Unique Smooth Solution Research Articles

Diffusive relaxation limits of compressible Euler–Maxwell equations

This work is concerned with compressible Euler–Maxwell equations, which take the form of Euler equations for the conservation laws of mass density, current density and energy density for electrons, coupled to Maxwellʼs equations for self-consistent electromagnetic field. We give a model hierarchy of non-isentropic Euler–Maxwell equations from the point of view of diffusive relaxation limits. More precisely, inspired by Maxwell-type iteration, we construct new approximations and show that periodic initial-value problems of a certain scaled Euler–Maxwell equations have unique smooth solutions in a time interval independent of momentum relaxation time and energy relaxation time. Furthermore, it is proved that smooth solutions converge to solutions of drift-diffusion models and energy-transport models in the process of combined diffusive relaxation limits, and the corresponding convergence rates are also obtained.

The Calabi metric for the space of Kähler metrics

Given any closed Kähler manifold we define, following an idea by Calabi (Bull. Am. Math. Soc. 60:167–168, 1954), a Riemannian metric on the space of Kähler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation are given.

Global Schrödinger maps in dimensions d>=2: Small data in the critical Sobolev spaces

We consider the Schrodinger map initial-value problem ¤ @ t = on R d R; (0) = 0; where : R d R! S 2 ,! R 3 is a smooth function. In all dimensions d 2, we prove that the Schrodinger map initial-value problem admits a unique global smooth solution 2 C(R : H 1 Q ), Q2 S 2 , provided that the data 02 H 1 Q is smooth and satises the smallness condition

A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures

An initial-boundary-value problem for the sixth order Cahn-Hilliard type equation in 3-D is studied. The problem describes phase transition dynamics in ternary oil-water-surfactant systems. It is based on the Landau-Ginzburg theory proposed for such systems by G. Gompper et al. We prove that the problem under consideration is well posed in the sense that it admits a unique global smooth solution which depends continuously on the initial datum.

A bifurcation theorem for the p-logistic equation

We consider a p-logistic equation with an equidiffusive reaction. Using variational methods and truncation techniques, we show that there is a critical parameter value λ ∗ > 0 such that for λ > λ ∗ the problem has a unique positive smooth solution, and for λ ∈ (0, λ ∗] the problem has no positive solution.

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.

The diffusive relaxation limit of non-isentropic Euler–Maxwell equations for plasmas

This paper concerns the non-isentropic Euler–Maxwell equations for plasmas with short momentum relaxation time. With the help of the Maxwell-type iteration, it is obtained that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled non-isentropic Euler–Maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the non-isentropic Euler–Maxwell equations.

On a quasilinear hyperbolic system in blood flow modeling

This paper aims at an initial-boundary value problem on bounded domains for a one-dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that, for given smooth initial data close to a constant equilibrium state, there exists a unique global smooth solution to the model. Time asymptotically, it is shown that the solution converges to the constant equilibrium state exponentially fast as time goes to infinity due to viscous damping and boundary effects.

Cauchy problem for the Boltzmann-BGK model near a global Maxwellian

In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK model for a general class of collision frequencies. We prove that the Boltzmann-BGK model linearized around a global Maxwellian admits a unique global smooth solution if the initial perturbation is sufficiently small in a high order energy norm. We also establish an asymptotic decay estimate and uniform L2-stability for nonlinear perturbations.

On the decay and stability of global solutions to the 3‐D inhomogeneous Navier‐Stokes equations

AbstractIn this paper, we investigate the large‐time decay and stability to any given global smooth solutions of the 3‐D incompressible inhomogeneous Navier‐Stokes equations. In particular, we prove that given any global smooth solution (a,u) of (1.2), the velocity fieldudecays to 0 with an explicit rate, which coincides with theL2norm decay for the weak solutions of the 3‐D classical Navier‐Stokes system [26,29] astgoes to ∞. Moreover, a small perturbation to the initial data of (a,u) still generates a unique global smooth solution to (1.2), and this solution keeps close to the reference solution (a,u) fort> 0. We should point out that the main results in this paper work for large solutions of (1.2). © 2010 Wiley Periodicals, Inc.

Global wellposedness of the 3-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem, in the setting that the interface tends to the horizontal plane, the velocity and acceleration on the interface tend to zero at spatial infinity. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate 1/t.

