Conditions For Uniqueness Of Solutions Research Articles

Equilibrium for fragmentation with immigration

This paper introduces stochastic processes that describe the evolution of systems of particles in which particles immigrate according to a Poisson measure and split according to a self-similar fragmentation. Criteria for existence and absence of stationary distributions are established and uniqueness is proved. Also, convergence rates to the stationary distribution are given. Linear equations which are the deterministic counterparts of fragmentation with immigration processes are next considered. As in the stochastic case, existence and uniqueness of solutions, as well as existence and uniqueness of stationary solutions, are investigated.

Oscillatory Dynamics of CO Oxidation on Platinum-Group Metal Catalysts

Rate oscillations are theoretically studied for the CO + O2 reaction proceeding according to a modified Langmuir-Hinshelwood mechanism on a platinum-group metal catalyst. A new hierarchical system of consistent mathematical models is suggested for the identification of oscillatory regimes in the stochastic model. This system includes the stochastic model based on the Monte Carlo method and a point deterministic model in the medium field approximation. Three fundamentally different types of oscillatory behavior of the stochastic model are revealed and studied. These are kinetic oscillations corresponding to autooscillations of the point model, fluctuation-induced oscillations occurring in an excitable medium in the region of the unique stable stationary solution of the point model, and fluctuation-induced random phase transitions between stable stationary solutions of the point model in the bistability region. The effect of internal fluctuations (which are inherent in stochastic models) on the oscillatory dynamics of the reaction is studied.

A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds

An important component of Hamilton’s program for the Ricci flow on compact 3-manifolds is the classification of singularities which form under the flow for certain initial metrics. In particular, Type I singularities, where the evolving metrics have curvatures whose maximums are inversely proportional to the time to blow-up, are modelled on the 3-sphere and the cylinder S × R and their quotients. On the other hand, Type II singularities (the complementary case) are much more difficult to understand. Despite this, it is known from the work of Hamilton that their singularity models are stationary solutions to the Ricci flow. This uses several techniques, including Harnack inequalities of Li-Yau-Hamilton type, the strong maximum principle for systems, dimension reduction, and the study of the geometry at infinity of noncompact stationary solutions (see§§1426 of [ H2].) In terms of Hamilton’s program, at least two obstacles remain: obtaining an injectivity radius estimate for Type II solutions and ruling out the so-called cigar soliton (the unique complete stationary solution on a surface with positive curvature) as the dimension reduction of a Type II singularity model. ∗Research partially supported by NSF grant DMS-9971891. 1In fact, Hamilton and Yau have announced informally that these are the only two obstacles and that they both would follow from obtaining a suitable differential matrix Harnack inequality of Li-Yau-Hamilton type for arbitrary solutions of the Ricci flow on compact 3manifolds. 2Added in proof: Very recently, Perelman [P1] has given a proof of the Little Loop Conjecture in all dimensions without curvature restriction. See also [P2] for some further developments.

Uniqueness for plane crack problems in dipolar gradient elasticity and in couple-stress elasticity

The present work deals with the uniqueness theorem for plane crack problems in solids characterized by dipolar gradient elasticity. The theory of gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory employed here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. These cases are also treated in the present study. We consider an anisotropic material response of the cracked plane body, within the linear version of gradient elasticity, and conditions of plane-strain or anti-plane strain. It is emphasized that, for crack problems in general, a uniqueness theorem more extended than the standard Kirchhoff theorem is needed because of the singular behavior of the solutions at the crack tips. Such a theorem will necessarily impose certain restrictions on the behavior of the fields in the vicinity of crack tips. In standard elasticity, a theorem was indeed established by Knowles and Pucik [Knowles, J.K., Pucik, T.A., 1973. Uniqueness for plane crack problems in linear elastostatics. J. Elast. 3, 155–160], who showed that the necessary conditions for solution uniqueness are a bounded displacement field and a bounded body-force field. In our study, we show that the additional (to the two previous conditions) requirement of a bounded displacement-gradient field in the vicinity of the crack tips guarantees uniqueness within the general form of the theory of dipolar gradient elasticity. In the specific cases of couple-stress elasticity and pure strain-gradient elasticity, the additional requirement is less stringent. This only involves a bounded rotation field for the first case and a bounded strain field for the second case.

Ergodicity of stochastic differential equations driven by fractional Brownian motion

We study the ergodic properties of finite-dimensional syste ms of SDEs driven by non-degenerate additive fractional Brownian motion with arbitrary Hurst parameter H 2 (0, 1). A general framework is constructed to make precise the notions of “invariant measure” and “stationary state” for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solut ion towards the stationary one is (at least) algebraic. A lower bound on the exponent is also given.

Non-existence of one-dimensional stress problems in solid–solid phase transitions and uniqueness conditions for incompressible phase-transforming materials

We show that for solid–solid phase transitions one-dimensional stress problems do not exist. The lack of uniqueness of solutions in modeling dynamical phase transitions was an unsolved issue. For a slender circular cylinder composed of an incompressible phase-transforming material we establish the proper model equation. From our model equation we establish three relations for three unknowns across the phase boundary, which provide the uniqueness conditions for solutions. Our results seem to resolve the long outstanding issue of nonuniqueness of solutions in modeling dynamical problems of phase-transforming materials. To cite this article: H.-H. Dai, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

A Note on the Solution of a Stochastic Partial Differential Equation

A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.

