Related papers: Global Mild Solutions to a BGK Model for Barotropi…
We use guiding principles from nonequilibrium thermodynamics to develop an admissible set of 13 moment equations for rarefied gas flows. The main benefits of our thermodynamic approach are an explicit entropy expression fulfilling an $H$…
We consider the (complete) Euler system describing the motion of a compressible perfect fluid. We propose a platform suitable for constructing the statistical solutions. The main ingredients of our approach include: 1. The concept of…
We investigate the initial-value problem for the relativistic Euler equations governing isothermal perfect fluid flows, and generalize an approach introduced by LeFloch and Shelukhin in the non-relativistic setting. We establish the…
We show how to obtain general nonlinear aggregation-diffusion models, including Keller-Segel type models with nonlinear diffusions, as relaxations from nonlocal compressible Euler-type hydrodynamic systems via the relative entropy method.…
Entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics. Through a modified Glimm scheme, an approximate solution sequence is constructed, and then a priori estimates are…
We consider a class of nonlinear, spatially inhomogeneous kinetic equations of BGK-type with density dependent collision rates. These equations share the same superlinearity as the Boltzmann equation, and fall into the class of run and…
We investigate the global existence and exponential decay of mild solutions for the Boussinesq systems in $L^p$-phase spaces on the framework of real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$, where $d \geqslant 2$ and $1<p\leq d$. We…
Multi-species BGK models describe the dynamics of rarefied gases with constituent particles of different elements or compounds with potentially non-trivial velocity distributions. In this paper, moment equations for the bulk velocities,…
This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits.…
In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [4] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in a open…
We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $L^r_v L^\infty_t L^\infty_x $ spaces where $r \in (1,\infty]$ by using the excess…
We study hypocoercivity for a class of linear and linearized BGK models for discrete and continuous phase spaces. We develop methods for constructing entropy functionals that prove exponential rates of relaxation to equilibrium. Our…
This minicourse contains a description of recent results on the modelling of rarefied gases in weakly non equilibrium regimes, and the numerical methods used to approximate the resulting equations. Therefore this work focuses on BGK type…
We study BV solutions for a $2\times2$ system of hyperbolic balance laws. We show that when initial data have small total variation on $(-\infty,\infty)$ and small amplitude, and decay sufficiently fast to a constant equilibrium state as…
Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical…
We prove the existence of global weak solutions with finite energy to some two-fluid systems with magnetic field and the results suit for corresponding two-fluid systems. The proof method is mainly inspired by Novotn\'y et al. and Vasseur…
We establish the global existence of weak entropy solutions for 1D isentropic gas dynamics with general pressure laws ($\gamma > 1$). To address vacuum degeneracy, we introduce a novel structural regularization via a "Synchronized Dual…
Generalized hydrodynamics (GHD) was proposed recently as a formulation of hydrodynamics for integrable systems, taking into account infinitely-many conservation laws. In this note we further develop the theory in various directions. By…
An exact closure for hydrodynamic variables is rigorously derived from the linear Boltzmann kinetic equation. Our approach, based on spectral theory, structural properties of eigenvectors and the theory of slow manifolds, allows us to…
The generalized hydrodynamics (GHD) equation is the equivalent of the Euler equations of hydrodynamics for integrable models. Systems of hyperbolic equations such as the Euler equations usually develop shocks and are plagued by problems of…