Negative Sobolev spaces and the two-species Vlasov–Maxwell–Landau system in the whole space

A global solvability result of the Cauchy problem of the two-species Vlasov–Maxwell–Landau system near a given global Maxwellian is established by employing an approach different than that of [2]. Compared with that of [2], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker. Our analysis does not rely on the decay of the corresponding linearized system and the Duhamel principle and thus it can be used to treat the one-species Vlasov–Maxwell–Landau system for the case of γ>−3$\gamma >-3$ and the one-species Vlasov–Maxwell–Boltzmann system for the case of −1<γ≤1$-1<\gamma \le 1$ to deduce the global existence results together with the corresponding temporal decay estimates.     

The non-cutoff boltzmann equation with potential force in the whole space

This paper is concerned with the non-cutoff Boltzmann equation for full-range interactions with potential force in the whole space. We establish the global existence and optimal temporal convergence rates of classical solutions to the Cauchy problem when initial data is a small perturbation of the stationary solution. The analysis is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [1–3, 15] and the non-cutoff Vlasov-Poisson-Boltzmann system [6].     

Spatially homogeneous solutions of the Vlasov–Nordström–Fokker–Planck system

The Vlasov–Nordström–Fokker–Planck system describes the evolution of self-gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation. We study the spatially-homogeneous system and prove global existence and uniqueness of solutions for the corresponding initial value problem in three momentum dimensions. Additionally, we study the long time asymptotic behavior of the system and prove that even in the absence of friction, solutions possess a non-trivial asymptotic profile. An exact formula for the long time limit of the particle density is derived in the ultra-relativistic case.     

High-electric-field limit for the Vlasov–Maxwell–Fokker–Planck system

In this paper we derive the high-electric-field limit of the three-dimensional Vlasov–Maxwell–Fokker–Planck system. We use the relative entropy method which requires the smoothness of the solution of the limit problem. We obtain convergences of the electro-magnetic field, charge and current densities.     

A PIC method for solving a paraxial model of highly relativistic beams

Solving the Vlasov–Maxwell problem can lead to very expensive computations. To construct a simpler model, Laval et al. [G. Laval, S. Mas-Gallic, P.A. Raviart, Paraxial approximation of ultrarelativistic intense beams, Numer. Math. 69 (1) (1994) 33–60] proposed to exploit the paraxial property of the charged particle beams, i.e the property that the particles of the beam remain close to an optical axis. They so constructed a paraxial model and performed its mathematical analysis. In this paper, we investigate how their framework can be adapted to handle the axisymmetric geometry, and its coupling with the Vlasov equation. First, one constructs numerical schemes and error estimates results for this discretization are reported. Then, a Particle In Cell (PIC) method, in the case of highly relativistic beams is proposed. Finally, numerical results are given. In particular, numerical comparisons with the Vlasov–Poisson model illustrate the possibilities of this approach.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

Decay of the two-species Vlasov–Poisson–Boltzmann system

We establish the time decay rates of the solution to the Cauchy problem for the two-species Vlasov–Poisson–Boltzmann system near Maxwellians via a refined pure energy method. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov–Poisson–Boltzmann system or the Navier–Stokes–Poisson equations in which the electric field decays at the optimal algebraic rate, and compared to the Vlasov–Boltzmann system in which the disparity between two species decays at the optimal algebraic rate. Our achievement heavily relies on a reformulation of the problem which well displays the cancelation property of the two-species system, and our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.     

Stationary solutions of non-autonomous Maxwell–Dirac systems

The Maxwell–Dirac system describes the interaction of a particle with its self-generated electromagnetic field. In this paper, we study the existence of least energy stationary solutions for non-autonomous Maxwell–Dirac system with a nonlinear term in (3+1)$(3+1)$-Minkowski space–time via variational methods. This problem is strongly indefinite and presents a lack of compactness. To overcome these difficulties, we will use the linking and concentration compactness arguments.     

