relaxation limit from a one-velocity Baer-Nunziato system to a Kapila model as the pressure-relaxation parameter tends to zero, in a unif...

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in $\dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and for any $0 < \tau <1$. Such result improves the existing literature, where the asymptotic stability is proved for initi...

Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper (Crin-Barat and Danchin in Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case...

We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole space Rd with d≥2. Following our recent work dedicated to the one-dimensional case [11], we establish the existence of global strong solutions and decay estimates in the critical regularity setting whenever the system under consideration satisfies...

We study the time-asymptotic behavior of linear hyperbolic systems under partial dissipation which is localized in suitable subsets of the domain. More precisely, we recover the classical decay rates of partially dissipative systems satisfying the stability condition (SK) with a time-delay depending only on the velocity of each component and the si...

Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper [10] to a functional framework where the low frequencies of the solution are only bounded in L p-type spaces with p larger than 2....

An Euler-type hyperbolic-parabolic system of chemotactic aggregation describing the vascular network formation is investigated in the critical regularity setting. For initial data near a constant equilibrium state, the global well-posedness of the classical solution to the Cauchy problem with general pressure laws is proved in critical hybrid Besov...

Cette thèse est consacrée à l'étude de la classe des systèmes hyperboliques partiellement dissipatifs satisfaisant la condition de stabilité de Shizuta-Kawashima (souvent appelée "condition (SK)'') et à un modèle multi-fluide compressible proche de cette classe mais ne vérifiant pas cette condition, le tout dans un cadre à régularité critique.Dans...

In this paper we study a singular limit problem in the context of partially dissipative first order quasilinear systems. This problem arises in multiphase fluid mechanics. More precisely, taking into account dissipative effects for the velocity, we show that the so-called Kapila system is obtained as a relaxation limit from the Baer-Nunziato (BN) s...

We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole space $\mathbb{R}^d$ with $d\geq2$. Following our recent work [10] dedicated to the one-dimensional case, we establish the existence of global strong solutions and decay estimates in the critical regularity setting whenever the system under conside...

Here we develop a method for investigating global strong solutions of partially dissipative hyperbolic systems in the critical regularity setting. Compared to the recent works by Kawashima and Xu, we use hybrid Besov spaces with different regularity exponent in low and high frequency. This allows to consider more general data and to track the exact...