Well-posedness of the Hydrodynamic Model for Semiconductors

This paper concerns the well-posedness of the hydrodynamic model for semiconductor devices, a quasi-linear elliptic–parbolic–hyperbolic system. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary conditions for the hyperbolic equations are assumed to be well-posed in L² sense. Maximally strictly dissipative boundary conditions for the hyperbolic equations satisfy the assumption of well-posedness in L² sense. The well-posedness of the model under the boundary conditions is demonstrated.     

The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case

In this article we consider the 3D Primitive Equations (PEs) of the ocean, without viscosity and linearized around a stratified flow. As recalled in the Introduction, the PEs without viscosity ought to be supplemented with boundary conditions of a totally new type which must be nonlocal. In this article a set of boundary conditions is proposed for which we show that the linearized PEs are well-posed. The proposed boundary conditions are based on a suitable spectral decomposition of the unknown functions. Noteworthy is the rich structure of the Primitive Equations without viscosity. Our study is based on a modal decomposition in the vertical direction; in this decomposition, the first mode is essentially a (linearized) Euler flow, then a few modes correspond to a stationary problem partly elliptic and partly hyperbolic; finally all the other modes correspond to a stationary problem fully hyperbolic.     

Exact Controllability for Nonautonomous First Order Quasilinear Hyperbolic Systems

By means of the theory on the semi-global C1 solution to the mixed initial-boundary value problem (IBVP) for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary conditions.     

On the completeness of root vectors generated by systems of coupled hyperbolic equations

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non‐selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary‐value problem (IBVP) for a non‐homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with a one‐parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non‐singular and singular; (c) IBVP for a three‐dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (e) IBVP for a coupled Euler‐Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending‐torsion vibration model); (f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double‐walled carbon nanotube. Problems have been discussed in the first paper of the aforementioned set. Problems are discussed in the present paper.     

Decaying modes for the wave equation in the exterior of an obstacle

In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λ_k} of Δ on the geometry of the obstacle ??. In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle ??, both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres—for which the eigenvalues {λ_k} can be determined as roots of special functions—upper and lower bounds for the density of the real {λ_k}, and upper and lower bounds for λ₁, the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on ??. Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono-energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.     

Global well-posedness for the microscopic FENE model with a sharp boundary condition

We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement. The well-posedness of the FENE model that consists of the incompressible Navier–Stokes equation and the Fokker–Planck equation has been studied intensively, mostly with the zero flux boundary condition. In this article, we show that for the well-posedness of the microscopic FENE model ($b>2$) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function. Under this condition, it is shown that there exists a unique weak solution in a weighted Sobolev space. Moreover, such a condition still ensures that the distribution is a probability density. The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero no faster than the distance function.     

The Discontinuous Riemann–Hilbert Problem for Elliptic Complex Equations of First Order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.     

On a problem of Diophantus for rationals

Let q be a nonzero rational number. We investigate for which q there are infinitely many sets consisting of five nonzero rational numbers such that the product of any two of them plus q is a square of a rational number. We show that there are infinitely many square-free such q and on assuming the Parity Conjecture for the twists of an explicitly given elliptic curve we derive that the density of such q is at least one half. For the proof we consider a related question for polynomials with integral coefficients. We prove that, up to certain admissible transformations, there is precisely one set of non-constant linear polynomials such that the product of any two of them except one combination, plus a given linear polynomial is a perfect square.     

On the Dirichlet and Lidstone Problems for a Higher-Order Linear Hyperbolic Equation

For a higher-order linear hyperbolic equation with nonsmooth coefficients, we consider the Dirichlet and Lidstone problems in a rectangle with nonclassical boundary conditions and prove that these problems are equivalent to the classical Dirichlet and Lidstone problems, respectively.     

