Motion of the Ginzburg-Landau vortices for the mixed flow with convective forcing

Mean-field dynamics for Ginzburg-Landau vortices with pinning and forcing

We consider the time-dependent 2D Ginzburg-Landau equation in the whole plane with terms modeling impurities and applied currents. The Ginzburg-Landau vortices are then subjected to three forces: their mutual repulsive Coulomb-like interaction, the applied current pushing them in a fixed direction, and the pinning force attracting them towards the impurities. The competition between the three is expected to lead to complicated glassy effects. We rigorously study the limit in which the number of vortices $N_ε$ blows up as the inverse Ginzburg-Landau parameter $ε$ goes to $0$, and we derive via …

Ginzburg-Landau vortex dynamics with pinning and strong applied currents

We study a mixed heat and Schrödinger Ginzburg-Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and "pins" them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg-Landau parameter, or $\ep \…

Global well-posedness for the two-dimensional stochastic complex Ginzburg-Landau equation

We study the stochastic complex Ginzburg-Landau equation (SCGL) with an additive space-time white noise forcing on the two-dimensional torus. This equation is singular and thus we need to renormalize the nonlinearity in order to give proper meaning to the equation. Unlike the real-valued stochastic quantization equation, SCGL is complex valued and hence we are forced to work with the generalized Laguerre polynomials for the sake of renormalization. In handling nonlinearities of arbitrary degree, we derive a useful algebraic identity on the renormalization in the complex-valued setting and prov…

Space-Time Invariant Measures, Entropy, and Dimension for Stochastic Ginzburg-Landau Equations

We consider a randomly forced Ginzburg-Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the corresponding Markov process and we define the spatial densities of topological entropy, of measure-theoretic entropy, and of upper box-counting dimension. We prove inequalities relating these different quantities. The proof of existence of an invariant measure uses the compact embedding of some space of uniformly smooth functions into the space of locally …

关于具有对流换热效应的非局部两相流

On a nonlocal two-phase flow with convective heat transfer

本文研究具有Caginalp型对流换热的非局部Cahn-Hilliard-Navier-Stokes两相流系统，证明了弱解的全局存在性，并证明当卷积核适当选取时非局部解收敛到局部模型。

能量临界复金兹堡－朗道方程的整体正则性与一般系数奇异极限

Singular limits for general energy-critical complex Ginzburg-Landau equations

该研究解决了Cazenave的能量临界复金兹堡－朗道方程公开问题，构建了系数一致的临界稳定性框架，建立了无粘极限下的系数一致Strichartz估计。