Kinetic & Related Models
December 2014 , Volume 7 , Issue 4
Issue on nonlinear elliptic and parabolic partial differential equations
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We consider the time decay rates of the solution to the Cauchy problem for the compressible Euler equations with damping. We prove the optimal decay rates of the solution as well as its higher-order spatial derivatives. The damping effect on the time decay estimates of the solution is studied in details.
Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.
We review some classical and more recent results on the mean field limit and propagation of chaos for systems of many particles, leading to Vlasov or macroscopic equations.
This paper presents stability and convergence analysis of a finite volume scheme for solving aggregation, breakage and the combined processes by showing consistency of the method and Lipschitz continuity of numerical fluxes. It is investigated that the finite volume scheme is second order convergent independently of the meshes for pure breakage problem while for pure aggregation and coupled problems, it indicates second order convergence on uniform and non-uniform smooth meshes. Furthermore, it gives only first order accuracy on non-uniform meshes. The mathematical results of convergence analysis are also demonstrated numerically for several test problems.
We study the convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible model with ill-prepared initial data in critical Besov spaces. Under the condition that the initial data is small in some norm, we show that the convergence holds globally as the Mach number goes to zero. Moreover, we also obtain the convergence rate.
N. N. Bogolyubov discovered that the Boltzmann--Enskog kinetic equation has microscopic solutions. They have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these solutions by introduction of a special regularization of the delta-functions. The crucial observation is that the collision integral of the Boltzmann--Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres if the special regularization to the delta-functions is applied. This allows to reduce the nonlinear Boltzmann--Enskog equation to the BBGKY hierarchy of linear equations in this particular case.
Also we show that similar functions are exact smooth solutions for the recently proposed generalized Enskog equation. They can be referred to as ``particle-like'' or ``soliton-like'' solutions and are analogues of multisoliton solutions of the Korteweg--de Vries equation.
We study the magnetohydrodynamics system, generalized via a fractional Laplacian. When the domain is in $N-$dimension, $N$ being three, four or five, we show that the regularity criteria of its solution pair may be reduced to $(N-1)$ many velocity field components with the improved integrability condition in comparison to the result in . Furthermore, we extend this result to the three-dimensional magneto-micropolar fluid system.
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