Optimal time decay of the quantum Landau equation in the whole space

We are concerned with the Cauchy problem of the quantum Landau equation in the whole space. The existence of local in time nearby quantum Maxwellian solutions is proved by the iteration method and generalized maximum principle. Based on Kawashima?s compensating function and nonlinear energy estimates, the global existence and the optimal time decay rate of those solutions are obtained under some conditions on initial data.     

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 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.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

Diffusive expansion for solutions of the Boltzmann equation in the whole space

This paper is concerned with the diffusive expansion for solutions of the rescaled Boltzmann equation in the whole space

(0.1)     

Hypocoercivity for Kolmogorov backward evolution equations and applications

In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser in [16]. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. This equation e.g. also appears in applied mathematics as the so-called fiber lay-down process. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and perturbation theory.     

Regularizing effects for the classical solutions to the Landau equation in the whole space

The Landau equation describes the binary collisional effects (through long range coulombian interaction) in a plasma. In this paper, we prove that the known classical solutions to the Landau equation near Maxwellian in the whole space have a regularizing effect in all (time, space and velocity) variables, that is, become immediately smooth with respect to all variables.     

Hypocoercivity for degenerate Kolmogorov equations and applications to the Langevin dynamics

We present an extension of the powerful Hilbert space hypocoercivity method that was developed originally by Dolbeault, Mouhot and Schmeiser. We focus attention on including important domain issues that have not been considered before. The setting can be used to provide a complete elaboration of the hypocoercivity theorem for the degenerate Langevin dynamics. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local solvability of equations of magnetohydrodynamics on the whole space

Using the technique of expanding domains, we prove the existence of a weak, local in time solution to the equations of magnetohydrodynamics, derived from the equations for viscous, compressible and heat-conducting fluids, on the whole space under special assumptions on pressure and entropy. Compared with the same approach for barotropic compressible fluids, we show how to overcome loss of the global integrability of temperature and velocity fields in corresponding spaces.     

Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators

We prove explicit coercivity estimates for the linearized Boltzmann and Landau operators, for a general class of interactions including any inverse-power law interactions, and hard spheres. The functional spaces of these coercivity estimates depend on the collision kernel of these operators. They cover the spectral gap estimates for the linearized Boltzmann operator with Maxwell molecules, improve these estimates for hard potentials, and are the first explicit coercivity estimates for soft potentials (including in particular the case of Coulombian interactions). We also prove a regularity property for the linearized Boltzmann operator with non locally integrable collision kernels, and we deduce from it a new proof of the compactness of its resolvent for hard potentials without angular cutoff.     

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.     

Stability of stationary solutions of parabolic equations and of the Navier-Stokes system in the whole space

Rostov on Don. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 1, pp. 151–158, January–February, 1988.     

Steady state solution of the relativistic Boltzmann transport equation

Résumé Depuis les travaux deJüttner [1], nous connaissons la forme relativiste de la fonction de répartition maxwellienne des vitesses. Le traité deJüttner est basé sur une généralisation de la méthode de l'extension en phases, de sorte que le théorème de Liouville est valable par rapport aux équations relativistes de mouvement. La fonction de répartition est déterminée par la méthode combinatoire usuelle.Ici, nous proposons de suivre une autre voie. Partant d'une forme relativiste de l'équation de transport de Boltzmann donnée parClemmow etWillson [3] etKen-iti Goto [4], nous proposons une expression relativiste de l'intégrale de collision qui complète cette équation, et nous donnons ensuite sa solution pour l'état stationnaire.

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A part of these calculations was carried out while the author was the guest of the Osservatorio Astrofisico di Arcetri, Firenze whose kind hospitality is gratefully acknowledged.     

Second‐order elliptic and parabolic equations with BMO coefficients in the whole space

In this paper, we obtain the global regularity estimates in Orlicz spaces for second‐order divergence elliptic and parabolic equations with BMO coefficients in the whole space. In fact, the global result can follow from the local estimates. As a corollary we obtain L^p‐type regularity estimates for such equations. Copyright © 2011 John Wiley & Sons, Ltd.     

The Boltzmann equation without angular cutoff in the whole space: I,Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and a possible gain of weight in the velocity variable. By defining and analyzing a non-isotropic norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross-sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.     

Optimal large‐time behavior of the Vlasov‐Maxwell‐Boltzmann system in the whole space

In this paper we study the large‐time behavior of classical solutions to the two‐species Vlasov‐Maxwell‐Boltzmann system in the whole space \input amssym ${\Bbb R}^3$ . The existence of global‐in‐time nearby Maxwellian solutions is known from Strain in 2006. However, the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work on time decay for the simpler Vlasov‐Poisson‐Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of O(t^{−3/2 + 3/(2r)}) in the L (L)‐norm for any 2 ≤ r ≤ ∞ if initial perturbation is smooth enough and decays in space velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to 0 are also provided. © 2011 Wiley Periodicals, Inc.     

Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space

In this paper a quasi-linear elliptic equation in the whole Euclidean space is considered. The nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of Trudinger–Moser type inequality. Under some assumptions on the potential and the nonlinearity, it is proved that there is a nontrivial positive weak solution to this equation. Also it is shown that there are two distinct positive weak solutions to a perturbation of the equation. The method of proving these results is combining Trudinger–Moser type inequality, Mountain-pass theorem and Ekeland?s variational principle.     

Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations

We consider the Riemann problem of three-dimensional relativistic Euler equations with two discontinuous initial states separated by a planar hypersurface. Based on the detailed analysis on the Riemann solutions, special relativistic effects are revealed, which are the variations of limiting relative normal velocities and intermediate states and thus the smooth transition of wave patterns when the tangential velocities in the initial states are suitably varied. While in the corresponding non-relativistic fluid, these special relativistic effects will not occur.     

Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations

We consider the Riemann problem of three-dimensional relativistic Euler equations with two discontinuous initial states separated by a planar hypersurface. Based on the detailed analysis on the Riemann solutions, special relativistic effects are revealed, which are the variations of limiting relative normal velocities and intermediate states and thus the smooth transition of wave patterns when the tangential velocities in the initial states are suitably varied. While in the corresponding non-relativistic fluid, these special relativistic effects will not occur.