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 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

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For the physical vacuum free boundary problem with the sound speed being C^1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.

We study homogenization of G-equation with a ow straining term (or the strain G-equation) in two dimensional periodic cellular ow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton-Jacobi equation. The main objective is to investigate how the ow induced straining (the nonconvex term) in uences front propagation as the ow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent ame speed sT (effective Hamiltonian) is well-defined for any periodic ow with small divergence and is enhanced by the cellular ow as $s_T \ge O(A/logA)$. In the second regime where A is slightly above the critical value, homogenization breaks down, and $s_T$ is not well defined along any direction. Solutions become a mixture of fast moving part and a stagnant part. When $A$ is sufficiently large, the whole ame front ceases to propagate forward due to the flow induced straining. In particular, along directions $p = (1; 0)$ and $(0;1)$, $s_T$ is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player dierential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.