We study the vanishing viscosity limit of the compressible NavierStokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible NavierStokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the NavierStokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.

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.

A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

We consider the nonlinear Schr{ö}dinger equation with a defocusing nonlinearity which is mass-(super)critical and energy-subcritical. We prove uniform in time error estimates for the Lie-Trotter time splitting discretization. This uniformity in time is obtained thanks to a vectorfield which provides time decay estimates for the exact and numerical solutions. This vectorfield is classical in scattering theory, and requires several technical modifications compared to previous error estimates for splitting methods.

The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 14271442] for the L 1 stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A