Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

We study the boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws in one space dimension for the relaxation system proposed in [6]. First, it is shown that for initial and boundary data satisfying a strict version of the subcharacteristic condition, there exists a unique global (in time) solution, (u^ε, v^ε), to the relaxation system (1.4) for each ε > 0. The spatial total variation of (u^ε, v^ε) is shown to be bounded independently of ε, and consequently, a subsequence of (u^ε, v^ε) converges to a limit (u, v) as ε → 0⁺. Furthermore, u(x, t) is a weak solution to the scalar conservation law (1.5) and v = f(u). Next, we prove that for data that are suitably small perturbations of a nontransonic state, the relaxation limit function satisfies the boundary-entropy condition (2.11). Finally, the weak solutions to (1.5) with the boundary-entropy condition (2.11) is shown to be unique. Consequently, the relaxation limit of solutions to (1.4) is unique, and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layer occurs only in the u-component in the sense that v^ε(0, ·) converges strongly to γ ○ v = f(γ ○ u), the trace of f(u) on the t-axis. © 1998 John Wiley & Sons, Inc.     

First-order L1-convergence for relaxation approximations to conservation laws

We derive a first-order rate of L¹-convergence for stiff relaxation approximations to its equilibrium solutions, i.e., piecewise smooth entropy solutions with finitely many discontinuities for scalar, convex conservation laws. The piecewise smooth solutions include initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time, and interactions of all these patterns. A rigorous analysis shows that the relaxation approximations to approach the piecewise smooth entropy solutions have L¹-error bound of O(ε|log ε| + ε), where ε is the stiff relaxation coefficient. The first-order L¹-convergence rate is an improvement on the error bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to O(ε). © 1998 John Wiley & Sons, Inc.     

Exact analysis of asymmetric random polling systems with single buffers and correlated input process

We introduce a simple approach for modeling and analyzing asymmetric random polling systems with single buffers and correlated input process. We consider two variations of single buffers system: the conventional system and the buffer relaxation system. In the conventional system, at most one customer may be resided in any queue at any time. In the buffer relaxation system, a buffer becomes available to new customers as soon as the current customer is being served. Previous studies concentrate on conventional single buffer system with independent Poisson process input process. It has been shown that the asymmetric system requires the solution ofm 2 ^m –1) linear equations; and the symmetric system requires the solution of 2 ^m–1–1 linear equations, wherem is the number of stations in the system. For both the conventional system and the buffer relaxation system, we give the exact solution to the more general case and show that our analysis requires the solution of 2 ^m –1 linear equations. For the symmetric case, we obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, loss probability, throughput, and the expected delay observed by a customer.     

Global well-posedness and relaxation limits of a model for radiating gas

We study the initial value problem for a hyperbolic-elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L¹-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic-parabolic and the hyperbolic-hyperbolic relaxation limits for that system.     

Domain decomposition methods for hyperbolic problems

In this paper a method is developed for solving hyperbolic initial boundary value problems in one space dimension using domain decomposition, which can be extended to problems in several space dimensions. We minimize a functional which is the sum of squares of the L ² norms of the residuals and a term which is the sum of the squares of the L ² norms of the jumps in the function across interdomain boundaries. To make the problem well posed the interdomain boundaries are made to move back and forth at alternate time steps with sufficiently high speed. We construct parallel preconditioners and obtain error estimates for the method. The Schwarz waveform relaxation method is often employed to solve hyperbolic problems using domain decomposition but this technique faces difficulties if the system becomes characteristic at the inter-element boundaries. By making the inter-element boundaries move faster than the fastest wave speed associated with the hyperbolic system we are able to overcome this problem.     

Convergence properties of relaxation algorithms

Nonadaptive relaxation algorithms require strong continuity assumptions and adaptive relaxation algorithms are computationally costly. To remedy that situation, an anti-jamming procedure similar to the one used by the author for the method of feasible directions is proposed. The resulting algorithms are compared with the existing ones for solving unconstrained optimization problem inE ⁿ .     

The nonvacuum Einstein flow on surfaces of nonnegative curvature

We prove future nonlinear stability of homogeneous solutions to the Einstein–Vlasov system with massive particles on manifolds with topology M = ?×Σ, where Σ is either 𝕊² or 𝕋². For the sphere this implies the existence of an open subset of the initial data manifold with elements of strictly positive scalar curvature, whose developments are future geodesically complete. In combination with an earlier result for hyperbolic surfaces we conclude future completeness for the Einstein–Vlasov system in 2+1 dimensions independent of the compact spatial topology for an open set of initial data.     

