In this paper we investigate the asymptotic stability of viscous shock wave for a onedimensional isentropic model of viscous gas with density dependent viscosity by a weighted energy method developed in the papers of Under the condition that the viscosity coefficient is given as a function of the absolute temperature which is determined by the Chapman-Enskog expansion theory in rarefied gas dynamics, any viscous shock wave is shown to be asymptotically stable for small initial perturbations with integral zero. This generalizes the previous result of Matsumua-Nishihara (1985) where the viscosity coefficient is given by a constant and a restriction on the strength of the viscous shock wave is assumed. This also analytically assures the spectral stability in the Zumbrun's theory for any viscous shock wave in our specific case.

In this paper, we solve the Riemann problem for a coupled hyperbolic system of conservation laws, which arises as an intermediate model in the flux splitting method for the computation of Euler equations in gasdynamics. We study the properties of solutions involving shock and rarefaction waves, and establish their existence and uniqueness. We present numerical examples for different initial data, and finally discuss all possible elementary wave interactions; it is noticed that in certain cases the resulting wave pattern after interaction is substantially different from that which arises in isentropic gasdynamics.

In this paper, we study the zero dissipation limit problem for the Navier-Stokes equations of one-dimensional compressible viscous heat-conducting fluids. We prove that if the solution of the inviscid Euler equations is piecewise smooth with finitely many noninteracting shocks satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from shock discontinuities at a rate of 1 as the viscosity tend to zero, provided that the heat-conducting coefficient k = O().

We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrodinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKBtype approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared. We also show by an analysis of the limit hydrodynamical equaiton that, however, this WKB-type approximation breaks down before reaching the focal point: Nonlinear effects lead to the formation of singularity of the leading term of the phase function.

We study the convergence rate of solutions to the initial-boundary value problem for scalar viscous conservation laws on the half line. Especially, we deal with the case where the Riemann problem for the corresponding hyperbolic equation admits transonic rarefaction waves. In this case, it is known that the solution tends toward a linear superposition of the stationary solution and the rarefaction wave. We show that the convergence rate is (1 + t).1 2 (1. 1 p ) log2(2 + t) in Lp norm (1  p < 1) and (1 + t).1 2+. in L1 norm if the initial perturbation from the corresponding superposition is located in H1 L1. The proof is given by a combination of the weighted Lp energy method and the L1 estimate.

We study the solutions of a quasilinear Schr¨odinger equation which has been derived in many areas of physical modeling. Using the Morawetz Radial Identity, we show that the local energy of a solution is integrable in time and the local L2 norm of the solution approaches zero as time approaches the infinity.

In this paper, we establish a blow up criterion for strong solutions of the full compressible Navier-Stokes equations just in terms of the gradient of the velocity. It shows that the gradient of the velocity alone dominates the global existence of strong solutions.

A typical example is that f(u) = u−μ1u −μ2 with constants p > max(q, 0). This kind of semilinear equation appears in several applications in mechanics and physics. In particular, it has been used to model the dynamics of thin films for viscous liquids. Some detailed physics background can be found in [1]-[3], [9] and [12]-[14]. Some recent mathematical analysis can be found in [4, 6, 7, 8, 10, 11, 15, 17] and the references therein. It was proved in [8] (see also [11]) that in dimension N ≥ 3, for each α ∈ (0, t0), (1.1) has a unique positive solution uα. Moreover, uα oscillates around the constant t0, that is, there is an increasing positive sequence {rn α} such that {r ∈ (0,∞) : uα(r) = 0} = {rn α}, and limn→∞ r α = ∞. Here r α are local maxima of uα with uα(r 2i+1 α ) > t0 for any i ∈ N; while r α are local minima with uα(r α) 0 for r ∈ (0,∞) and f(u0) ∈ Lloc(R ). Moreover, any singular radial solution of (1.1) is oscillatory around t0 and converges to t0 as r → ∞.

Variational method is a useful mathematical tool in various areas of research, espe- cially in image processing. Recently, solving image processing problems on general manifolds using variational techniques has become an important research topic. In this paper, we solve several image processing problems on general Riemann surfaces using variational models defined on surfaces. We propose an explicit method to solve variational problems on general Riemann surfaces, using the conformal parameterization and covariant derivatives defined on the surface. To simplify the com- putation, the surface is firstly mapped conformally to the two dimensional rectangular domains, by computing the holomorphic 1-form on the surface. It is well known that the Jacobian of a conformal map is simply the scalar multiplication of the conformal factor. Therefore with the conformal para- meterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean differential operators, except for the scalar multiplication. As a result, any variational problem on the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well developed numerical scheme on the 2D domain. With the proposed method, we solve various image processing problems on surfaces using different variational models, which include image segmen- tation, surface denoising, surface inpainting, texture extraction and automatic landmark tracking. Experimental results show that our method can effectively solve variational problems and tackle image processing tasks on general Riemann surfaces.

The singular parabolic problem ut = ∆u − λf(x) (1+u)2 on a bounded domain Ω of R with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage —represented here by λ. As a byproduct, we prove that for sufficiently large λ, finitetime quenching must occur near the maximum point of the varying dielectric permittivity profile f(x).