On a phase-change problem arising from inductive heating

In this paper we study a phase-change problem arising from induction heating. The mathematical model consists of time-harmonic Maxwell’s system in a quasi-stationary field coupled with nonlinear heat conduction. The enthalpy form is used to characterize the phase-change in the material. It is shown that the problem has a global solution. Moreover, it is shown that the solution is unique and regular in one-space dimension even with an unbounded resistivity. This work is supported in part by a NSF grant: DMS-0102261     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

Existence-uniqueness and monotone approximation for a phytoplankton-zooplankton aggregation model

In this paper, we study a model that describes the dynamics of phytoplankton and zooplankton prey-predator system within the context of phytoplankton aggregation. Existence-uniqueness results of the solution are established via a comparison principle and the upper-lower solution technique. (Received: December 12, 2004)     

Asymptotic stability of viscous shock wave profiles for viscous conservation laws

This paper is concerned with the asymptotic stability of travelling wave solution to the two-dimensional steady isentropic irrotational flow with artificial viscosity. We prove that there exists a unique travelling wave solution up to a shift to the system if the end states satisfy both the Rankine–Hugoniot condition and Lax's shock condition, and that the travelling wave solution is stable if the initial disturbance is small.     

The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays

The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays.     

Regularity of weak solution to Maxwell's equations and applications to microwave heating

In this paper we first study the regularity of weak solution for time-harmonic Maxwell's equations in a bounded anisotropic medium Ω. It is shown that the weak solution to the linear degenerate system, , is Hölder continuous under the minimum regularity assumptions on the complex coefficients γ(x) and ξ(x). We then study a coupled system modeling a microwave heating process. The dynamic interaction between electric and temperature fields is governed by Maxwell's equations coupled with an equation of heat conduction. The electric permittivity, electric conductivity and magnetic permeability are assumed to be dependent of temperature. It is shown that under certain conditions the coupled system has a weak solution. Moreover, regularity of weak solution is studied. Finally, existence of a global classical solution is established for a special case where the electric wave is assumed to be propagating in one direction.     

Diffusive approximations for a class of nonlinear parabolic equations

We study a class of discrete velocity type approximations to nonlinear parabolic equations with source. After proving existence results and estimates on the solution to the relaxation system, we pass into the limit towards a weak solution, which is the unique entropy solution if the coefficients of the parabolic equation are constant.     

On the nonlinear Schrödinger limit of the magnetic Zakharov system

In this paper, we study the limit behavior of a smooth solution for the magnetic type Zakharov system. As the parameters tend to infinity, we prove that solutions of the magnetic Zakharov system converge to the solution of the nonlinear Schrödinger equation. Moreover, the detailed convergence rate is also investigated.     

Two-dimensional travelling waves over a moving bottom

Two-dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied. The bottom and the wave travel with a same speed. The exact Euler equations are formulated as a spatial dynamic system by using the stream function. A manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions. A homoclinic solution to the normal form of this reduced system persists when higher-order terms are added, which gives a generalized solitary wave—the homoclinic solution connecting a periodic solution.     

On the interaction of δ-shocks and contact vacuum states

We study the pressureless gas equations, with piecewise constant initial data. In the immediate solution, δ-shocks and contact vacuum states arise and even meet (interact) eventually. A solution beyond the “interaction” is constructed. It shows that the δ-shock will continue with the velocity it attained instantaneously before the time of interaction, and similarly, the contact vacuum state will move past the δ-shock with a velocity value prior to the interaction. We call this the “no-effect-from-interaction” solution. We prove that this solution satisfies a family of convex entropies (in the Lax’s sense). Next, we construct an infinitely large family of weak solutions to the “interaction”. Suppose further that any of these solutions satisfy a convex entropy, it is necessary and suffcient that these solutions reduce to only the “no-effect-from-interaction” solution. In [1], Bouchut constructed another entropy satisfying solution. As with other previous papers, it is obvious that it will not be sufficient that a “correct” solution satisfies a convex entropy, in a non-strictly hyperbolic conservation laws system. Research done in the University of Michigan-Ann Arbor, submission from Temasek Laboratories, National University of Singapore.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation

In this paper we consider the Yudovich type solution of the 2D inviscid Boussinesq system with critical and supercritical dissipation. For the critical case, we show that the system admits a global and unique Yudovich type solution; for the supercritical case, we prove the local and unique existence of Yudovich type solution, and the global result under a smallness condition of _θ0${\theta }_0$. We also give a refined blowup criterion in the supercritical case.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

Asymptotic behavior of solutions to a quasilinear nonuniform parabolic system modelling chemotaxis

In this paper, we study a quasilinear nonuniform parabolic system modelling chemotaxis and taking the volume-filling effect into account. The results on the existence of a unique global classical solution was obtained in Cie?lak (2007) [4]. However, the convergence to equilibrium was not considered in that paper. In this paper, we first obtain the crucial uniform boundedness of the solution. Then with the help of a suitable non-smooth Simon-?ojasiewicz approach we obtain the results on convergence to equilibrium and the decay rate.     

Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities

In this paper, the author proves the global structure stability of the Lax's Riemann solution , containing only shocks and contact discontinuities, of general n×n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C¹ solution u=u(t,x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

A regularity result of solution to the compressible Stokes equations on a convex polygon

A compressible Stokes problem is analyzed in a convex polygon D. The goal of this paper is to sort out a singularity of the pressure function at the corner and to establish the corresponding regularity result of the resulted remainder part of the solution. For this a solution formula is derived and the singular function of the Stokes problem is considered. It is seen that the lowest order of the regularity of the system is the same as that of the (incompressible) Stokes one.     

A system of partial differential equations modeling the competition for two complementary resources in flowing habitats

This paper examines a system of reaction-diffusion equations arising from a flowing water habitat. In this habitat, one or two microorganisms grow while consuming two growth-limiting, complementary (essential) resources. For the single population model, the existence and uniqueness of a positive steady-state solution is proved. Furthermore, the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values. Mathematical analysis for the two competing populations is carried out. More precisely, the long-time behavior is determined by using the monotone dynamical system theory when the semi-trivial solutions are both unstable. It is also shown that coexistence solutions exist by using the fixed point index theory when the semi-trivial solutions are both (asymptotically) stable.     

On the singular limit for slightly compressible fluids

In this paper we study the motion of slightly compressible inviscid fluids. We prove that the solution of the corresponding system of nonlinear partial differential equations converges (uniformly) in the strong norm (that of the data space) to the solution of the incompressible equations, as the Mach number goes to zero (see Theorem 1.2). Actually, our proof applies to a large class of singular limit problems as shown in the Theorem 2.2.     

A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. On the basis of the existence result for the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions, provided that the C¹ norm and the BV norm of the initial data are bounded but possibly large. In contrast to former results obtained by Liu and Zhou [J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479-500], ours do not require their assumption that the system is rich in the sense of Serre. Applications include that to the one-dimensional Born-Infeld system arising in string theory and high energy physics.     

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.