The time evolution of the distribution function for the charged particles in a dilute gas is governed by the VlasovPoissonBoltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial data is a small perturbation of a global Maxwellian. Moreover, the convergence rate in time to the global Maxwellian is also obtained through the energy method. The proof is based on the theory of compressible NavierStokes equations with forcing and the decomposition of the solutions to the Boltzmann equation with respect to the local Maxwellian introduced in [23] and elaborated in [31].

In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of $z$-weak solution is established for any such initial datum that itself and its vertical derivative belong to $L^2$. This not only extends the results in \cite{Cao5} from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be $H^2$ there.

This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m ≤ 2 − 2 / d, the L p space for both dynamic and steady solutions are detected with ${p:=\frac{d(2-m)}{2} }$ . Firstly, the global existence of the weak solution is proved for small initial data in L p . Moreover, when m > 1 − 2 / d, the weak solution preserves mass and satisfies the hyper-contractive estimates in L q for any p < q < ∞. Furthermore, for slow diffusion 1 < m ≤ 2 − 2/d, this weak solution is also a weak entropy solution which blows up at finite time provided by the initial negative free energy. For m > 2 − 2/d, the hyper-contractive estimates are also obtained. Finally, we focus on the L p norm of the steady solutions, it is shown that the energy critical exponent m = 2d/(d + 2) is the critical exponent separating finite L p norm and infinite L p norm for the steady state solutions.

We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.

We prove the convergence of vortex blob methods to classical weak solutions for the twodimensional incompressible Euler equations with initial data satisfying the conditions that the vorticity is a finite Radon measure of distinguished sign and the kinetic energy is locally bounded. This includes the important example of vortex sheets. The result is valid as long as the computational grid size h does not exceed the smoothing blob size E, i.e., h/~ 5 C. 0 1995 John Wiley & Sons, Inc.