On existence of solutions for a system of Boltzmann transport equations

We consider a linear system of Boltzmann transport equations. The system models charged particle transport in tissue, for example. Although only one species of particles, say photons, is invasing these particles mobilize electrons and positrons. Hence in realistic modelling of particle transport one needs a coupled system of three Boltzmann transport equations. The solution of this system must satisfy the inflow boundary condition. We show existence and uniqueness result of the solution applying coercitivity of the underlying linear operator and its adjoint operator. In addition, we consider existence of continuous solutions by iterative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

Solution to the Boltzmann kinetic equation for high-speed flows

The Boltzmann kinetic equation is solved by a finite-difference method on a fixed coordinate-velocity grid. The projection method is applied that was developed previously by the author for evaluating the Boltzmann collision integral. The method ensures that the mass, momentum, and energy conservation laws are strictly satisfied and that the collision integral vanishes in thermodynamic equilibrium. The last property prevents the emergence of the numerical error when the collision integral of the principal part of the solution is evaluated outside Knudsen layers or shock waves, which considerably improves the accuracy and efficiency of the method. The differential part is approximated by a second-order accurate explicit conservative scheme. The resulting system of difference equations is solved by applying symmetric splitting into collision relaxation and free molecular flow. The steady-state solution is found by the relaxation method.     

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.     

On the asymptotic analysis of the Boltzmann equation tending towards the Stokes equations

This work deals with the analysis of the asymptotic limit for the Boltzmann equation tending towards the linearized Navier–Stokes equations when the Knudsen number $\epsilon$ tends to zero. Global existence and uniqueness theorems are proven for regular initial fluctuations. As $\epsilon$ tends to zero, the solution converges strongly to the solution of the linearized Navier–Stokes systems.     

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.     

Explicit solutions to a coupled integrable lattice equation

A coupled integrable lattice equation is derived from a 4 × 4 matrix spectral problem, then with the help of a special Darboux matrix, explicit solutions of the aforementioned equation are given by means of gauge transformation between the Lax pair. Finally, the density profiles of these exact solutions are presented to illustrate these solutions.     

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.     

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)     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.     

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.     

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.     

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.     

On blowup for gain‐term‐only classical and relativistic Boltzmann equations

We show that deletion of the loss part of the collision term in all physically relevant versions of the Boltzmann equation, including the relativistic case, will in general lead to blowup in finite time of a solution and hence prevent global existence. Our result corrects an error in the proof given (Math. Meth. Appl. Sci. 1987; 9 :251–259), where the result was announced for the classical hard sphere case; here we give a simpler proof which applies much more generally. Copyright © 2004 John Wiley & Sons, Ltd.     

On the sizes of discrete velocity models of the Boltzmann equation for mixtures

The Boltzmann equation for a mixture of particles with different masses is modeled using symmetric discrete velocity models that involve energy interchange between the species of the mixture. The computational complexity of this problem is investigated. New discrete models are presented. Published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 6, pp. 1045–1054.     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.