A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension

The Vlasov–Fokker–Planck equation is a model for a collisional, electrostatic plasma. The approximation of this equation in one spatial dimension is studied. The equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function. The approximation method applied is the deterministic particle method described in Wollman and Ozizmir [Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. For the present linear problem an analysis of the stability and convergence of the numerical method is carried out. In addition, computations are done that verify the convergence of the numerical solution. It is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in Bouchut and Dolbeault [On long time asymptotics of the Vlasov–Fokker–Planck equation and of the Vlasov–Poisson–Fokker–Planck system with coulombic and Newtonian potentials, Differential Integral Equations 8(3) (1995) 487–514].     

Large-time behavior for the solution to the generalized Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space

In this paper, we consider the global existence as well as the optimal decay estimates of the Cauchy problem for the multi-dimensional Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space. And these results are obtained by Green?s function method, Fourier analysis method, energy estimates method combined with the time-frequency decomposition method.     

Weak turbulence plasma induced by two-scale homogenization

This paper is devoted to the homogenization of the quasilinear theory of the plasma turbulence described by the Vlasov–Poisson system. It is shown that the homogenization limit, in the sense of two-scale limit, of the distribution function satisfies the linear Vlasov–Poisson equations. Moreover, the limit distribution function can be decomposed into the mean and the fluctuation parts and the mean part (the equilibrium distribution function) is shown to be the solution of the nonlocal quasilinear velocity-space diffusion equation. We also investigate the Landau damping from the point of view of homogenization through the two-scale limit.     

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.     

Stability of stationary solutions for the non-isentropic Euler-Maxwell system in the whole space

In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R³. It is known in the authors’ previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method.     

Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds

We prove the Bochner–Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li–Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry–Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.     

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo–Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber–Krahn inequalities or localised Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies–Gaffney estimates.     

Global existence of strong solution for the Cucker–Smale–Navier–Stokes system

We present a global existence theory for strong solution to the Cucker–Smale–Navier–Stokes system in a periodic domain, when initial data is sufficiently small. To model interactions between flocking particles and an incompressible viscous fluid, we couple the kinetic Cucker–Smale model and the incompressible Navier–Stokes system using a drag force mechanism that is responsible for the global flocking between particles and fluids. We also revisit the emergence of time-asymptotic flocking via new functionals measuring local variances of particles and fluid around their local averages and the distance between local averages velocities. We show that the particle and fluid velocities are asymptotically aligned to the common velocity, when the viscosity of the incompressible fluid is sufficiently large compared to the sup-norm of the particles' local mass density.     

Non-existence of vortices in the small density region of a condensate

In this paper, we answer a question raised by Lev Pitaevskii and prove that the ground state of the Gross-Pitaevskii energy describing a Bose-Einstein condensate in a rotationally symmetric trap at low rotation does not have vortices in the low density region. Therefore, the first ground state with vortices has its vortices in the bulk. In fact we prove something stronger, which is that the ground state for the model at low and moderate rotations is equal to the ground state in a condensate with no rotation. This is obtained by proving that for small rotational velocities, the ground state is multiple of the ground state with zero rotation. We rely on sharp bounds of the decay of the wave function combined with weighted Jacobian estimates.     

Existence of charged vortices in a Maxwell–Chern–Simons model

In this paper, we prove the existence of charged vortex solitons in a Maxwell–Chern–Simons model. We establish the main existence theorem by a constrained minimization method applied on an indefinite action functional which is induced from the original field-theoretical Lagrangian. We also show that the solutions obtained are smooth.     

An Oseen scheme for the conduction–convection equations based on a stabilized nonconforming method

An Oseen iterative scheme for the stationary conduction–convection equations based on a stabilized nonconforming finite element method is given. The stability and error estimates are analyzed, which show that the presented method is stable and has good precision. Numerical results are shown to support the developed theory analysis and demonstrate the good effectiveness of the given method.