On the existence of multivortices in a generalized Bogomol'nyi system

An interesting feature of the generalized Abelian Higgs vortex model introduced by M. Lohe is that it allows a wider class of Higgs self-interaction potential functions, yet, noninteracting multivortices are still present in the critical coupling phase as the solutions of a modified Bogomol'nyi system. In this paper, we add two new classes of multivortex solutions to Lohe's model. We prove that, under the 't Hooft boundary condition, there are solutions realizing a periodic lattice structure and a quantized flux. A necessary and sufficient condition is obtained for the existence of such solutions. We also show that the Meissner effect occurs in the model. Moreover, we establish for any given vortex distribution in the plane, the existence of a one-parameter family of gauge-distinct solutions of infinite energy.     

On the Minimal Solutions of Variational Inequalities in Orlicz-Sobolev Spaces

In this paper, the author studies the existence of the minimal nonnegative solutions of some elliptic variational inequalities in Orlicz-Sobolev spaces on bounded or unbounded domains. She gets some comparison results between different solutions as tools to pass to the limit in the problems and to show the existence of the minimal solutions of the variational inequalities on bounded domains or unbounded domains. In both cases,coercive and noncoercive operators are handled. The sufficient and necessary conditions for the existence of the minimal nonnegative solution of the noncoercive variational inequality on bounded domains are established.     

Energy solutions for polymer aqueous solutions in two dimension

The aim of this article is to study a nonlinear system modeling a Non-Newtonian fluid of polymer aqueous solutions. We are interested here in the existence of weak solutions for the stationary problem in a bounded plane domain or in two-dimensional exterior domain. Due to the third order of derivatives in the non-linear term, it’s difficult to obtain solutions satisfying energy inequality. But with a good choice of boundary conditions, an adapted special basis and the use of the good properties of the trilinear form associated to the non-linear term, we obtain energy solutions. The problem in bounded domains is treated and the more difficult problem on non bounded domains too.     

Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy

The Chern–Simons–Higgs energy serves as a model for high temperature superconductivity. We show the existence of weak solutions to the CSH equations that are minimizers of the CSH energy. The solutions are vortexless for an applied magnetic field h _ex below the critical field strength, whereas vortices appear when h _ex exceeds the critical field strength. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. X. Yan was supported in part by NSF grants DMS-0700966 and DMS-0401048.     

一般有界区域上次线性椭圆方程改进的正解估计

在一般有界区域上，建立了次线性椭圆方程广义Ｄｉｒｉｃｈｌｅｔ问题改进的正解估计和有界强解的存在性定理．     

Coupled system of Korteweg–de Vries equations type in domains with moving boundaries

We consider the initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for the coupled system of equations of Korteweg–de Vries (KdV)-type modelling strong interactions between internal solitary waves. Finite domains of wave propagation changing in time arise naturally in certain practical situations when the equations are used as a model for waves and a numerical scheme is needed. We prove a global existence and uniqueness for strong solutions for the coupled system of equations of KdV-type as well as the exponential decay of small solutions in asymptotically cylindrical domains. Finally, we present a numerical scheme based on semi-implicit finite differences and we give some examples to show the numerical effect of the moving boundaries for this kind of systems.     

Non-topological solutions in the generalized self-dual Chern-Simons-Higgs theory

In this paper we construct a non-topological multivortex solution of a generalized version of the relativistic self-dual Chern-Simons-Higgs system in that makes the energy functional finite. Our method of proof is an extension of the previous argument used by the authors to prove the existence of general type of non-topological multivortex solutions of the relativistic Chern-Simons-Higgs system, using an implicit function theorem argument with features similar to the Liapunov-Schmidt decomposition. Received: 20 January 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002 This research supported partially by BSRI-MOE, KOSEF(2000-2-10200-002-5).     

Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and smooth bounded and unbounded domains

Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.     

