The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials

In this paper, the existence of boundary layer solutions to the Boltzmann equation for hard potential with mixed boundary condition, i.e., a linear combination of Dirichlet boundary condition and diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. As for the problem with Dirichlet boundary condition (Chen et al., 2004 [5]), the existence of a solution highly depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given.     

Nonlinear stability of boundary layers for the Boltzmann equation with cutoff soft potentials

The study on the boundary layer is important in both mathematics and physics. This paper considers the nonlinear stability of boundary layer solutions for the Boltzmann equation with cutoff soft potentials when the Mach number of the far field is less than −1. Unlike the collision frequency is strictly positive in the hard potential or hard sphere model, the collision frequency has no positive lower bound for the cutoff soft potentials, so the decay in time cannot be expected. Instead, the present paper proves that the solution will always be in a small region around the boundary layer by noticing the decay property of collision operator in velocity.     

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.     

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.     

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L¹‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L¹‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L¹‐spaces. For soft potentials, in L¹‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.     

Well-posedness of the spatially homogeneous Landau equation for soft potentials

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderately soft potentials, we assume additionally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka [H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Geb. 46 (1) (1978-1979) 67-105].     

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.     

On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: Global existence of weak solutions

For general initial data we prove the global existence and weak stability of weak solutions of the Boltzmann equation for Fermi-Dirac particles in a periodic box for very soft potentials (−5<γ?−3) with a weak angular cutoff. In particular the Coulomb interaction (γ=−3) with the weak angular cutoff is included. The conservation of energy and moment estimates are also proven under a further angular cutoff. The proof is based on the entropy inequality, velocity averaging compactness of weak solutions, and various continuity properties of general Boltzmann collision integral operators.     

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.     

Large time behavior of the solutions to the Boltzmann equation with specular reflective boundary condition

In this paper a half space problem for the one-dimensional Boltzmann equation with specular reflective boundary condition is investigated. It is shown that the solution of the Boltzmann equation time-asymptotically converges to a global Maxwellian under some initial conditions. Furthermore, a time-decay rate is also obtained.     

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

The stationary Boltzmann equation for soft forces in the context of a two‐component gas is considered in the slab. An existence theorem is proved when one component satisfies a given indata profile and the other component satisfies diffuse reflection at the boundaries in a renormalized sense. Weak L¹ compactness is extracted from the control of the entropy production term. Trace at the boundaries is also controlled. Copyright © 2008 John Wiley & Sons, Ltd.     

Existence of nonlinear boundary layer solution to the Boltzmann equation with physical boundary conditions

In this paper, the existence of boundary layer solutions to the Boltzmann equation with two physical boundary conditions for hard sphere model is considered. The boundary condition is first imposed on incoming particles of diffuse reflection type and the solution tends 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. 236 (3) (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian, and there is an implicit solvability conditions yielding the co-dimensions of the boundary data. At last, the specular reflection boundary condition is considered and the similar conclusions are obtained.     

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.     

Particular solutions of the linearized Boltzmann equation for a binary mixture of rigid spheres

Particular solutions that correspond to inhomogeneous driving terms in the linearized Boltzmann equation for the case of a binary mixture of rigid spheres are reported. For flow problems (in a plane channel) driven by pressure, temperature, and density gradients, inhomogeneous terms appear in the Boltzmann equation, and it is for these inhomogeneous terms that the particular solutions are developed. The required solutions for temperature and density driven problems are expressed in terms of previously reported generalized (vector-valued) Chapman–Enskog functions. However, for the pressure-driven problem (Poiseuille flow) the required particular solution is expressed in terms of two generalized Burnett functions defined by linear integral equations in which the driving terms are given in terms of the Chapman–Enskog functions. To complete this work, expansions in terms of Hermite cubic splines and a collocation scheme are used to establish numerical solutions for the generalized (vector-valued) Burnett functions.     

The temperature-jump problem based on the CES model of the linearized Boltzmann equation

An analytical discrete-ordinates method is used to solve the temperature-jump problem as defined by a synthetic-kernel model of the linearized Boltzmann equation. In particular, the temperature and density perturbations and the temperature-jump coefficient defined by the CES model equation are obtained (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented for general values of the accommodation coefficient to yield numerical results that compare well with solutions derived from more computationally intensive techniques.     

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.     

A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion

Abstract In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jinand Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weaklyparabolic, has a linearly hyperbolic convection part, and is endowed with a generalized eotropy inequality. Itagrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system forthe Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme,and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and bythe extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments showthat the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equationobtained by the DSMC, for a range of Mach numbers for hypersonic flows, th     

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.     

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

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