Long time behavior of the Fokker-Planck-Boltzmann equation

This paper is devoted to the existence and long time behavior of the global classical solution to Fokker-Planck-Boltzmann equation with initial data near the absolute Maxwellian.     

On the trend to equilibrium for the Vlasov-Poisson-Boltzmann equation

The dynamics of dilute electrons and plasma can be modeled by Vlasov-Poisson-Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is O(t−∞), by using a method developed for the Boltzmann equation without external force in [L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 (2005) 245-316]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro-micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components.     

Regularizing effects for the classical solutions to the Landau equation in the whole space

The Landau equation describes the binary collisional effects (through long range coulombian interaction) in a plasma. In this paper, we prove that the known classical solutions to the Landau equation near Maxwellian in the whole space have a regularizing effect in all (time, space and velocity) variables, that is, become immediately smooth with respect to all variables.     

A FLUID-PARTICLE MODEL WITH ELECTRIC FIELDS NEAR A LOCAL MAXWELLIAN WITH RAREFACTION WAVE

The paper is concerned with time-asymptotic behavior of solution near a local Maxwellian with rarefaction wave to a fluid-particle model described by the Vlasov-Fokker-Planck equation coupled with the compressible and inviscid fluid by Euler-Poisson equations through the relaxation drag frictions, Vlasov forces between the macroscopic and microscopic momentums and the electrostatic potential forces. Precisely, based on a new micro-macro decomposition around the local Maxwellian to the kinetic part of the fluid-particle coupled system, which was first developed in [16], we show the time-asymptotically nonlinear stability of rarefaction wave to the one-dimensional compressible inviscid Euler equations coupled with both the Vlasov-Fokker-Planck equation and Poisson equation.     

Existence of Infinite Energy Solution to the Inelastic Boltzmann Equation with External Force

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy. More precisely, under the assumptions on the bicharacteristic generated by external force, we prove the global existence of solution for small initial data compared to the local Maxwellian exp{–p|x – v|²}, which has infinite mass and energy.     

Asymptotic stability of the relativistic Maxwellian

Global classical solutions near the relativistic Maxwellian are constructed for the relativistic Boltzmann equation in both a periodic box and the whole space. For both cases, we are able to get the non‐negativity of the global solution under less restriction than in (Publ. Res. Inst. Math. Sci. 1993; 29 :301–347) on the scattering kernel. Copyright © 2006 John Wiley & Sons, Ltd.     

Global classical solutions to the initial value problem for the relativistic Landau equation

Global in time classical solutions near the relativistic Maxwellian are constructed for the relativistic Landau equation in the whole space. The construction of global solutions is based on refined energy analysis.     

Global existence and decay estimates of the Boltzmann equation with frictional force: The general results

In this paper, we give the existence theory and the optimal time convergence rates of the solutions to the Boltzmann equation with frictional force near a global Maxwellian. We generalize our previous results on the same problem for hard sphere model into both hard potential and soft potential case. The main method used in this paper is the classic energy method combined with some new time–velocity weight functions to control the large velocity growth in the nonlinear term for the case of interactions with hard potentials and to deal with the singularity of the cross-section at zero relative velocity for the soft potential case.     

Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman–Enskog picture of near-equilibrium flow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

Stability of the Equilibrium to the Boltzmann Equation with Large Potential Force

The Boltzmann equation with external potential force exists a unique equilibrium—local Maxwellian. The author constructs the nonlinear stability of the equilibrium when the initial datum is a small perturbation of the local Maxwellian in the whole space R~3. Compared with the previous result [Ukai, S., Yang, T. and Zhao, H.-J.,Global solutions to the Boltzmann equation with external forces, Anal. Appl.(Singap.), 3,2005, 157–193], no smallness condition on the Sobolev norm H~1 of the potential is needed in our arguments. The proof is based on the entropy-energy inequality and the L~2-L~∞ estimates.     

