The development of shocks in nonlinear hyperbolic conservation laws may be regularized through either diffusion or relaxation. However, we have observed surprisingly that for some physical problems, when both of the smoothing factors - diffusion and relaxation - coexist, under appropriate asymptotic assumptions, the dispersive waves are enhanced. This phenomenon is studied asymptotically in the sense of the Chapman-Enskog expansion and demonstrated numerically.

The weak well-posedness of strong damping wave equations defined by fractal Laplacians is proved by using Galerkin method. These fractal Laplacians are defined by self-similar measures with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio, the three-fold convolution of the Cantor measure, and a class of self-similar measures that we call essentially of finite type. In general, the structure of self-similar measures with overlap are complicated and intractable. However, some important information about the structure of the three measures above can be obtained. We make use of these information to set up a framework for one-dimensional measures to discretize the equations, and use the finite element and central difference methods to obtain numerical approximations of the weak solutions. We also show that the numerical solutions converge to the actual solution and obtain the rate of convergence.

In this paper, we study the global well-posedness of the 2D compressible NavierStokes equations with large initial data and vacuum. It is proved that if the shear viscosity<i></i> is a positive constant and the bulk viscosity <i></i> is the power function of the density, that is, <i></i>(<i></i>) = <i></i> ^<i></i> with <i></i> &gt; 3, then the 2D compressible NavierStokes equations with the periodic boundary conditions on the torus T 2 admit a unique global classical solution (<i>, u</i>) which may contain vacuums in an open set of T 2 . Note that the initial data can be arbitrarily large to contain vacuum states.

This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the nonequilibrium part of the distri-bution function. This equation is solved only locally and is related to the fluid equation through a downscaling effect. The method does not need to find an interface condition as do usual domain decomposition methods to match fluid and kinetic representations. We show our approach applies to problems that have a hydrodynamic time scale as well as to problems with diffusion time scale. Simple numerical schemes are proposed to discretize our models, and several numerical examples are used to validate the method.

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin lm equation in one-dimensional case ht + @x(h n @xxxh) + @x(h n+2@xh) = 0; where n  1. There exists a critical mass Mc = 2 p 6 3 found by Witelski etal. (2004 Euro. J. of Appl. Math. 15, 223{256) for n = 1. We obtain global existence of a non-negative entropy weak solution if initial mass is less than Mc. For n  4, entropy weak solutions are positive and unique. For n = 1,a nite time blow-up occurs for solutions with initial mass larger than Mc.For the Cauchy problem with n = 1 and initial mass less than Mc, we show that at least one of the following long-time behavior holds: the second moment goes to innity as the time goes to innity or h(
; tk) * 0 in L1 (R) for some subsequence tk ! 1.