Relative Entropies for Kinetic Equations in Bounded Domains (Irreversibility, Stationary Solutions, Uniqueness)

The relative-entropy method describes the irreversibility of the Vlasov-Poisson and Vlasov-Boltzmann-Poisson systems in bounded domains with incoming boundary conditions. Uniform-in-time estimates are deduced from the entropy. In some cases, these estimates are sufficient to prove the convergence of the solution to a unique stationary solution, as time goes to infinity. The method is also used to analyse other types of boundary conditions such as mass- and energy-preserving diffuse-reflection boundary conditions, and to prove the uniqueness of stationary solutions for some special collision terms.

UNIQUENESS OF STATIONARY SOLUTIONS WITH VACUUM OF EULER-POISSON EQUATIONS

In this paper, the uniqueness of stationary solutions with vacuum of Euler-Poisson equations is considered. Through a nonlinear transformation which is a function of density and entropy, the corresponding problem can be reduced to a semilinear elliptic equation with a nonlinear source term consisting of a power function, for which the classical theory[4],[9] of the elliptic equations leads the authors to the uniqueness result under some assumptions on the entropy function S(x). As an example, the authors get the uniqueness of stationary solutions with vacuum of Euler-Poisson equations for S(x) = |x|θ and θ∈{0}≤∪[2(N−2),+∞.

Uniqueness of semilinear elliptic inverse problem

We consider the uniqueness of the inverse problem for a semilinear elliptic differential equation with Dirichlet condition. The necessary and sufficient condition of unique solution is obtained. We improved the results obtained by Isakov and Sylvester (1994) for the same problem.

Doubly stochastic Hilbertian processes

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (Xn) with several regimes. We suppose that the underlying process (In) which drives the evolution of (Xn) is stationary. Under some dependence assumptions on (In), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.

Doubly stochastic Hilbertian processes

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (X n ) with several regimes. We suppose that the underlying process (I n ) which drives the evolution of (X n ) is stationary. Under some dependence assumptions on (I n ), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.

ASYMPTOTIC BEHAVIOR OF GLOBAL SMOOTH SOLUTIONS TO THE FULL 1D HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

In this paper, we study the asymptotic behavior of smooth solutions to the initial boundary value problem for the full one-dimensional hydrodynamic model for semiconductors. We prove that the solution to the problem converges to the unique stationary solution time asymptotically exponentially fast.

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C1 and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C2 and C1 nonlinear complementarity problems.

SOME INTERESTING SPECIAL CASES OF A NON-LOCAL PROBLEM MODELLING OHMIC HEATING WITH VARIABLE THERMAL CONDUCTIVITY

AbstractThe non-local equation$$ u_t=(u^3u_x)_x+\frac{\lambda f(u)}{(\int_{-1}^1f(u)\,\rd x)^{2}} $$is considered, subject to some initial and Dirichlet boundary conditions. Here $f$ is taken to be either $\exp(-s^4)$ or $H(1-s)$ with $H$ the Heaviside function, which are both decreasing. It is found that there exists a critical value $\lambda^*=2$, so that for $\lambda>\lambda^{*}$ there is no stationary solution and $u$ ‘blows up’ (in some sense). If $0\lt\lambda\lt\lambda^{*}$, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.AMS 2000 Mathematics subject classification: Primary 35B30; 35B35; 35B40; 35K20; 35K55; 35K99

Sufficient conditions for unique solutions within a certain class of curve resolution models

Sufficient conditions for unique solutions within a certain class of curve resolution models

THE ASYMPTOTIC BEHAVIOR OF GLOBAL SMOOTH SOLUTIONS TO THE MACROSCOPIC MODELS FOR SEMICONDUCTORS

The authors study the asymptotic behavior of the smooth solutions to the Cauchy problems for two macroscopic models (hydrodynamic and drift-diffusion models) for semiconductors and the related relaxation limit problem. First, it is proved that the solutions to these two systems converge to the unique stationary solution time asymptotically without the smallness assumption on doping profile. Then, very sharp estimates on the smooth solutions, independent of the relaxation time, are obtained and used to establish the zero relaxation limit.

Asymptotics of Initial Boundary Value Problems for Hydrodynamic and Drift Diffusion Models for Semiconductors

In this paper, we study the asymptotic behavior of the solutions to the boundary value problems for hydrodynamic and drift diffusion models for semiconductors. Under the insulating boundary condition and equal mass condition on electron and doping profile, we prove that the solutions to these two systems converge to the unique stationary solution time asymptotically.

Solvability of Inverse Extremal Problems for Stationary Heat and Mass Transfer Equations

We consider inverse extremal problems for the stationary system of heat and mass transfer equations describing the propagation of a substance in a viscous incompressible heat conducting fluid in a bounded domain with Lipschitz boundary. The problems consist in finding some unknown parameters of a medium or source densities from a certain information of a solution. We study solvability of the direct boundary value problem and the inverse extremal problem, justify application of the Lagrange principle, introduce and analyze the optimality systems, and establish sufficient conditions for uniqueness of solutions.

Relative entropies for the Vlasov–Poisson system in bounded domains

Abstract Using relative entropies we study irreversibility for the Vlasov–Poisson system with injection conditions on the boundary with or without collisions. If the solution converges for large times, this allows to deduce that the limit has a given trace on the boundary. In the one-dimensional collisionless case and under strong regularity assumptions, this is possible only for the unique stationary solution.