Solutions of the Boltzmann equation with infinite energy: Existence, stability and long time behavior

This paper is devoted to studying a class of solutions to the nonlinear Boltzmann equation having infinite kinetic energy, these solutions have an upper Maxwellian bound with infinite kinetic energy. Firstly, the existence and stability of this kind of solutions are established near vacuum. Secondly, it is proved that this kind of solutions are stable for any initial data, as a consequence, the Boltzmann equation has at most one solution with infinite kinetic energy. Finally, the long time behavior of the solutions is also established.     

Quantitative Lower Bounds for the Full Boltzmann Equation,Part I: Periodic Boundary Conditions

ABSTRACT

We prove the appearance of an explicit lower bound on the solution to the full Boltzmann equation in the torus for a broad family of collision kernels including in particular long-range interaction models, under the assumption of some uniform bounds on some hydrodynamic quantities. This lower bound is independent of time and space. When the collision kernel satisfies Grad's cutoff assumption, the lower bound is a global Maxwellian and its asymptotic behavior in velocity is optimal, whereas for noncutoff collision kernels the lower bound we obtain decreases exponentially but faster than the Maxwellian. Our results cover solutions constructed in a spatially homogeneous setting, as well as small-time or close-to-equilibrium solutions to the full Boltzmann equation in the torus. The constants are explicit and depend on the a priori bounds on the solution.     

Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics

This paper is part of our efforts to show how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the convergence to equilibrium of the solutions of the Kac equation. Here, we consider convergence with respect to the following metrics: Kolmogorov’s uniform metric; 1 and 2 Gini’s dissimilarity indices (widely known as 1 and 2 Wasserstein metrics); χ-weighted metrics. Our main results provide new bounds, or improvements on already well-known ones, for the corresponding distances between the solution of the Kac equation and the limiting Gaussian (Maxwellian) distribution. The study is conducted both under the necessary assumption that initial data have finite energy, without assuming existence of moments of order greater than 2, and under the condition that the (2 + δ)-moment of the initial distribution is finite for some δ > 0.     

A Petrov-Galerkin method for solving the generalized equal width (GEW) equation

The generalized equal width (GEW) equation is solved numerically by the Petrov-Galerkin method using a linear hat function as the test function and a quadratic B-spline function as the trial function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. Finally, the Maxwellian initial condition pulse is studied.     

The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in L_p-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.     

Existence of local solutions for the Boltzmann equation without angular cutoff

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable. To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Conditions at infinity for boltzmann's equation.

We consider here realistic conditions at infinity for solutions of the Boltzmann's equation, such as a pure Maxwellian equilibrium at infinity possibly with suitable boundary conditions on an exterior domain, different Maxwellian equilibria at +∞ and -;∞ in a tube-like situation and more generally conditions at infinity obtained from a fixed solution. In order to adapt the recent global existence and compactness results due to R.J. DiPerna and the author, we have to obtain some local a priori estimates on the mass, kinetic energy and entropy. And this is precisely what we achieve here by two different and new methods. The first one consists in using the relative entropy of solutions with respect to a fixed, possibly local, Maxwellian. This method allows to treat general collision kernels with angular cut-off and some of the conditions at infinity mentioned above. The second method is based upon a L¹ estimate and an extension of the entropy identity which uses a truncated H-functional. This method requires a “uniform integrability” condition on the collision kernel but allows to consider the most general conditions at infinity.     

Anisotropic hypoelliptic estimates for Landau-type operators

We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various linear inhomogeneous kinetic equations, we establish for linear Landau-type operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion.     

Eternal solutions of the Boltzmann equation near travelling Maxwellians

It is shown in this paper that the Cauchy problem of the Boltzmann equation, with a cut-off soft potential and an initial datum close to a travelling Maxwellian, has a unique positive eternal solution. This eternal solution is exponentially decreasing at infinity for all t∈(−∞,∞), consequently the moments of any order are finite. This result gives a negative answer to the conjecture of Villani in the spatially inhomogeneous case.