Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

Strong solutions to the equations of a ferrofluid flow model

We study the differential system introduced by M.I. Shliomis to describe the motion of a ferrofluid driven by an external magnetic field. The system is a combination of the Navier-Stokes equations, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of strong solutions to the system.     

Global strong solutions to the Shliomis system for ferrofluids in a bounded domain

We consider the partial differential equations proposed by Shliomis to model the dynamics of an incompressible viscous ferrofluid submitted to an external magnetic field. The Shliomis system consists of the incompressible Navier‐Stokes equations, the magnetization equations, and the magnetostatic equations. The magnetization equations is of Bloch type, and no regularizing term is added. We prove the global existence of unique strong solution to the initial boundary value problem for the system in a bounded domain, with the small initial data and external magnetic field but without any restrictions on the physical parameters. The novelty of the analysis is to introduce a linear combination of magnetic fields.     

Strong solutions to the equations of electrically conductive magnetic fluids

We study the equations of flow of an electrically conductive magnetic fluid, when the fluid is subjected to the action of an external applied magnetic field. The system is formed by the incompressible Navier–Stokes equations, the magnetization relaxation equation of Bloch type and the magnetic induction equation. The system takes into account the Kelvin and Lorentz force densities. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions. We also establish a blow-up criterion for the local strong solution.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.     

Global weak solutions to a ferrofluid flow model

We are concerned with the global solvability of the differential system introduced by Shliomis to describe the flow of a colloidal suspension of magnetized nanoparticles in a nonconducting liquid, under the action of an external magnetic field. The system is a combination of the Navier–Stokes equations, the magnetization equation, and the magnetostatic equations. We prove, by using a method of regularization, the existence of global‐in‐time weak solutions with finite energy to an initial boundary‐value problem and establish the long‐time behaviour of such solutions. The main difficulty is due to the singularity of the gradient magnetic force and the torque. Copyright © 2007 John Wiley & Sons, Ltd.     

Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Travelling Waves with a Singularity in Magnetic Nanowires

We model the evolution of the magnetization in an infinite cylinder by harmonic map heat flow with an additional external field. Using variational methods, we prove the existence of corotationally symmetric travelling wave solutions with a moving vortex. We moreover show that for weak and strong fields the travelling waves connect the original state anti-parallel to the external magnetic field with the totally reversed state in direction of the external field. Our results match numeric simulations. For thicker wires several groups have found a reversal mode where a domain wall with a corotational symmetry and a vortex is propagating through the wire.     

Homogenizing the viscoelasticity problem with long-term memory

The system of integro-differential equations describing the small oscillations of an ?-periodic viscoelastic material with long-term memory is considered. Using the two-scale convergencemethod, we construct the systemof homogenized equations and prove the strong convergence as ? → 0 of the solutions of prelimit problems to the solution of the homogenized problem in the norm of the space L ².     

Existence and decay of solutions in full space to Navier-Stokes equations with delays

We consider the Navier-Stokes equations with delays in Rⁿ,2≤n≤4. We prove existence of weak solutions when the external forces contain some hereditary characteristics and uniqueness when n=2. Moreover, if the external forces satisfy a time decay condition we show that the solution decays at an algebraic rate.     

A system of elliptic equations arising in Chern-Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R². We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R² limit from the bounded-domain solutions to obtain a topological solution in R².     

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR² or IR³. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.     

Weighted Spectral Gap for Magnetic Schrödinger Operators with a Potential Term

We prove that singular Schrödinger equations with external magnetic field admit a representation with a positive Lagrangian density whenever their “nonmagnetic” counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L ^p -norm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of a magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrödinger equation with the periodic magnetic field.     

Global solutions of the Navier–Stokes equations for isentropic flow with large external potential force

We prove the global-in-time existence of weak solutions to the Navier–Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent γ ≥ 1. Initial data and solutions are small in L ² around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when γ = 1. A great deal of information about partial regularity and large-time behavior is obtained.     

Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control

In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the H¹-energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in H²-sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Global Weak Solutions to the Compressible Navier-Stokes Equations in the Exterior Domain with Spherically Symmetric Data

In this paper, we prove the global existence of weak solutions to the full compressible Navier-Stokes equations in the domain exterior to a ball in R ⁿ (n=2,3) and with spherically symmetric data.     

Three-dimensional vortices in Abelian gauge theories

In this paper we consider an Abelian Gauge Theory in R⁴${\mathbb{R}}^4$ equipped with the Minkowski metric. This theory leads to a system of equations, the Klein–Gordon–Maxwell equations, which provide models for the interaction between the electromagnetic field and matter. A three-dimensional vortex is a finite energy solution of these equations in which the magnetic field looks like the field created by a finite solenoid. Under suitable assumptions, we prove the existence of vortex solutions.     

On some nonlinear evolution equations with strong dissipation,III

In this paper we continue the existence theories of classical solutions of nonlinear evolution equations with strong dissipation studied in previous papers [5, 6], where we proved the existence of global classical solutions with small data applying small energy techniques. This time, we prove the existence of a set of initial values which guarantees the solution to be global. We know the set is not bounded in the escalated energy spaces (Sobolev spaces). For the purpose, we establish approximate equations with another dissipative term which give a devised penalty to the solutions and lead the solutions to be bounded for all t > 0. Therefore we give an improvement to existence theories of equations describing a local statement of balance of momentum for materials for which the stress is related to strain and strain rate. These have been studied by many authors (cf. Greenberg et al. 19], Greenberg [10], Davis [3], Clements [2], Andrews [1], Yamada [12], Webb [13], etc.).     

On the evolution equation for magnetic geodesics

Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations (PDEs) whose solutions describe the orbits of closed p-branes under the effect of a “generalized physical force”. For the corresponding evolution equation, which is a system of parabolic nonlinear PDEs associated to the elliptic PDE, we can establish existence of short time solutions.