The dynamics of dilute electrons can be modelled by the fundamental VlasovPoissonBoltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any smooth perturbation of a given global Maxwellian leads to a unique global-in-time classical solution when either the mean free path is small or the background charge density is large. Moreover, the solution converges to the global Maxwellian when time tends to infinity. The analysis combines the techniques used in the study of conservation laws with the decomposition of the Boltzmann equation introduced in [17, 19] by obtaining new entropy estimates for this physical model.

We consider the self-adjoint Smilansky Hamiltonian H in L2 R2) associated with the formal differential expression -x2-1/2(y2+y2)-2y(x) in the sub-critical regime, (0, 1). We demonstrate the existence of resonances for H on a countable subfamily of sheets of the underlying Riemann surface whose distance from the physical sheet is finite. On such sheets, we find resonance free regions and characterize resonances for small > 0. In addition, we refine the previously known results on the bound states of H in the weak coupling regime (0+). In the proofs we use Birman-Schwinger principle for H, elements of spectral theory for Jacobi matrices, and the analytic implicit function theorem.

Abstract We study the GinzburgLandau equation on the plane with initial data being the product of n wellseparated+ 1 vortices and spatially decaying perturbations. If the separation distances are O ( 1), l, we prove that the n vortices do not move on the time scale O(^2),=o(1\over); instead, they move on the time scale O(^-21\over) according to the law j= xj W, W= l j log| xl xj|, xj=(j, j) 2, the location of the jth vortex. The main ingredients of our proof consist of estimating the large space behavior of solutions, a monotonicity inequality for the energy density of solutions, and energy comparisons. Combining these, we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law. John & Wiley Sons, Inc.

This paper is concerned with the Cauchy problem of the one-dimensional compressible Navier--Stokes equations with degenerate temperature dependent transport coefficients which satisfy conditions from the consideration in kinetic theory. A result on the existence and uniqueness of a globally smooth nonvacuum solution is obtained provided that the -norm of the initial perturbation) for some positive constant independent of . Here is the adiabatic gas constant. This is a Nishida--Smoller type global solvability result with large data.

In this paper, we study the strong convergence of a sequence of uniform<i>L^p</i>_loc(<b>R</b><b>R</b>⁺) bounded approximate solutions {<i>u</i>(<i>x</i>,<i>t</i>)} to the following scalar conservation laws[formula]with initial data[formula]Without the convexity assumption and growth condition at infinity for<i>f</i>(<i>x</i>,<i>t</i>,<i>u</i>), we prove strong convergence of a subsequence of {<i>u</i>(<i>x</i>,<i>t</i>)}. Under a more general growth condition than those in the previous work, we prove the existence of weak solution for the equation. The result obtained here generalizes those in earlier work. Some applications of the results are also given at the end of this paper.