Spectral and phase space analysis of the linearized non-cutoff Kac collision operator

The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven.     

Global well posedness for the Maxwell–Navier–Stokes system in 2D

We prove global existence of regular solutions to the full MHD system (or more precisely the Maxwell–Navier–Stokes system) in 2D. We also provide an exponential growth estimate for the Hs norm of the solution when the time goes to infinity.     

Mapping Properties of the Boltzmann Collision Operator

In the present paper we study the mapping properties of the nonlinear Boltzmann collision operator on a scale of weighted Bessel potential spaces.     

Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equation for soft and Coulomb potentials

We give an explicit bound for the Wasserstein distance with quadratic cost between the solutions of the Boltzmann and Landau equations in the case of soft and Coulomb potentials. This gives an explicit rate of convergence for the grazing collisions limit. Our result is local in time for very soft and Coulomb potentials and global in time for moderately soft potentials.     

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.     

Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem

We determine the shape which minimizes, among domains with given measure, the first eigenvalue of a nonlocal operator consisting of a perturbation of the standard Dirichlet Laplacian by an integral of the unknown function. We show that this problem displays a saturation behaviour in that the corresponding value of the minimal eigenvalue increases with the weight affecting the average up to a (finite) critical value of this weight, and then remains constant. This critical point corresponds to a transition between optimal shapes, from one ball as in the Faber–Krahn inequality to two equal balls.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

Analyticity estimates for the Navier–Stokes equations

We study spatial analyticity properties of solutions of the three-dimensional Navier–Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier–Stokes equations with data in Hr, r?1/2, and prove a stability result for the analyticity radius.     

An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory

Summary. We propose a finite difference scheme to approximate the Fokker-Planck collision operator in 3 velocity dimensions. The principal feature of this scheme is to provide a decay of the numerical entropy. As a consequence, it preserves the collisional invariants and its stationary solutions are the discrete Maxwellians. We consider both the whole velocity-space problem and the bounded velocity problem. In the latter case, we provide artificial boundary conditions which preserve the decay of the entropy. Received October 18, 1993     

Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations

We consider in this paper the relativistic Euler equations in isentropic fluids with the equation of state p = κ²ρ, where κ, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm’s method.     

Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds

For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt–Caffarelli–Friedman and Caffarelli–Jerison–Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace–Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

The relaxation-time limit in the compressible Euler–Maxwell equations

The aim of this paper is to study multidimensional Euler–Maxwell equations for plasmas with short momentum relaxation time. The convergence for the smooth solutions to the compressible Euler–Maxwell equations toward the solutions to the smooth solutions to the drift–diffusion equations is proved by means of the Maxwell iteration, as the relaxation time tends to zero. Meanwhile, the formal derivation of the latter from the former is justified.     

Singular linear space and its applications

As a generalization of attenuated spaces, the concept of singular linear spaces was introduced in [K. Wang, J. Guo, F. Li, Association schemes based on attenuated spaces, European J. Combin. 31 (2010) 297–305]. This paper first gives two anzahl theorems in singular linear spaces, and then discusses their applications to the constructions of Deza digraphs, quasi-strongly regular graphs, lattices and authentication codes.     

Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.     

Operator ergodic theory for one-parameter decomposable groups

After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.     

On the computation of parameter derivatives of solutions of linear difference equations

A method is given to compute the parameter derivatives of recessive solutions of second-order inhomogeneous linear difference equations. The case of difference equations in which all solutions have the same rate of growth is also discussed.     

Liouville type theorems for the Euler and the Navier–Stokes equations

We prove Liouville type theorems for weak solutions of the Navier–Stokes and the Euler equations. In particular, if the pressure satisfies p∈L¹(0,T;L¹(RN)) with , then the corresponding velocity should be trivial, namely v=0 on RN×(0,T). In particular, this is the case when p∈L¹(0,T;Hq(RN)), where Hq(RN), q∈(0,1], the Hardy space. On the other hand, we have equipartition of energy over each component, if p∈L¹(0,T;L¹(RN)) with . Similar results hold also for the magnetohydrodynamic equations.     

Uniform stability estimates for solutions and their gradients to the Boltzmann equation: A unified approach

As to the Cauchy problem for the spatially inhomogeneous Boltzmann equation with cut-off, we prove uniform stability estimates for solutions and their gradients in a unified and elementary way.     

Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform

We extend the Lp-theory of the Boltzmann collision operator by using classical techniques based in the Carleman representation and Fourier analysis, allied to new ideas that exploit the radial symmetry of this operator. We are then able to greatly simplify existent technical proofs in this theory, extend the range, and obtain explicit sharp constants in some convolution-like inequalities for the gain part of the Boltzmann collision operator.