Global existence and blow-up for harmonic map heat flow

For degree-one equivariant maps on bounded domains, the question of finite-time blow-up vs. global existence of solutions to the harmonic map heat flow has been well studied. In this paper we study the Cauchy problem for degree-m equivariant harmonic map heat flow from (2+1)-dimensional space-time into the 2-sphere with initial energy close to the energy of harmonic maps. It is proved that solutions are globally smooth for m?4, whereas for m=1, we show that finite-time singularities can form for this class of data.     

Dirichlet Boundary Conditions for Degenerate and Singular Nonlinear Parabolic Equations

We study existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To this purpose some barrier functions are properly introduced and used.     

Optimal Control of the Instationary Three Dimensional Navier-Stokes-Voigt Equations

We study an optimal control problem with quadratic objective functional for the three dimensional Navier-Stokes-Voigt equations in bounded domains. We show the existence of optimal solutions, the necessary optimality conditions and the sufficient optimality conditions. The second-order optimality conditions obtained in the article seem to be optimal.     

Weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies

We consider a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions and prove existence of weak solutions for it. In contrast to earlier contributions, we study a model with a singular nonlocal free energy, which controls the H^α/2-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

Smoothing estimates of 2d incompressible Navier–Stokes equations in bounded domains with applications

Motivated by the study on the uniqueness problem of the coupled model, in this paper, we revisit 2d incompressible Navier–Stokes equations in bounded domains. In fact, we establish some new smoothing estimates to the Leray solution based on the spectral analysis of Stokes operator. To understand well these estimates, on one hand, we establish some new Brezis–Waigner type inequalities in general domain and in any dimension and disclose the connection between both of them. On the other hand, we show that these new estimates can be applied to prove the existence and uniqueness of the weak solutions for two coupled models: Boussinesq system with partial viscosity (no dissipation for the temperature) and Fluid/Particle system, in two dimension and in bounded domains.     

Global existence for quasilinear parabolic systems with homogeneous Neumann boundary conditions

We present a result on the global existence of classical solutions for quasilinear parabolic system in bounded domains with homogenous Neumann boundary conditions.     

Peak solutions without non-degeneracy conditions

We prove the existence of peak solutions of nonlinear elliptic equations on bounded domains when the diffusion is small. We greatly weaken the non-degeneracy condition usually assumed.     

Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system

In this paper, we study the stationary flow for a one-dimensional isentropic bipolar Euler-Poisson system (hydrodynamic model) for semiconductor devices. This model consists of the continuous equations for the electron and hole densities, and their current densities, coupled the Poisson equation of the electrostatic potential. In a bounded interval supplemented by the proper boundary conditions, we first show the unique existence of stationary solutions of the one-dimensional isentropic hydrodynamic model, based on the Schauder fixed-point principle and the careful energy estimates. Next, we investigate the zero-electron-mass limit, combined zero-electron mass and zero-hole mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit, respectively. We also show the strong convergence of the sequence of solutions and give the associated convergence rates.     

On compressible Navier–Stokes equations with density dependent viscosities in bounded domains

The present note extends to smooth enough bounded domains recent results about barotropic compressible Navier–Stokes systems with density dependent viscosity coefficients. We show how to get the existence of global weak solutions for both classical Dirichlet and Navier boundary conditions on the velocity, under appropriate constraints on the initial density profile and domain curvature. An additional turbulent drag term in the momentum equation is used to handle the construction of approximate solutions.     

On the lack of exponential stability for an elastic–viscoelastic waves interaction system

In this paper, we consider an interaction system in which a wave and a viscoelastic wave equation evolve in two bounded domains, with natural transmission conditions at a common interface. We show the lack of uniform decay of solutions in general domains. The method is based on the construction of ray-like solutions by means of geometric optics expansions and a careful analysis of the transfer of the energy at the interface.     

Global existence for fully parabolic boundary value problems

We present some results on the global existence of classical solutions for quasilinear parabolic equations with nonlinear dynamic boundary conditions in bounded domains with a smooth boundary.