Optimal Decay Estimates on the Linearized Boltzmann Equation with Time Dependent Force and their Applications

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L ^p -L ^q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n ≥ 3, the global existence and decay rates of solutions to the Cauchy problem are obtained under the condition that the force and source decay in time with some rates. This time decay restriction can be removed for space dimension n ≥ 5. Moreover, the existence and asymptotic stability of the time periodic solution are given for space dimension n ≥ 5 when the force and source are time periodic with the same period.     

Relaxation of the ideal Bose gas

For the ideal Bose gas we study the approach to equilibrium. Above the critical temperature we prove exponential behaviour, with a relaxation time of the order (T-T _c)^-2 around T _c. For T _c we find the t ^-1 law for the two-point function.     

Robust Exponential Decay of Correlations for Singular-Flows

We construct open sets of C ^k (k ≥ 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure.     

On the Boltzmann Equation for Diffusively Excited Granular Media

We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L²(^N) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.     

On global solutions of the Maxwell-Dirac equations

We prove, for the Maxwell-Dirac equations in 1+3 dimensions, that modified wave operators exist on a domain of small entire test functions of exponential type and that the Cauchy problem, inR ⁺×R ³, has a unique solution for each initial condition (att=0) which is in the image of the wave operator. The modification of the wave operator, which eliminates infrared divergences, is given by approximate solutions of the Hamilton-Jacobi equation, for a relativistic electron in an electromagnetic potential. The modified wave operator linearizes the Maxwell-Dirac equations to their linear part.Dedicated to Walter Thirring on his 60^th birthdayThis work is dedicated to Walter Thirring upon the occasion of his sixtieth birthday with appreciation and friendship     

Approach to Equilibrium in a Microscopic Model of Friction

We consider the time evolution of a disk under the action of a constant force and interacting with a free gas in the mean-field approximation. Letting V₀>0 be the initial velocity of the disk and V_∞>0 its equilibrium velocity, namely the one for which the external field is balanced by the friction force exerted by the background, we show that, if V_∞−V₀ is positive and sufficiently small, then the disk reaches V_∞ with the power law t^−(d+2), d=1,2,3 being the dimension of the physical space. The reason for this behavior is the long tail memory due to recollisions. Any Markovian approximation (or simply neglecting the recollisions) yields an exponential approach to equilibrium.     

Harnack Inequalities and Discrete—Continuous Error Estimates for a Chain of Atoms with Two—Body Interactions

We consider deformations in ℝ³ of an infinite linear chain of atoms where each atom interacts with all others through a two-body potential. We compute the effect of an external force applied to the chain. At equilibrium, the positions of the particles satisfy an Euler–Lagrange equation. For large classes of potentials, we prove that every solution is well approximated by the solution of a continuous model when applied forces and displacements of the atoms are small. We establish an error estimate between the discrete and the continuous solution based on a Harnack lemma of independent interest. Finally we apply our results to some Lennard-Jones potentials.     

Existence,Non-existence,and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2

We consider a semilinear heat equation with exponential nonlinearity in ?². We prove that local solutions do not exist for certain data in the Orlicz space exp L ²(?²), even though a small data global existence result holds in the same space exp L ²(?²). Moreover, some suitable subclass of exp L ²(?²) for local existence and uniqueness is proposed.

     

Exponential decay of Green's functions for a class of long range Hamiltonians

We consider a class of long range Hamiltonians with diagonal disorder onl ² (Z). For anyergodic potentialV with non-empty essential range, we prove the exponential decay of the Green's functions for energies in the essential range. IfV is independent identically distributed, we obtain the exponential decay of the Green's functions for all coupling constant >0. Moreover the Hamiltonian has only pure point spectrum.     

