This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the strident restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version]

An introduction to the relativistic kinetic theory on curved spacetimes

This article provides a self-contained pedagogical introduction to the relativistic kinetic theory of a dilute gas propagating on a curved spacetime manifold (M,g) of arbitrary dimension. Special emphasis is made on geometric aspects of the theory in order to achieve a formulation which is manifestly covariant on the relativistic phase space. Whereas most previous work has focussed on the tangent bundle formulation, here we work on the cotangent bundle associated with (M,g) which is more naturally adapted to the Hamiltonian framework of the theory. In the first part of this work we discuss the relevant geometric structures of the cotangent bundle T*M, starting with the natural symplectic form on T*M, the one-particle Hamiltonian and the Liouville vector field, defined as the corresponding Hamiltonian vector field. Next, we discuss the Sasaki metric on T*M and its most important properties, including the role it plays for the physical interpretation of the one-particle distribution function. In the second part of this work we describe the general relativistic theory of a collisionless gas, starting with the derivation of the collisionless Boltzmann equation for a neutral simple gas. Subsequently, the description is generalized to a charged gas consisting of several species of particles and the general relativistic Vlasov-Maxwell equations are derived for this system. The last part of this work is devoted to a transparent derivation of the collision term, leading to the general relativistic Boltzmann equation on (M,g). The meaning of global and local equilibrium and the stringent restrictions for the existence of the former on a curved spacetime are discussed. We close this article with an application of our formalism to the expansion of a homogeneous and isotropic universe filled with a collisional simple gas and its behavior in the early and late epochs. [abbreviated version] 

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In this paper, we study the local-in-time validity of the Hilbert expansion for the relativistic Landau equation. We justify that solutions of the relativistic Landau equation converge to small classical solutions of the limiting relativistic Euler equations as the Knudsen number shrinks to zero in a weighted Sobolev space. The key diﬃculty comes from the temporal and spatial derivatives of the local Maxwellian, which produce momentum growth terms and are uncontrollable by the standard L 2 -based energy and dissipation. We introduce novel time-dependent weight functions to generate additional dissipation terms to suppress the large momentum. The argument relies on a hierarchy of energy-dissipation structures with or without weights. As far as the authors are aware of, this is the ﬁrst result of the Hilbert expansion for the Landau-type equation.

Hilbert Expansion for the Relativistic Landau Equation

Although the compressible fluid limit of the Boltzmann equation with cutoff has been well investigated in [6] and [33], it still remains largely open to obtain analogous results in case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, essentially due to the velocity diffusion effect of collision operator such that L∞ estimates are hard to obtain without using Sobolev embeddings. In the paper, we are concerned with the compressible Euler and acoustic limits of the Landau equation for Coulomb potentials in the whole space. Specifically, over any finite time interval where the full compressible Euler system admits a smooth solution around constant states, we construct a unique solution in a high-order weighted Sobolev space for the Landau equation with suitable initial data and also show the uniform estimates independent of the small Knudsen number ε > 0, yielding the O(ε) convergence of the Landau solution to the local Maxwellian whose fluid quantities are the given Euler solution. Moreover, the acoustic limit for smooth solutions to the Landau equation in an optimal scaling is also established. For the proof, by using the macro-micro decomposition around local Maxwellians together with techniques for viscous compressible fluid and properties of Burnett functions, we design an ε-dependent energy functional to capture the dissipation in the compressible fluid limit with feature that only the highest order derivatives are most singular.

Compressible fluid limit for smooth solutions to the Landau equation

Two fundamental models in plasma physics are given by the Vlasov–Maxwell–Boltzmann system and the compressible Euler–Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number [Formula: see text] tends to zero. In this paper, we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in [Formula: see text] over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new [Formula: see text]-dependent energy estimates basing on the macro–micro decomposition to characterize the asymptotic limit in the compressible setting.

Compressible Euler–Maxwell limit for global smooth solutions to the Vlasov–Maxwell–Boltzmann system

Consider the relativistic Vlasov–Maxwell–Boltzmann system describing the dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent works Germain and Masmoudi (Ann Sci Ec Norm Super 47(3):469–503, 2014), Guo et al. (J Math Phys 55(12):123102, 2014) and Deng et al. (Arch Ration Mech Anal 225(2):771–871, 2017), we establish the global-in-time validity of its Hilbert expansion and derive the limiting relativistic Euler–Maxwell system as the mean free path goes to zero. Our method is based on the $$L^2-L^{\infty }$$
 framework and the Glassey–Strauss Representation of the electromagnetic field, with auxiliary $$H^1$$
 estimates and $$W^{1,\infty }$$
 estimates to control the characteristic curves and corresponding $$L^{\infty }$$
 norm.

Global Hilbert Expansion for the Relativistic Vlasov–Maxwell–Boltzmann System

Consider the relativistic Vlasov-Maxwell-Boltzmann system describing the dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent works \cite{Germain-Masmoudi-ASENS-2014, Guo-Ionescu-Pausader-JMP-2014} and \cite{Deng-Ionescu-Pausader-ARMA-2017}, we establish the global-in-time validity of its Hilbert expansion and derive the limiting relativistic Euler-Maxwell system as the mean free path goes to zero. Our method is based on the $L^2-L^{\infty}$ framework and the Glassey-Strauss Representation of the electromagnetic field, with auxiliary $H^1$ estimate and $W^{1,\infty}$ estimates to control the characteristic curves and corresponding $L^{\infty}$ norm.

Qinghua Xiao

Papers

Global Hilbert Expansion for the Relativistic Vlasov–Maxwell–Boltzmann System

Global Hilbert expansion for the relativistic Vlasov-Maxwell-Boltzmann system.