A note on the optimal temporal decay estimates of solutions to the Cahn–Hilliard equation

This paper is concerned with the optimal temporal decay estimates on the solutions of the Cauchy problem of the Cahn–Hilliard equation. It is shown in Liu, Wang and Zhao (2007) [11] that such a Cauchy problem admits a unique global smooth solution u ( t , x ) provided that the smooth nonlinear function φ ( u ) satisfies a local growth condition. Furthermore if φ ( u ) satisfies a somewhat stronger local growth condition, the optimal temporal decay estimates on u ( t , x ) are also obtained in Liu, Wang and Zhao (2007) [11]. Thus a natural question is how to deduce the optimal temporal decay estimates on u ( t , x ) only under the local growth condition which is sufficient to guarantee the global solvability of the corresponding Cauchy problem and the main purpose of this paper is devoted to this problem. Our analysis is motivated by the technique developed recently in Ukai, Yang and Zhao (2006) [15] with a slight modification.

ENERGY-TRANSPORT LIMIT OF THE HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

This paper is mainly devoted to study the energy-transport limit of a non-isentropic hydrodynamic model with momentum relaxation time τ and energy relaxation time σ. Inspired by the Maxwell iteration, we construct a new approximation under the assumption τσ = 1, and show that periodic initial-value problems of a certain scaled hydrodynamic model have unique smooth solutions in a finite time interval independent of τ. Furthermore, it is also obtained that as τ tends to zero, the smooth solutions converge to the smooth solutions of energy-transport models at the rate of τ2. The proof of these results is based on a continuation principle.

The combined semi-classical and relaxation limit in a quantum hydrodynamic semiconductor model

We discuss the combined semi-classical and relaxation limit of a one-dimensional isentropic quantum hydrodynamical model for semiconductors. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density, including the quantum potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration, we prove that, as the relaxation time and Planck constant tend to zero, periodic initial-value problems of a scaled one-dimensional isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the classical drift-diffusion model from the quantum hydrodynamic model.

Global existence for the Seiberg–Witten flow

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in C(infinity) up to gauge to a critical point of the Seiberg-Witten functional.

Hyperbolic Conservation Laws

Hyperbolic Conservation Laws

Global smooth solutions for the one-dimensional spin-polarized transport equation

The existence of global smooth solutions of spin-polarized transport equation in dimension one is studied. We prove that for smooth initial data, the equation admits a unique global smooth solution without smallness restriction. When the Gilbert damping term vanishes ( γ = 0 ) , refined a priori estimates are made and the global existence and uniqueness of smooth solutions are also obtained.

Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation

The paper studies the existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation arising in the model in variational form for the neo–Hookean elastomer rod where k1, k2>0 are real numbers, g(s) is a given nonlinear function. When g(s)=sn (where n⩾2 is an integer), by using the Fourier transform method we prove that for any T>0, the Cauchy problem admits a unique global smooth solution u∈C∞((0, T]; H∞(R))∩C([0, T]; H3(R))∩C1([0, T]; H−1(R)) as long as initial data u0∈W4, 1(R)∩H3(R), u1∈L1(R)∩H−1(R). Moreover, when (u0, u1)∈H2(R) × L2(R), g∈C2(R) satisfy certain conditions, the Cauchy problem has no global solution in space C([0, T]; H2(R))∩C1([0, T]; L2(R))∩H1(0, T; H2(R)). Copyright © 2009 John Wiley & Sons, Ltd.

THE GENERALIZED RIEMANN PROBLEM FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS I

In this paper, we consider a generalized Riemann problem of the first order hyperbolic conservation laws. For the case that excludes the centered wave, we prove that the generalized Riemann problem admits a unique piecewise smooth solution u = u(t, x), and this solution has a structure similar to the similarity solution u = <TEX>$U{(\frac{x}{t})}$</TEX> of the correspondin Riemann problem in the neighborhood of the origin provided that the coefficients of the system and the initial conditions are sufficiently smooth.

From a multidimensional quantum hydrodynamic model to the classical drift-diffusion equation

In the paper, we discuss the combined semiclassical and relaxation-time limits of a multidimensional isentropic quantum hydrodynamical model for semiconductors with small momentum relaxation time and Planck constant. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohn potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration, we prove that, as the Planck constant and the relaxation time tend to zero, periodic initial-value problems of a scaled isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion models have smooth solutions.