Convergence of Weighted Empirical Measures

This article considers a system with infinitely many interacting particles, starting with a system of n interacting particles that is described by a system of n stochastic differential equations for the time-varying locations and weights. Any particle in the system interacts with others through the weighted empirical measure Uⁿ formed by the sum of weighted Dirac measures on the n particles. Weak convergence of the weighted empirical measure is studied under suitable conditions, such as bounded initial values, and linear growth of drift and diffusion coefficients. Thereafter, the limit of the weighted empirical measures is identified to be a martingale solution of the infinite interacting system.     

Stability of travelling wave solutions to a semilinear hyperbolic system with relaxation

We study a semilinear hyperbolic system with relaxation and investigate the asymptotic stability of travelling wave solutions with shock profile. It is shown that the travelling wave solution is asymptotically stable, provided the initial disturbance is suitably small. Moreover, we show that the time convergence rate is polynomially (resp. exponentially) fast as t→∞ if the initial disturbance decays polynomially (resp. exponentially) for x→∞. Our proofs are based on the space–time weighted energy method. Copyright © 2008 John Wiley & Sons, Ltd.     

The interior of charged black holes and the problem of uniqueness in general relativity

We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein‐Maxwell‐scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space‐time is extendible as a C⁰ metric but along which the Hawking mass blows up identically; thus, the space‐time is inextendible as a C¹ metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C⁰ extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C⁰ formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc.     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions g_i(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.     

Existence of a Weak Solution in L p to the Vortex-Wave System

The vortex-wave system is a coupling of the two-dimensional vorticity equation with the point-vortex system. It is a model for the motion of a finite number of concentrated vortices moving in a distributed vorticity background. In this article, we prove existence of a weak solution to this system with an initial background vorticity in L ^p , p>2, up to the time of first collision of point vortices.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium

In this paper, we prove the global existence of smooth solutions to the three‐dimensional incompressible magnetohydrodynamical system with initial data close enough to the equilibrium state, (e₃,0). Compared with previous works by Lin, Xu, and Zhang and by Xu and Zhang, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is nondegenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in the earlier works. By using the Frobenius theorem and anisotropic Littlewood‐Paley theory for the Lagrangian formulation of the system, we achieve the global L¹‐in‐time Lipschitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large‐time decay rate of the solution is also obtained.© 2016 Wiley Periodicals, Inc.     

Approach to equilibrium of a rotating sphere in a Stokes flow

We study the unsteady rotary motion of a sphere immersed in a Stokes fluid. The equation of motion for the sphere leads to an integro-differential equation, and we are interested in the asymptotic behavior in time of the solution. Preparing initially the system (sphere + fluid) as a stationary state, we prove that the angular velocity of the sphere slows down with a law t ^−3/2 if no other forces than the one exerted by the fluid act on the sphere, while if the sphere is subject also to an elastic torque the asymptotic behavior of the angular position of the sphere is t ^−γ , with γ = 5/2 if the initial angular velocity is zero, γ = 3/2 otherwise. This behavior is due to the memory effect of the surrounding fluid. We discuss briefly other initial preparations of the system.     

Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion

The global small solutions of the tropical climate model are obtained with the fractional dissipative terms Λ^αu in the equation of the barotropic mode u and Λ^αv in the equation of the first baroclinic mode v. More precisely, we prove for 1<α ≤ 2 that the couple system has global unique strong solutions for small initial data with critical regularities. Moreover, the smallness assumption imposed on the initial barotropic mode of the velocity can be removed if α=2. We also study the large time behavior of the constructed solutions and obtain optimal time decay rates by a pure energy argument.     

Generalized dynamic programming methods in integer programming

When regarded as a shortest route problem, an integer program can be seen to have a particularly simple structure. This allows the development of an algorithm for finding thek ^th best solution to an integer programming problem with max{O(kmn), O(k logk)} operations. Apart from its value in the parametric study of an optimal solution, the approach leads to a general integer programming algorithm consisting of (1) problem relaxation, (2) solution of the relaxed problem parametrically by dynamic programming, and (3) generation ofk ^th best solutions until a feasible solution is found. Elementary methods based on duality for reducingk for a given problem relaxation are then outlined, and some examples and computational aspects are discussed.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.