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- Changyou Wang:
On the Landau-Lifschitz equation in dimensions at most 4
ABSTRACT: For $n\le 4$ and any bounded domain $\Omega\subset R^n$, we establish the existence of a global weak solution for the Landau- Lifschitz equation on $\Omega$ with respect to any smooth initial-boundary date, which is smooth away from a closed set with locally finite $n$-dimensional parabolic Hausdoff measure. The approach is based on the Ginzburg-Landau approximation and the $\eta$-compactness type apriori estimate under the smallness of renormalized Ginzburg-Landau energy. - Changfeng Gui:
Quadruple Solutions in Three Dimensional Spaces
Abstract: We consider a vector-valued Allen-Cahn equation in the entire threee dimensional spaces. The equation describes the local behavior of quadruple junctions in phase seperation. In the talk I will show the construction of an entire solution and its asymptotic behavior. - Chun Liu:
Energy Variational Approaches in Complex Fluids
- Dingping
Li:
Vortex phase diagram with disorder -Ginzburg Landau approach
Abstract: A metastable supercooled homogeneous vortex liquid state exists down to zero !0fluctuation temperature!1 in systems of mutually repelling objects. The zero temperature liquid state therefore serves as a (pseudo) ''fixed point'' controlling the properties of vortex liquid below and even around melting point. There exists Madelung constant for the liquid in the limit of zero temperature which is higher than that of the solid by an amount approximately equal to the latent heat of melting. Based on this picture a quantitative theory of vortex melting in type II superconductors in the framework of Ginzburg - Landau approach is presented. The melting line location is determined and magnetization and specific heat jumps are calculated. The pointlike disorder shifts the line downwards and makes it very complicated: several Kauzmann points exist. There are though just two static phases and a single (universal) line between them. Disorder induces irreversible effects via replica symmetry breaking. The irreversibility line is calculated. Therefore the generic phase diagram contains four phases: liquid, solid, vortex glass and Bragg glass. - Fanghua Lin:
- Qiang Du:
- Shijin Ding:
Ginzburg-Landau Vortices
in Superconducting Thin Films
- Weizhu Bao:
Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrodinger equation
- Taichia Lin:
Bound state solutions of coupled nonlinear Schrodinger equations
Abstract: In this lecture, I'll introduce spikes and domain walls coming from bound state solutions of coupled nonlinear Schrodinger equations which describe multispecies Bose-Einstein condensates. Due to Feshbach resonance, the sign and quantity of coupling constants may change in a large region. Hence some interesting phenomena like spikes and domain walls are observed by physical experiments. By mathematical analysis, we may obtain such bound state solutions from coupled nonlinear Schrodinger equations. - Tiezheng
Qian:
Phase Slips in Thin Superconducting Wires:
An Accurate Numerical Evaluation Using the String Method
- Yisong Yang:
Mathematics of the Gap Equations in Superconductivity
- Xiaobing Feng:
p-Harmonic Map Heat Flows: Analysis, Numerics, and Applications
- Zuhan Liu:
Vortices
Set and the Applied Magnetic Field for
Superconductivity in dimension 3.
Abstract: In this paper, the structure of vortices set of the Ginzburg-Landau system of the superconductivity in dimension 3 was studied when applied magnetic field $|h_{ex}|=O(|\ln \varepsilon|)$. This singularities set is 1-dimensional rectifiable. Its generalied mean curvature was given. - Xinbing Pan:
On a Problem Related to Vortex Nucleation of 3-Dimensional
Superconductors
Abstract: Type II superconductors in an increasing applied magnetic field undergo phase transitions from the Meissner state to the mixed state as the applied field increases to the superheating field. A simplified model in the form of a system of partial differential equations involving the operator $curl^2$ was derived by Chapman and has been used to describe vortex nucleation in superconductors. For a cylindrical sample in an applied field parallel to the axis, this system is reduced to a single equation and has been analyzed. However, for a bulk superconductor occupying a bounded 3-dimensional domain, the system has not been well-understood. In this paper we examine the system in a 3-dimensional domain, and derive a priori $C^{2+\alpha}$ estimates for weak solutions. Then we show that, if the penetration length is small, the solutions concentrate in a boundary layer at the surface of the domain, and outside the boundary layer the solutions decay to zero. Moreover, the current is maximal at a subset of the surface where the tangential component of the applied field is maximal. In particular, if the sample is subjected to a homogeneous magnetic field, the current is maximal at the surface of the sample where the applied field is tangential to the surface. This suggests the location of vortex nucleation in the samples as the applied fields increases to the superheating field. This is a joint work with Peter Bates of Michigan State University.
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