The Navier-Stokes limit of the stationary boltzmann equation for hard potentials

In this paper we extend recent results on the hydrodynamic Navier-Stokes limit of the stationary Boltzmann equation for the flow of a gas of hard spheres in a channel in the presence of an external force to the case of a hard intermolecular potential with Grad angular cutoff. We prove the convergence of the solution, for small Knudsen numbers, to the Maxwellian with parameters solving the corresponding Navier-Stokes equation. In the present case we only get polynomial decay of the solution for large velocities, instead of the exponential decay which holds for hard spheres.     

Solvation forces in fluids

Using a formalism derived by us in an earlier paper we obtain the density profile and solvation force between two identical hard plates separated by a fluid characterized by (i) an Orstein-Zernike correlation function and (ii) a Percus-Yevick sticky sphere correlation function. Example (i) yields an exponential or hyperbolic density profile characterized by a correlation length ξ that diverges at the critical point as (T-T_c )^-v . The corresponding force is weak and attractive following an exponential law at large separations. Example (ii) yields an attractive force in the vicinity of the critical region and an oscillatory force at higher densities and/or higher temperatures which can significantly modify forces between colloid particles.     

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast . If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast .     

Stability of Global Solution for the Relativistic Enskog Equation near Vacuum

The Cauchy problem of the relativistic Enskog equation with near-vacuum data is considered in this paper. Under the same assumption as that in Jiang (J. Stat. Phys. 127:805–812, 2007) for the relativistic Enskog equation, we obtain the uniform L ^∞-stability of the solution. What’s more important, is that for two new types of the scattering cross section σ, we give the global existence and L ¹(x,v)-stability for mild solution when the initial data lies in the space L ¹(x,v). As a corollary, we have a BV-type estimate. It is worth mentioning that the stability results in this paper can be applied to the case in Jiang (J. Stat. Phys. 127:805–812, 2007).     

Exponential bounds and semi-finiteness of point spectrum forN-body Schrödinger operators

For a large class ofN-body Schrödinger operatorsH, we prove that eigenvalues ofH cannot accumulate from above at any threshold ofH. Our proof relies onL ² exponential upper bounds for eigenfunctions ofH with explicit constants obtained by modifying methods of Froese and Herbst.Bantrell Research Fellow in Mathematical Physics     

On the free boundary problem for stationary compressible Navier-Stokes equations

We consider the equations which describe a stationary motion of a viscous compressible barotropic fluid in a bounded domain in ³ with a free boundary determined by the surface tension. By means of some a priori estimates we prove the existence of rotationally symmetric solutions (in reality with some additional symmetry) for a sufficiently small external force and in the case of rotationally symmetric force and domain (where also we need more symmetry, respectively).     

弹性矩形板方程在非线性边界条件下整体解的存在唯一性

考虑了在非线性边界条件下的弹性矩形板方程.利用Galerkin方法,首先证明了该方程在非线性边界(a)及初值w⁰∈W,w¹∈W的条件下初边值问题存在唯一整体弱解w(t).其次证明了该方程在非线性边界(b)及初值w⁰∈W₁,w¹∈W₁的条件 关键词： 弹性矩形板方程 非线性边界条件 初边值问题 整体解     

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

 We consider the quasi-geostrophic equation with the dissipation term, κ (-Δ)^α θ, In the case , Constantin-Cordoba-Wu [6] proved the global existence of strong solution in H ¹ and H ² under the assumption of small L ^∞ -norm of initial data. In this paper, we prove the global existence in the scale invariant Besov space, B ^2−2α _2,1 , for initial data small in the B ^2−2α _2,1 norm. We also prove a global stability result in B ¹ _2,1 . Received: 24 April 2002 / Accepted: 29 July 2002 Published online: 10 December 2002 Communicated by P. Constantin     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997     

Supports of Measure Solutions for Spatially Homogeneous Boltzmann Equations

We prove that the support of the unique measure solution for the spatially homoge-neous Boltzmann equation in R3 is the whole space, if the initial distribution is not a Dirac measure and has 4-order moment. More precisely, we obtain the lower bound of exponential type for the probability of any small ball in ℝ³ relative to the measure solution.