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Kinetic and Related Models

August 2022 , Volume 15 , Issue 4

Special Issue on nonlinear waves and kinetic theory dedicated to the memory of Bob Glassey. Part II

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Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $
David Hoff
2022, 15(4): 535-550 doi: 10.3934/krm.2021037 +[Abstract](491) +[HTML](200) +[PDF](387.35KB)
Abstract:

We derive pointwise bounds for the Green's function and its derivatives for the Laplace operator on smooth bounded sets in \begin{document}$ {\bf R}^3 $\end{document} subject to Neumann boundary conditions. The proofs require only ordinary calculus, scaling arguments and the most basic facts of \begin{document}$ L^2 $\end{document}-Sobolev space theory.

Lagrangian dual framework for conservative neural network solutions of kinetic equations
Hyung Ju Hwang and Hwijae Son
2022, 15(4): 551-568 doi: 10.3934/krm.2021046 +[Abstract](429) +[HTML](187) +[PDF](1775.3KB)
Abstract:

In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.

Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus
Jin Woo Jang, Robert M. Strain and Tak Kwong Wong
2022, 15(4): 569-604 doi: 10.3934/krm.2021039 +[Abstract](633) +[HTML](220) +[PDF](689.04KB)
Abstract:

Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system.

Kinetic description of stable white dwarfs
Juhi Jang and Jinmyoung Seok
2022, 15(4): 605-620 doi: 10.3934/krm.2021033 +[Abstract](517) +[HTML](237) +[PDF](426.57KB)
Abstract:

In this paper, we study fermion ground states of the relativistic Vlasov-Poisson system arising in the semiclassical limit from relativistic quantum theory of white dwarfs. We show that fermion ground states of the three dimensional relativistic Vlasov-Poisson system exist for subcritical mass, the mass density of such fermion ground states satisfies the Chandrasekhar equation for white dwarfs, and that they are orbitally stable as long as solutions exist.

Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D
Boyan Jonov, Paul Kessenich and Thomas C. Sideris
2022, 15(4): 621-649 doi: 10.3934/krm.2021038 +[Abstract](569) +[HTML](216) +[PDF](481.46KB)
Abstract:

The initial value problem for incompressible Hookean viscoelastic motion in three space dimensions has global strong solutions with small displacements.

A second look at the Kurth solution in galactic dynamics
Markus Kunze
2022, 15(4): 651-662 doi: 10.3934/krm.2021028 +[Abstract](641) +[HTML](315) +[PDF](416.16KB)
Abstract:

The Kurth solution is a particular non-isotropic steady state solution to the gravitational Vlasov-Poisson system. It has the property that by means of a suitable time-dependent transformation it can be turned into a family of time-dependent solutions. Therefore, for a general steady state \begin{document}$ Q(x, v) = \tilde{Q}(e_Q, \beta) $\end{document}, depending upon the particle energy \begin{document}$ e_Q $\end{document} and \begin{document}$ \beta = \ell^2 = |x\wedge v|^2 $\end{document}, the question arises if solutions \begin{document}$ f $\end{document} could be generated that are of the form

for suitable functions \begin{document}$ R $\end{document}, \begin{document}$ P $\end{document} and \begin{document}$ B $\end{document}, all depending on \begin{document}$ (t, r, p_r, \beta) $\end{document} for \begin{document}$ r = |x| $\end{document} and \begin{document}$ p_r = \frac{x\cdot v}{|x|} $\end{document}. We are going to show that, under some mild assumptions, basically if \begin{document}$ R $\end{document} and \begin{document}$ P $\end{document} are independent of \begin{document}$ \beta $\end{document}, and if \begin{document}$ B = \beta $\end{document} is constant, then \begin{document}$ Q $\end{document} already has to be the Kurth solution.

This paper is dedicated to the memory of Professor Robert Glassey.

Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach
Zhiwu Lin
2022, 15(4): 663-679 doi: 10.3934/krm.2021048 +[Abstract](441) +[HTML](150) +[PDF](474.05KB)
Abstract:

We consider linear stability of steady states of 1\begin{document}$ \frac{1}{2} $\end{document} and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.

The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation
Sebastian Günther and Gerhard Rein
2022, 15(4): 681-719 doi: 10.3934/krm.2021040 +[Abstract](649) +[HTML](188) +[PDF](578.02KB)
Abstract:

We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in maximal areal coordinates. The latter coordinates have been used both in analytical and numerical investigations of the Einstein-Vlasov system [3,8,18,19], but neither a local existence theorem nor a suitable continuation criterion has so far been established for the corresponding nonlinear system of PDEs. We close this gap. Although the analysis follows lines similar to the corresponding result in Schwarzschild coordinates, essential new difficulties arise from to the much more complicated form which the field equations take, while at the same time it becomes easier to control the necessary, highest order derivatives of the solution. The latter observation may be useful in subsequent investigations.

On time decay for the spherically symmetric Vlasov-Poisson system
Jack Schaeffer
2022, 15(4): 721-727 doi: 10.3934/krm.2021021 +[Abstract](760) +[HTML](380) +[PDF](256.9KB)
Abstract:

A collisionless plasma is modeled by the Vlasov-Poisson system. Solutions in three space dimensions that have smooth, compactly supported initial data with spherical symmetry are considered. An improved field estimate is presented that is based on decay estimates obtained by Illner and Rein. Then some estimates are presented that ensure only particles with sufficiently small velocity can be found within a certain (time dependent) ball.

Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation
Fedor Nazarov and Kevin Zumbrun
2022, 15(4): 729-752 doi: 10.3934/krm.2022012 +[Abstract](152) +[HTML](52) +[PDF](554.38KB)
Abstract:

We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$ (0, +\infty) $\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $\end{document} stable manifolds of such equations, showing that \begin{document}$ L^2_{loc} $\end{document} solutions that remain sufficiently small in \begin{document}$ L^\infty $\end{document} (i) decay exponentially, and (ii) are \begin{document}$ C^\infty $\end{document} for \begin{document}$ t>0 $\end{document}, hence lie eventually in the \begin{document}$ H^1 $\end{document} stable manifold constructed by Pogan and Zumbrun.

2021 Impact Factor: 1.398
5 Year Impact Factor: 1.685
2021 CiteScore: 2.7

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