GLOBAL EXISTENCE AND LARGE TIMEBEHAVIOR OF SOLUTION TO REACTING FLOW MODEL WITH BOUNDARY EFFECT

1IntroductionThegoalofthispaperistoinvestigatetheglobalexistenceandlargetimebehaviorofsolutionstoareactingflowwithboundaryeffectsast-oo.ThesystemillEulerianformcallbewrittenaswhichwasproposedbyR.J.LeVequeandothersin[8]tomodelthemotiollofreacti11ggaswithtwomodes.Wl1ere,p7'isthedensityofthemajormodeandpscorrespolldstotllellli1lormode,r s=l.itisthevelocity,andp=pc'(r Ps)isthepressllrewllichcallbederivedbyAvogadro'sLaw.Here,cisthesouudspeedoftl1emajorn1ode.Thepara1lleterPprovidessometenuousliu…     

Hyperbolic equations with Dirichlet boundary feedback via position vector: Regularity and almost periodic stabilization—Part I

A hyperbolic equation defined on a bounded domain is considered, with input acting in theDirichlet boundary condition and expressed as a specifiedfeedback of theposition vector only. Two main results are established. First, we prove a well-posedness and regularity result of the feedback solutions. Second, we specialize our equation to the case when the original differential operator withzero boundary conditions is self-adjoint and unstable. Here, under certain natural algebraic conditions based on the finitely many unstable eigenvalues, we establish the existence ofboundary vectors, for which the corresponding feedback solutions have the same desirablestructural property of astable free system: They can be expressed as an infinite linear combination of sines and cosines (special case of almost periodicity). A cosine operator approach is employed.This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-78-3350 (I.L.) and Grant AFOSR-77-3338 (R.T.) through ISU.This research was performed while the author was visiting the Department of Systems Science, University of California, Los Angeles.     

Absorbing Boundary Conditions for Hyperbolic Systems

<正>This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions.We prove the strict well-posedness of the resulting initial boundary value problem in 1D.Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme.Hereby,we have to extend the classical proofs,since the(discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.     

A Mixed Problem for a Hyperbolic System on the Plane with Delay in the Boundary Conditions

We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.     

Lp-solvability of a full superconductive model

In this article the mean-field vortex model arising from the II-type superconductivity is investigated. The vortex model is reduced to a nonlinear hyperbolic–elliptic system of PDEs in a bounded domain. Motivated by experiments, we consider physical boundary conditions, which describe a flux of superconducting vortices through the boundary of the domain. We prove the global solvability for the system. To show the solvability result we take a vanishing “viscosity” limit in an approximated parabolic–elliptic system. Since the approximated solutions do not have a compactness property, we justify this limit transition, using a kinetic formulation of our problem. The main trick is that instead of the nonlinear system, we have to investigate a linear transport equation.     

On coercivity and regularity for linear elliptic systems

We introduce a nonlinear method to study a ??universal?? strong coercivity problem for monotone linear elliptic systems by compositions of finitely many constant coefficient tensors satisfying the Legendre?CHadamard strong ellipticity condition. We give conditions and counterexamples for universal coercivity. In the case of non-coercive systems we give examples to show that the corresponding variational integral may have infinitely many nowhere C ¹ minimizers on their supports. For some universally coercive systems we also present examples with affine boundary values which have nowhere C ¹ solutions.     

Linear Elliptic Boundary Value Problems with Non‐Smooth Data: Campanato Spaces of Functionals

In this paper linear elliptic boundary value problems of second order with non‐smooth data L^∞‐coefficients, sets with Lipschitz boundary, regular sets, non‐homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev‐Campanato spaces of functions and functionals, respectively.     

Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

In this article (which is divided in three parts) we investigate the non‐linear initial boundary value problems (1.2) and (1.3). In both cases we consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1 at hand, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (1.2) using the so‐called energy method. In the above sense, the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to the non‐linear initial boundary value problem (1.3). In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Infinitely many bounded solutions for the p-Laplacian with nonlinear boundary conditions

By applying a method due to Saint Raymond, we prove the existence of infinitely many weak solutions for a quasilinear elliptic partial differential equation, involving the p-Laplacian operator, coupled with a nonlinear boundary condition. Our main assumption is a suitable oscillatory behaviour of the nonlinearity either at infinity or at zero.     

n-Tuples of positive integers with the same second elementary symmetric function value and the same product

In this paper, by using the theory of elliptic curves, we prove that for every k, there exist infinitely many primitive sets of k n-tuples of positive integers with the same second elementary symmetric function value and the same product.