Optimal time decay of the Boltzmann equation with frictional force

In this paper, we use the combination of energy method and Fourier analysis to obtain the optimal time decay of the Boltzmann equation with frictional force towards equilibrium. Precisely speaking, we decompose the equation into macroscopic and microscopic partitions and perform the energy estimation. Then, we construct a special solution operator to a linearized equation without source term and use Fourier analysis to obtain the optimal decay rate to this solution operator. Finally, combining the decay rate with the energy estimation for nonlinear terms, the optimal decay rate to the Boltzmann equation with frictional force is established.     

Optimal decay rates and small global solutions to the dissipative Boussinesq equation

This work is devoted to the small amplitude solutions for the initial value problem of the multidimensional dissipative Boussinesq equation. We firstly derive the pointwise estimates of the fundamental solutions by the energy method in the Fourier space. We give the asymptotic profiles of solutions to the corresponding linear problem to get the optimal decay rate for the -norm of solutions in all space dimensions. Under smallness assumptions on the initial data, we study the global existence and uniqueness of solutions by the contractive mapping principle in the solution spaces with time weighted norm.     

Global existence and time decay of the non-cutoff Boltzmann equation with hard potential

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces $H^N\left({\mathbb{R}}_{x,v}^6\right)$ with $N\ge 2$. But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000).     

Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f₀(ξ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f(ξ,t)∈H^∞(R³) for all t>0. Furthermore, for any t₀>0 and s?0 the Hs norm of f(ξ,t) is bounded for t?t₀. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition.     

Viscosity analysis on the Boltzmann equation

This paper is devoted to investigating the asymptotic properties of the renormalized solution to the viscosity equation tfε + v ·▽xfε = Q (fε,fε ) + εΔvfε as ε→ 0+ . We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L1 ((0 , T ) × RN × RN ). The proof is based on compactness analysis and velocity averaging theory.     

The Boltzmann equation with frictional force

The Euler equations with frictional force have been extensively studied. Since the Boltzmann equation is closely related to the equations of gas dynamics, we study, in this paper, the Boltzmann equation with frictional force when the external force is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro-micro decomposition for the Boltzmann equation introduced in Liu et al. [Energy method for the Boltzmann equation, Physica D 188 (3-4) (2004) 178-192] and Liu and Yu [Boltzmann equation: micro-macro-decompositions and positivity of shock profiles, Comm. Math. Phys. 246(1) (2004) 133-179] through energy estimates.     

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.     

Acoustic limit for the Boltzmann equation in the whole space

This paper is devoted to the following rescaled Boltzmann equation in the acoustic time scaling in the whole space

(0.1)     

The Cauchy problem for the generalized Boltzmann equation with dissipative collisions

We prove the global existence, uniqueness, and positivity of solutions to the Cauchy problem, with general initial data, for a class of generalized Boltzmann models with dissipative collisions.     

A solution of the complex Ginzburg-Landau equation with a continuum of decay rates

We prove the existence of a uniform initial datum whose solution decays, in various Lpspaces, at different rates along different time sequences going to infinity, for complex Ginzburg-Landau equation on RN, of various parameters θ and γ.     

A new lattice Boltzmann model for the laplace equation

In this paper, a new lattice Boltzmann equation which is independent of time is proposed. Based on the new lattice Boltzmann equation, some steady problems can be modeled by the lattice Boltzmann method. In the further study, the Laplace equation is investigated with the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different space scales. The numerical results show that the new method is effective.     

The Boltzmann equation with potential force in the whole space

In this paper, we consider the Cauchy problem of the Boltzmann equation with potential force in the whole space. When some more natural assumptions compared with those of the previous works are made on the potential force, we can still obtain a unique global solution to the Boltzmann equation even for the hard potential cases by energy method, if the initial data are sufficiently close to the steady state. Moreover, the solution is uniformly stable. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

The stationary Boltzmann equation for a two‐component gas in the slab

The stationary Boltzmann equation for hard forces in the context of a two‐component gas is considered in the slab. An L¹ existence theorem is proved when one component satisfies a given indata profile and the other component satisfies diffuse reflection at the boundaries. Weak L¹ compactness is extracted from the control of the entropy production term. Trace at the boundaries are also controlled. Copyright © 2007 John Wiley & Sons, Ltd.     

A lattice Boltzmann model for the Fokker-Planck equation

In this paper, a lattice Boltzmann model is presented for solving one and two-dimensional Fokker-Planck equations with variable coefficients. In particular, it is efficient to simulate one-dimensional stochastic processes governed by the Fokker-Planck equation. Numerical results agree well with the exact solutions, which indicates that the proposed model is suitable for solving the Fokker-Planck equation.     

Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules

We prove propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for any value of the coefficient of restitution. The result follows from the uniform in time control of the tails of the Fourier transform of the solution, normalized in order to have constant energy. By standard arguments this implies the convergence of the scaled solution towards the stationary state in Sobolev and L¹$L^1$ norms in the case of regular initial data as well as the convergence of the original solution to the corresponding self-similar cooling state. In the case of weak inelasticity, similar results have been established by Carlen, Carrillo and Carvalho (2009) in [11] via a precise control of the growth of the Fisher information.