On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system

In this paper, we study the relativistic Vlasov-Fokker-Planck-Maxwell system in one space variable and two momentum variables. This non-linear system of equations consists of a transport equation for the phase space distribution function combined with Maxwell's equations for the electric and magnetic fields. It is important in modelling distribution of charged particles in the kinetic theory of plasma. We prove the existence of a classical solution when the initial density decays fast enough with respect to the momentum variables. The solution which shares this same decay condition along with its first derivatives in the momentum variables is unique.     

Analytic Solutions and Singularity Formation for the Peakon b-Family Equations

This paper deals with the well-posedness of the b-family equation in analytic function spaces. Using the Abstract Cauchy-Kowalewski theorem we prove that the b-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to H ^s with s>3/2, and the momentum density u ₀?u _0,xx does not change sign, we prove that the solution stays analytic globally in time, for b≥1. Using pseudospectral numerical methods, we study, also, the singularity formation for the b-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex plane as the singularity approaches the real axis, estimating the rate of decay of the Fourier spectrum.     

Particules collantes signées et lois de conservation scalaires 1DSigned sticky particles and 1D scalar conservation laws

We present an approximation of the entropy solution of a 1D scalar conservation law based on signed sticky particles when the variation of the initial condition is bounded. This method is a generalization of the one studied by Brenier and Grenier [2] in case the initial condition is monotone. When they collide, particles with the same sign stick together with conservation of the momentum whereas particles with opposite sign are destroyed. We prove the convergence of the approximate solution to the entropy solution when the initial number of particles goes to +∞. To cite this article: B. Jourdain, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 233–238.     

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

奇数维可压缩液晶流方程解的逐点估计

首先, 本文利用标准的能量估计方法得到高维(3 维及以上) 的液晶流方程组小初值经典解的整体存在性. 然后, 本文运用Green 函数方法, 得到奇数维情形(3 维及以上) 该解的逐点估计. 该结果表明, 密度ρ和动量m同Navier-Stokes 方程组一样满足一般Huygens 原理, 而单位向量场d则没有这种现象, 其有着与热方程的解类似的时空估计.     

Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

关于Camassa-Holm方程动量密度的紧支集在半轴上的估计

主要考虑在半轴上Camassa-Holm方程解的动量密度紧支集大小的估计,方法是根据区间长度与区间特征值的关系,通过估计第一Dirichlet特征值来估计动量密度紧支集的长度.因为知道动量密度紧支集外解的性态,所以通过估计动量密度支集的大小可以得到方程解的更多信息.     

On the dynamics of a fluid-particle interaction model: The bubbling regime

This article deals with the issues of global-in-time existence and asymptotic analysis of a fluid-particle interaction model in the so-called bubbling regime. The mixture occupies the physical space Ω⊂R³ which may be unbounded. The system under investigation describes the evolution of particles dispersed in a viscous compressible fluid and is expressed through the conservation of fluid mass, the balance of momentum and the balance of particle density often referred as the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually by the action-reaction principle. We show that solutions exist globally in time under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, we prove the large-time stabilization of the system towards a unique stationary state fully determined by the masses of the initial density of particles and fluid and the external potential.     

Optimal time-decay rates of the Boltzmann equation

We consider the optimal time-convergence rates of the global solution to the Cauchy problem for the Boltzmann equation in R3.We show that the global solution tends to the global Maxwellian at the optimal time-decay rate(1+t)-3/4,where the macroscopic density,momentum and energy decay at the optimal rate(1+t)-3/4 and the microscopic part decays at the optimal rate(1+t)-5/4.We also show that the solution tends to the Maxwellian at the optimal time-decay rate(1+t).5/4 in the case of the macroscopic part of the initial data is zero,where the macroscopic density,momentum and energy decay at the optimal rate(1+t)-5/4 and the microscopic part decays at the optimal rate(1+t)-7/4.These convergence rates are shown to be optimal for the Boltzmann equation.     

