In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier-Stokes equations with constant viscosity, the main difficulty here lies in the fact that the L^1 in time Lipschitz estimate of the velocity field can not be obtained by energy method. Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid, namely, striated regularity can help to get the L^/infty boundedness of the double Riesz transform, we derive the a priori L^1 in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space.

In his famous work on resolution of singularities, Slodowy pointed out that the McKay correspondence could be generalized to all affine Dynkin diagrams and showed this for most of the cases. We will first thoroughly explain the McKay-Slodowy correspondence using finite group theory, including the missing cases of A_{2n}^{(2)} and A_2^{(2)} in the literature. After the McKay-Slodowy correspondence is firmly established, we then show that the Poincare series for the tensor algebra of the induction and restriction of the fundamental modules realize all exponents of affine Lie algebras except A_{2n}^{(1)}. This is joint work with Danxia Wang and Honglian Zhang.

We are concerned with the three-dimensional time-dependent incompressible magnetohydrodynamic (MHD) equations with magnetic vector potential formulations. Compared with the traditional B formulations, the new MHD system has the advantage that it can ensure a direct exact discrete divergence-free magnetic induction in practical numerical computations. Using a mixed finite element approach, we discretize the velocity and pressure by stable finite elements, and the mag

Exciton diffusion length plays a vital role in the function of opto-electronic devices. Oftentimes, the domain occupied by a organic semiconductor is subject to surface measurement error. In many experiments, photoluminescence over the domain is measured and used as the observation data to estimate this length parameter in an inverse manner based on the least square method. However, the result is sometimes found to be sensitive to the surface geometry of the domain. We propose an asymptotic-based method as an approximate forward solver whose accuracy is justified both theoretically and numerically.

This talk is concerned with a type of nonlinear Schrodinger equation with potential possessing non-isolated critical points. we obtain the necessary condition, existence and local uniqueness of the positive single peak solution with concentrating at this kind of points. These types of results will also be mentioned for the BEC model. Here the main difficulty is the degeneracy and inhomogeneity of the potantial at the concentrating point.

In combustion theory, the propagation of premixed flames is usually described by the conventional thermal-diffusional model with standard Arrhenius kinetics. Formal asymptotic methods based on large activation energy have allowed simpler descriptions, especially when the thin flame zone is replaced by a free interface, called the flame front, which separates burned and unburned gases. At the flame front, the temperature and mass fraction gradients are discontinuous.

We get estimates on the higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, in terms of the eigenvalues of the Laplace–Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result (Alexandrov, 2007) of Alexandrov.