In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.
We present in this paper an estimate which bounds from below the entropy dissipation D(f) of the Landau operator with Coulomb interaction by a weighted H^1 norm of the square root of f. As a consequence, we get a weighted L^1_t(L^3_v) estimate for the solutions of the spatially homogeneous Landau equation with Coulomb interaction, and the propagation of L^1 moments of any order for this equation. We also present an application of our estimate to the Landau equation with (moderately) soft potentials, providing thus a new proof of some recent results of Kung-Chien Wu
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 addition…
We revisit the grazing collision limit connecting the Boltzmann equation to the Landau(-Fokker-Planck) equation from their recent reinterpretations as gradient flows. Our results are in the same spirit as the $Γ$-convergence of gradient flows technique introduced by Sandier and Serfaty. In this setting, the grazing collision limit reduces to showing the lower semi-continuous convergence of the Boltzmann entropy-dissipation to the Landau entropy-dissipation.
Hydrodynamic limit of rarefaction wave for the Vlasov-Maxwell-Landau system with Coulomb potential
School of Mathematics, Guangxi University（广西大学数学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；Center for Applied Mathematical of Guangxi (Guangxi University)（广西应用数学中心（广西大学））
In this paper, we study the hydrodynamic limits of both the Landau equation and the Vlasov-Maxwell-Landau system in the whole space. Our main purpose is two-fold: the first one is to give a rigorous derivation of the compressible Euler equations from the Landau equation via the Hilbert expansion; while the second one is to prove, still in the setting of Hilbert expansion, that the unique classical solution of the Vlasov-Maxwell-Landau system converges, which is shown to be globally in time, to the resulting global smooth solution of the Euler-Maxwell system, as the Knudsen number goes to zero.…
We describe a time-dependent functional involving the relative entropy and the $\dot{H}^1$ seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functionial sheds light on the competition between the dissipation and the nonlinearity for this equation. It enables to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.
We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) \in L^1 ([0,T]; L^\infty)$. The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (arXiv:1806.08720)
We examine the phenomenon of Landau Damping in relativistic plasmas via a study of the relativistic Vlasov-Poisson system (both on the torus and on $\mathbb{R}^3$) linearized around a sufficiently nice, spatially uniform kinetic equilibrium. We find that exponential decay of spatial Fourier modes is impossible under modest symmetry assumptions. However, by assuming the equilibrium and initial data are sufficiently regular functions of velocity for a given wavevector (in particular that they exhibit a kind of Gevrey regularity), we show that it is possible for the mode associated to this waveve…
We examine the phenomenon of Landau Damping in relativistic plasmas via a study of the relativistic Vlasov-Poisson system (rVP) on the torus for initial data sufficiently close to a spatially uniform steady state. We find that if the steady state is regular enough (essentially in a Gevrey class of degree in a specified range) and that the deviation of the initial data from this steady state is small enough in a certain norm, the evolution of the system is such that its spatial density approaches a uniform constant value sub-exponentially fast (i.e. like $\exp(-C|t|^{\overlineν})$ for $\overlin…
In this paper, we present new estimates for the entropy dissipation of the Landau-Fermi-Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.