Exact solution of the one-dimensional time-dependent Schrödinger equation with a rectangular well/barrier potential and its applications

We obtain an exact one-dimensional time-dependent solution for a wave function ψ(x, t) of a particle moving in the presence of a rectangular well or barrier. We present the solution, which holds for both the well and the barrier, in terms of the integrals of elementary functions; it is the sum of forward- and backward-moving components of the wave packet. We consider and numerically visualize the relative contribution of these components and of their interference to the probability density |ψ(x, t)|² and the particle arrival time and dwell time for the narrow and broad energy (momentum) distributions of the initial Gaussian wave packet. We show that in the case of a broad initial wave packet, the quantum mechanical counterintuitive effect of the influence of the backward-moving components on the considered quantities becomes essential.     

Relativistic effects on information measures for hydrogen-like atoms

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p. Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.     

Unicité de la solution faible à support compact de l’équation de Vlasov-Poisson

We study here the 3-dimensional Vlasov-Poisson equation of stellar dynamics. It is well known that this equation has weak solutions for every bounded initial density with finite kinetic energy. In [6], Lions and Perthame prove a uniqueness result under a Lipschitz continuity assumption on the initial datum. Using the moment estimates of [6], we can easily see that if the initial datum is compactly supported, the solution will remain compactly supported for ever. We prove here the uniqueness of the compactly supported weak solution. Our proof is an adaptation of that of Youdovitch (see [8]) for the 2-dimensional Euler equation.     

A Kerr metric solution in tetrad theory of gravitation

Using an axial parallel vector field we obtain two exact solutions of a vacuum gravitational field equations. One of the exact solutions gives the Schwarzschild metric while the other gives the Kerr metric. The parallel vector field of the Kerr solution have an axial symmetry. The exact solution of the Kerr metric contains two constants of integration, one being the gravitational mass of the source and the other constant h is related to the angular momentum of the rotating source, when the spin density S_ij^μ of the gravitational source satisfies ∂_μS_ij^μ=0. The singularity of the Kerr solution is studied.     

带自由边界的可压缩欧拉与欧拉-泊松方程组径向对称解的爆破北大核心CSCD

本文在R^(N)(N=2,3)中研究描述流向外部真空的可压缩流体的欧拉与欧拉-泊松方程组径向对称解的爆破.在分离流体与真空的连续自由边界条件下考虑其自由边值问题.对于径向对称的欧拉方程组,证明若初始流平均向外流动,则其光滑解将在有限时刻爆破.对于带有斥力与弛豫项的单极与双极径向对称欧拉-泊松方程组,证明若某个与初始动量有关的加权泛函适当大,则其光滑解将在有限时刻爆破。     

Relaxation limit and initial layer to hydrodynamic models for semiconductors

We study a relaxation limit of a solution to the initial-boundary value problem for a hydrodynamic model to a drift-diffusion model over a one-dimensional bounded domain. It is shown that the solution for the hydrodynamic model converges to that for the drift-diffusion model globally in time as a physical parameter, called a relaxation time, tends to zero. It is also shown that the solutions to the both models converge to the corresponding stationary solutions as time tends to infinity, respectively. Here, the initial data of electron density for the hydrodynamic model can be taken arbitrarily large in the suitable Sobolev space provided that the relaxation time is sufficiently small because the drift-diffusion model is a coupled system of a uniformly parabolic equation and the Poisson equation. Since the initial data for the hydrodynamic model is not necessarily in “momentum equilibrium”, an initial layer should occur. However, it is shown that the layer decays exponentially fast as a time variable tends to infinity and/or the relaxation time tends to zero. These results are proven by the decay estimates of solutions, which are derived through energy methods.     

Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source

In this paper, a multidimensional nonisentropic hydrodynamic model for semiconductors with the nonconstant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. Global existence to the Cauchy problem for the multidimensional nonisentropic hydrodynamic semiconductor model with the small perturbed initial data is established, and the asymptotic behavior of these smooth solutions is investigated, namely, that the solutions converge to the general steady-state solution exponentially fast as t→+∞ is obtained. Moreover, the existence and uniqueness of the stationary solutions are investigated.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

Evolutions of the momentum density,deformation tensor and the nonlocal term of the Camassa–Holm equation

Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Dynamical behaviors for 1D compressible Navier-Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T_∗>0, which is compressed at an algebraic rate and vanishes at the time T_∗, meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.