arXiv subjects

Bernard Ducomet

Publications and source records attributed to Bernard Ducomet.

13 recordsLinked to original sources

arXiv2025

On an Euler-Schr\"{o}dinger system appearing in laser-plasma interaction

We consider the Cauchy problem for the barotropic Euler system coupled to a vector Schr\"{o}dinger equation in the whole space. Assuming that the initial density and vector potential are small enough, and that the initial velocity is close to some reference vector field $u_0$ such that the spectrum of $Du_0$ is bounded away from zero, we prove the existence of a global-in-time unique solution with (fractional) Sobolev regularity. Moreover, we obtain some algebraic time decay estimates of the solution. Our work extends the papers by D. Serre and M. Grassin [11, 13, 19] and previous works by B. Ducomet and co-authors [4, 8] dedicated to the compressible Euler-Poisson system.

math.AP↗

arXiv2024

Global solutions of Euler-Maxwell equations with dissipation

We consider the Cauchy problem for a damped Euler-Maxwell system with no ionic background. For smooth enough data satisfying suitable so-called dispersive conditions, we establish the global in time existence and uniqueness of a strong solution that decays uniformly in time. Our method is inspired by the works of D. Serre and M. Grassin dedicated to the compressible Euler system.

math.AP↗

arXiv2020

Low Mach and thin domain limit for the compressible Euler system

We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer $Ω_δ=(0,δ)\times\mathbb{R}^2, \ \ δ>0$. In the framework of dissipative measure-valued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and $δ\rightarrow0$.

math.AP↗

arXiv2019

The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field u0 such that the spectrum of Du0 is bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends the papers by D. Serre and M. Grassin [20, 21, 43] dedicated to the compressible Euler system without coupling and integer regularity exponents.

math.AP↗

arXiv2017

Global existence of a radiative Euler system coupled to an electromagnetic field

We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely the 3D radiative compressible Euler system coupled to an electromagnetic field. Assuming smallness hypotheses for the data, we prove that the problem admits a unique global smooth solution and study its asymptotics.

math.AP↗

arXiv2017

Derivation of the Navier - Stokes - Poisson system with radiation for an accretion disk

We study the 3-D compressible barotropic radiation fluid dynamics system describing the motion of the compressible rotating viscous fluid with gravitation and radiation confined to a straight layer. We show that weak solutions in the 3-D domain converge to the strong solution of the rotating 2-D Navier-Stokes-Poisson system with radiation for all times less than the maximal life time of the strong solution of the 2-D system when the Froude number is small or to the strong solution of the rotating pure 2-D Navier- Stokes system with radiation.

math.AP↗

arXiv2016

The rotating Navier- Stokes- Fourier- Poisson system on thin domains

We consider the compressible Navier - Stokes - Fourier - Poisson system describing the motion of a viscous heat conducting rotating fluid confined to a straight layer $ Ω_ε = ω\times (0,ε) $, where $ω$ is a 2-D domain. The aim of this paper is to show that the weak solutions in the 3D domain converge to the strong solution of the 2-D Navier - Stokes - Fourier - Poisson system $ω$ as $ε\to 0$ on the time interval, where the strong solution exists. We consider two different regimes in dependence on the asymptotic behaviour of the Froude number.

math.AP↗

arXiv2015

Weak and strong solutions of equations of compressible magnetohydrodynamics

This article proposes a review of the analysis of the system of magnetohydrodynamics (MHD). First, we give an account of the modelling asumptions. Then, the results of existence of weak solutions, using the notion of renormalized solutions. Then, existence of strong solutions in the neighbourhood of equilibrium states is reviewed, in particular with the method of Kawashima and Shizuta. Finally, the special case of dimension one is highlighted : the use of Lagrangian coordinates gives a simpler system, which is solved by standard techniques.

math.AP↗

arXiv2015

Diffusive limits for a barotropic model of radiative flow

Here we aim at justifying rigorously different types of physically relevant diffusive limits for radiative flows. For simplicity, we consider the barotropic situation, and adopt the so-called P1-approximation of the radiative transfer equation. In the critical functional framework, we establish the existence of global-in-time strong solutions corresponding to small enough data, and exhibit uniform estimates with respect to the coefficients of the system. Combining with standard compactness arguments, this enables us to justify rigorously the convergence of the solutions to the expected limit systems. Our results hold true in the whole space as well as in a periodic box in dimension n $\ge$ 2.

math.AP↗

arXiv2013

A splitting higher order scheme with discrete transparent boundary conditions for the Schr\"odinger equation in a semi-infinite parallelepiped

An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schr\"odinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a P\"{o}schl-Teller-like potential-barrier and a rectangular potential-well are also included.

math.NA↗

arXiv2013

The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schr\"odinger equation on a semi-infinite strip

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrodinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method.

math.NA↗

arXiv2012

On a regularization of the magnetic gas dynamics system of equations

A brief derivation of a specific regularization for the magnetic gas dynamic system of equations is given in the case of general equations of gas state (in presence of a body force and a heat source). The entropy balance equation in two forms is also derived for the system. For a constant regularization parameter and under a standard condition on the heat source, we show that the entropy production rate is nonnegative.

math.AP↗

arXiv2002

Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state

We consider the Navier-Stokes system describing motions of viscous compressible heat-conducting and "self-gravitating" media. We use the state function of the form $p(η,θ)=p_0(η)+p_1(η)θ$ linear with respect to the temperature $θ$, but we admit rather general nonmonotone functions $p_0$ and $p_1$ of $η$, which allows us to treat various physical models of nuclear fluids (for which $p$ and $η$ are the pressure and specific volume) or thermoviscoelastic solids. For an associated initial-boundary value problem with "fixed-free" boundary conditions and possibly large data, we prove a collection of estimates independent of time interval for solutions, including two-sided bounds for $η$, together with its asymptotic behaviour as $t\to \infty$. Namely, we establish the stabilization pointwise and in $L^q$ for $η$, in $L^2$ for $θ$, and in $L^q$ for $v$ (the velocity), for any $q\in[2,\infty)$.

math-ph↗