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

We simulate the random front solutions of a nonlinear solute transport equation with spatial random coefficients modeling inhomogeneous sorption sites in porous media. The nonlinear sorption function is chosen to be Langmuir type, and the random coefficients are two independent stationary processes with fast decay of correlations. The model equation is in conservation form, and the random fronts are similar to random viscous shocks. We find that the average front speed is given by an ensemble averaged explicit RankineHugoniot relation, and the front position fluctuates about its mean. Our numerical calculations show that the standard deviation is of the order O( \sqrt t ) for large time, and the front fluctuation scaled by \sqrt t converges to a Gaussian random variable wih mean zero. We come up with a formal theory of front fluctuation, yielding an explicit expression of the root <i>t</i> normalized front standard

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].

We show that in a tubular domain with sufficiently small width, the normal and tangential gradients of a harmonic function have almost the same L 2 norm. This estimate yields a sharp estimate of the pressure in terms of the viscosity term in the Navier-Stokes equation with no-slip boundary condition. By consequence, one can analyze the Navier-Stokes equations simply as a perturbed vector diffusion equation instead of as a perturbed Stokes system. As an application, we describe a rather easy approach to establish a new isomorphism theorem for the non-homogeneous Stokes system.

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.