THE LOCAL WELL-POSEDNESS OF A CHEMOTAXIS-SHALLOW WATER SYSTEM WITH VACUUM

In this paper we prove the local well-posedness of strong solutions to a chemotaxisshallow water system with initial vacuum in a bounded domainΩ■R^(2)without the standard compatibility condition for the initial data.This improves some results obtained in[J.Differential Equations 261(2016),6758-6789].

On base manifolds of Lagrangian fibrations

We consider base spaces of Lagrangian fibrations from singular symplectic varieties. After defining cohomologically irreducible symplectic varieties, we construct an example of Lagrar, gian fibration whose base space is isomorphic to a quotient of the projective space. We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.

Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions

This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t →-∞ in the energy norm, and to show it has a free profile as t →＋∞. Our approach is based on the work of [11]. Namely we use a weighted L^∞ norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper.

The Gauss maps of Demoulin surfaces with conformal coordinates

Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space.

On fluorescence imaging:The diffusion equation model and recovery of the absorption coefficient of fluorophores

To quantify fluorescence imaging of biological tissues,we need to solve an inverse problem for the coupled radiative transfer equations which describe the excitation and emission fields in biological tissues.We begin by giving a concise mathematical argument to derive coupled diffusion equations with the Robin boundary condition as an approximation of the radiative transfer system.Then by using this coupled system of equations as a model for the fluorescence imaging,we have a nonlinear inverse problem to identify the absorption coefficient in this system.The associated linearized inverse problem is to ignore the absorbing effect on the excitation field.We firstly establish the estimates of errors on the excitation field and the solution to the inverse problem,which ensures the reasonability of the model approximation quantitatively.Some numerical verification is presented to show the validity of such a linearizing process quantitatively.Then,based on the analytic expressions of excitation and emission fields,the identifiability of the absorption coefficient from the linearized inverse problem is rigorously analyzed for the absorption coefficient in the special form,revealing the physical difficulty of the3-dimensional imaging model by the back scattering diffusive system.

NUMERICAL IMPLEMENTATION FOR A 2-D THERMAL INHOMOGENEITY THROUGH THE DYNAMICAL PROBE METHOD

In this paper, we present the theory and numerical implementation for a 2-D thermal inhomogeneity through the dynamical probe method. The main idea of the dynamical probe method is to construct an indicator function associated with some probe such that when the probe touch the boundary of the inclusion the indicator function will blow up. From this property, we can get the shape of the inclusion. We will give the numerical reconstruction algorithm to identify the inclusion from the simulated Neumann-to-Dirichlet map.

Local Well-Posedness for the Compressible Nematic Liquid Crystals Flow with Vacuum

In this paper we prove the local well-posedness of strong solutions to the compressible nematic liquid crystals flow with vacuum in a bounded domainΩ■R3.

Uniform Regularity for the Isentropic Compressible Magnetohydrodynamic System

In this paper we first establish the uniform regularity of smooth solutions with respect to the viscosity coefficients to the isentropic compressible magnetohydrodynamic system in a periodic domain T;.We then apply our result to obtain the isentropic compressible magnetohydrodynamic system with zero viscosity.

Two regularity criteria for 3D Navier-Stokes equations in a bounded domain

We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity.