We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation ()^<i></i>/2, 0 &lt; <i></i> 1). We show in this paper that if , or with is a weak solution of the 2D quasi-geostrophic equation, then <i></i> is a classical solution in . This result improves our previous result in [18].

In this paper we study the existence assertion of the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta e^{-\Delta u}$. This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in the equation can have a singular part in the sense of the Lebesgue decomposition theorem, and the exponential nonlinearity somehow “cancels” it out. The net result is that we obtain a solution $u$ that satisfies the equation and the initial boundary conditions in the almost everywhere (a.e.) sense.

We justify the smoothing effect for measure valued solutions to the space homogeneous Boltzmann equation of Maxwellian type cross sections. This is the first rigorous proof of the smoothing effect for any measure valued initial data except the single Dirac mass, which gives the optimal description on the regularity of solutions for positive time, caused by the singularity in the cross section. The main new ingredient in the proof is the introduction of a time degenerate coercivity estimate by using the microlocal analysis.

In this paper, we consider the Boltzmann equation with soft potentials and prove the stability of a class of non-trivial profiles defined as some given local Maxwellians. The method consists of the analytic techniques for viscous conservation laws, properties of Burnett functions and energy method through the micro-macro decomposition of the Boltzmann equation. In particular, one of the key observations is a detailed analysis of the Burnett functions so that the energy estimates can be obtained in a clear way. As an application of the main results in this paper, we prove the large time nonlinear asymptotic stability of rarefaction waves to the Boltzmann equation with soft potentials.

We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known BuckleyLeverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a ColeHopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the