Stokes‐Fourier and acoustic limits for the Boltzmann equation: Convergence proofs

We establish a Stokes‐Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension two or more. Appropriately scaled families of DiPerna‐Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes‐Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L² initial data of the Stokes‐Fourier equations. Both the motion and heat equations are both recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna‐Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna‐Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L² initial data. © 2001 John Wiley & Sons, Inc.     

Hydrodynamic limits: some improvements of the relative entropy method

The present paper is devoted to the study of the incompressible Euler limit of the Boltzmann equation via the relative entropy method. It extends the convergence result for well-prepared initial data obtained by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47–80]. It explains especially how to take into account the acoustic waves and relaxation layer, and thus to obtain convergence results under weak assumptions on the initial data.     

Asymptotics of the solutions to theN-particle Kolmogorov-Feller equations and the asymptotics of the solution to the Boltzmann equation in the region of large deviations

We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space Γ(L ₁(ℝ ⁿ )) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation inL ₁(ℝ ⁿ ) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to Γ(L ₁(ℝ ⁿ )) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent characteristic equation. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 694–709, November, 1995. The author is deeply grateful to Prof. A. M. Chebotarev, whose assistance has made the writing of this paper possible. This work was financially supported by the International Science Foundation under grants Nos. MFO000 and MFO300.     

Propagation of and Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann equation

We consider the n-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of L¹-Maxwellian weighted estimates, and consequently, the propagation L^∞-Maxwellian weighted estimates to all derivatives of the initial value problem associated to the afore mentioned equation. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagation of L¹-Maxwellian weighted estimates as originally developed Bobylev [A.V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys. 88 (1997) 1183–1214] in the case of hard spheres in 3 dimensions. To achieve this goal we implement the program presented in Bobylev–Gamba–Panferov [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equation with inelastic interactions, J. Statist. Phys. 116 (5–6) (2004) 1651–1682], which includes a full analysis of the moments by means of sharp moment inequalities and the control of L¹-exponential bounds, in the case of stationary states for different inelastic Boltzmann related problems with ‘heating’ sources where high energy tail decay rates depend on the inelasticity coefficient and the type of ‘heating’ source. More recently, this work was extended to variable hard potentials with angular cutoff by Gamba–Panferov–Villani [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press] in the elastic case collision case where the L¹-Maxwellian weighted norm was shown to propagate if initial states have such property. In addition, we also extend to all derivatives the propagation of L^∞-Maxwellian weighted estimates, proven in [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press], to solutions of the initial value problem to the Boltzmann equations for elastic collisions for variable hard potentials with Grad (angular) cutoff.     

Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t=0) as ?→0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.     

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.     

L ∞ Variational Problems with Running Costs and Constraints

Various approaches are used to derive the Aronsson–Euler equations for L ^∞ calculus of variations problems with constraints. The problems considered involve holonomic, nonholonomic, isoperimetric, and isosupremic constraints on the minimizer. In addition, we derive the Aronsson–Euler equation for the basic L ^∞ problem with a running cost and then consider properties of an absolute minimizer. Many open problems are introduced for further study.     

Concentration-cancellation phenomena for the velocity fields in incompressible fluid flows

A more intuitive sufficient condition is given for the concentration cancellation phenomena in 2- or 3-D incompressible fluid flows; that is, if the projection of concentration set of the weak-star defect measure associated with the approximate solution sequence onto space ℝ^π _x (n = 2, 3) is a set with Hausdorff dimension less than 1, then the weak-L ² limit of the approximate solution sequence is a classical weak solution of Euler equation. Using this condition, an example is given to elucidate concentration-cancellation phenomena.     

L6 BOUND FOR BOLTZMANN DIFFUSIVE LIMIT

We consider diffusive limit of the Boltzmann equation in a periodic box. We establish L⁶ estimate for the hydrodynamic part Pf of particle distribution function, which leads to uniform bounds global in time.     

Central limit theorem for Maxwellian molecules and truncation of the wild expansion

We prove an L¹ bound on the error made when the Wild summation for solutions of the Boltzmann equation for a gas of Maxwellian molecules is truncated at the n^th stage. This gives quantitative control over the only constructive method known for solving the Boltzmann equation. As such, it has recently been applied to numerical computation but without control on the approximation made in truncation. We also show that our bound is qualitatively sharp and that it leads to a simple proof of the exponentially fast rate of relaxation to equilibrium for Maxwellian molecules along lines originally suggested by McKean. © 2000 John Wiley & Sons, Inc.     

The Incompressible Limit and the Initial Layer of the Compressible Euler Equation in ℝn+

We consider the incompressible limit of the compressible Euler equation in the half-space ℝⁿ₊. It is proved that the solutions of the non-dimensionalized compressible Euler equation converge to the solution of the incompressible Euler equation when the Mach number tends to zero. If the initial data v^∞₀ do not satisfy the condition ‘∇⋅v^∞₀=0’, then the initial layer will appear. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On a diffusion‐kinetic equation arising in extended kinetic theory

This paper is devoted to the study of solvability of an evolution equation obtained as a hydrodynamic limit of a linear Boltzmann equation with elastic and inelastic scattering terms. An interesting feature of this equation is that it combines the diffusion operator and a singular operator of a kinetic type. The study is carried out in the L₁ space which is natural from the physical point of view and allows to apply modifications of methods of the kinetic theory. Copyright © 2000 John Wiley & Sons, Ltd.     

A large data existence result for the stationary Boltzmann equation in a cylindrical geometry

AnL ¹-existence theorem is proved for the nonlinear stationary Boltzmann equation with hard forces and no small velocity truncation—only the Grad angular cut-off-in a setting between two coaxial rotating cylinders when the indata are given on the cylinders.     

On the Strong L1 Trend to Equilibrium for the Boltzmann Equation

This paper considers the Boltzmann equation for hard and soft (k > 2) forces having an angular cutoff. The main result is strong L¹ convergence to global Maxwellians when time tends to infinity..     

Strong convergence towards homogeneous cooling states for dissipative Maxwell models

We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogeneous cooling state present in the literature implies the strong convergence in Sobolev norms and in the L¹ norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the L¹ distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.     

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.     

Unstable Manifolds of Euler Equations

We consider a steady state v₀ of the Euler equation in a fixed bounded domain in ?ⁿ. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center‐stable subspaces. By rewriting the Euler equation as an ODE on an infinite‐dimensional manifold of volume‐preserving maps in W^{k, q} the unstable (and stable) manifolds of v₀ are constructed under a certain spectral gap condition that is satisfied for both two‐dimensional and three‐dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v₀ in the sense that arbitrarily small W^{k, q} perturbations can lead to L² growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R ³. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L ¹ topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.Mathematics Subject Classification (2000) 35Q35, 35Q30, 82C40     

Stationary solutions of the non-linear Boltzmann equation in a bounded spatial domain

A contraction mapping (or, alternatively, an implicit function theory) argument is applied in combination with the Fredholm alternative to prove the existence of a unique stationary solution of the non-linear Boltzmann equation on a bounded spatial domain under a rather general reflection law at the piecewise C¹ boundary. The boundary data are to be small in a weighted L_∞-norm.