Direct problem for a nonlinear evolutionary system

In this article, we consider the first initial boundary-value problem for an evolutionary system describing nonlinear interactions of electromagnetic and elastic waves. The system under study consists of three coupled differential equations, one of them is a hyperbolic equation (an analogue of the Lamé equations) and the other two equations form a parabolic system (an analogue of the diffusion Maxwell system). Existence and uniqueness results are established. We also prove the stability estimate of a weak solution.     

The first initial–boundary-value problem for a nonlinear model of the electrodynamics of vibrating elastic media

The first initial–boundary-value problem for nonlinear differential equations describing the interactions of a vibrating electroconductive body and the electromagnetic field is studied. We assume that the motion of the body occurs at velocities that are much smaller than the velocity of propagation of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations; one of them is the hyperbolic equation (an analogue of the Lamé system) and the other is the parabolic equation (an analogue of the diffusion Maxwell system). We prove an existence and uniqueness result. The proof is based on the classical Faedo–Galerkin method.     

On a nonclassical fractional boundary-value problem for the Laplace operator

In this paper we consider a boundary-value problem for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides dependent on time. We prove the one-valued solvability of this problem, and provide the coercive estimates of the solution.     

Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators

This paper is devoted to a proof of regularity, near the initial state, for solutions to the Cauchy-Dirichlet and obstacle problem for a class of second order differential operators of Kolmogorov type. The approach used here is general enough to allow us to consider smooth obstacles as well as non-smooth obstacles.     

Existence and uniqueness of the solution to a free boundary-value problem with thermocapillary convection in an unbounded domain

A free boundary-value problem for a nonisothermal flow of two heavy viscous incompressible capillary liquids descibing a slot coating process is mathematically studied. The problem is stationary and two-dimensional and the Boussinesq approximation for the Navier-Stokes equations is used. The existence and uniqueness of a weak solution is proved in weighted Holder spaces for small data.     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

On a new boundary-value problem for the Navier-Stokes system with discontinuity in the type of boundary conditions depending on time

An initial boundary-value problem for the nonlinear system of Navier-Stokes equations with timedependent discontinuity in the type of boundary conditions is proposed to describe the dynamics of a hurricane over solid ground. The tools for the solution of the stated problem are displayed in the case of both smooth and nonsmooth evolution of a hurricane.This paper continues the authors' work [1].     

Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field

Partially supported by NSF Grant DMS-9123742     

The Cauchy problem for the Schrödinger-KdV system

In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.     

A multi-frequency inverse source problem

This paper is concerned with an inverse source problem that determines the source from measurements of the radiated fields away at multiple frequencies. Rigorous stability estimates are established when the background medium is homogeneous. It is shown that the ill-posedness of the inverse problem decreases as the frequency increases. Under some regularity assumptions on the source function, it is further proven that by increasing the frequency, the logarithmic stability converts to a linear one for the inverse source problem.     

An initial boundary-value problem for a system of equations of magnetohydrodynamics

In the present paper we consider one initial boundary-value problem for a system of equations of magnetohydrodynamics in the case where it is necessary to take into account the displacement currents in the Maxwell system of equations. We prove a local (in time) unique solvability of this problem in the Sobolev spaces. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 228–254, April–June, 2000. Translated by R. Lapinskas     

High order compact Alternating Direction Implicit method for the generalized sine-Gordon equation

High order compact Alternating Direction Implicit scheme is given for solving the generalized sine-Gordon equation in a two-dimensional rectangular domain. We apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives, and we give a linearized three time level algorithm for solving the original nonlinear equation. Error estimate is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this algorithm.     

An inverse source problem for Maxwell's equations in anisotropic media

In this article, we consider nonstationary Maxwell's equations in an anisotropic medium in the (x ₁,?x ₂,?x ₃)-space, where equations of the divergences of electric and magnetic flux densities are also unknown. Then we discuss an inverse problem of determining the x ₃-independent components of the electric current density from observations on the plane x ₃?=?0 over a time interval. Our main aim is, study conditional stability in the inverse problem provided the permittivity and the permeability are independent of x ₃. The main tool is a new Carleman estimate.     

On the Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system

     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

Nearly cloaking the full Maxwell equations: Cloaking active contents with general conducting layers

The regularized near-cloak via the transformation optics approach in the time-harmonic electromagnetic scattering is considered. This work extends the existing studies mainly in two aspects. First, it presents a near-cloak construction by incorporating a much more general conducting layer between the cloaked and cloaking regions. This might be of significant practical importance when production fluctuations occur. Second, it allows the cloaked contents to be both passive and active with an applied current inside. In assessing the near-cloaking performance, comprehensive and sharp estimates are derived for the scattering amplitude in terms of the asymptotic regularization parameter and the material tensors of the conducting layer. The scattering estimates are independent of the passive/active contents being cloaked, which implies that one could nearly cloak arbitrary contents by using the proposed near-cloak construction.     

Pressure integrability conditions for uniqueness of mild solutions of the Navier-Stokes system

We address uniqueness of mild solutions of the Navier-Stokes system in . We prove that solutions for which the pressure is locally square integrable are unique.     

An optimal control problem for microwave heating

In this paper, we study an optimization problem for a microwave/induction heating process. The cost function is defined such that the temperature profile at the final stage has a relative uniform distribution in the field. The control variable is the applied electric field on the boundary. We show that there exists an optimal electric field which minimizes the cost function. Moreover, a necessary condition for a special case is also derived.     

Mesoscopic model for ferromagnets with isotropic hardening

The proposed model combines tendency for minimization of Gibbs magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopic quantity of energy required to change one pole of a magnet to another. This energy can increase within the evolution which is the effect like hardening in plasticity. The microstructure is described on a mesoscopic level in terms of Young measures. Such a mesoscopic, distributed-parameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteretic macroscopic response, including effects as virgin curves and minor loops, is demonstrated.Received: December 3, 2002; revised: March 25, 2003