The contact discontinuity is one of the basic wave patterns in gas motions. The stability of contact discontinuities with general perturbations for the NavierStokes equations and the Boltzmann equation is a long standing open problem. General perturbations of a contact discontinuity may generate diffusion waves which evolve and interact with the contact wave to cause analytic difficulties. In this paper, we succeed in obtaining the large time asymptotic stability of a contact wave pattern with a convergence rate for the NavierStokes equations and the Boltzmann equation in a uniform way. One of the key observations is that even though the energy norm of the deviation of the solution from the contact wave may grow at the rate (1+ t) 1 4, it can be compensated by the decay in the energy norm of the derivatives of the deviation which is of the order of (1+ t) 1 4. Thus, this reciprocal order of decay rates for the time

The first part of this paper describes some important underlying themes in the mathematical theory of continuum mechanics that are distinct from formulating and analyzing governing equations. The main part of this paper is devoted to a survey of some concrete, conceptually simple, pretty problems that help illuminate the underlying themes. The paper concludes with a discussion of the crucial role of invariant constitutive equations in computation.

We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.

The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions

Let G=(V,E) be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on G (the Schrödinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.