Conditions For Uniqueness Of Solutions Research Articles

Asymptotic behavior of solutions for a three dimensional system of globally modified Navier–Stokes equations with a locally Lipschitz delay term

Existence, uniqueness, and continuity properties of solutions for a globally modified version of Navier–Stokes equations with finite delay terms within a locally Lipschitz operator are established. Moreover, we also analyze the stationary problem, and, under suitable assumptions, we prove that there exists a unique stationary solution, which is globally asymptotically exponentially stable.

Thermal runaway for a nonlinear diffusion model in thermal electricity

In this paper, we consider the phenomena of the thermal runaway andthe asymptotic runaway in a nonlocal nonlinear model, which israised from the thermal-electricity and it is so-called an Ohmicheating model. The model prescribes the dimensionless temperaturewhen the electric current flows through two conductors, subject to afixed potential difference. The electrical resistivity of the one ofthe conductors depends on the temperature and the other one remainsconstant. The problem will be mathematically formulated to aquasilinear nonlocal parabolic equation with Dirichlet boundarycondition. An analysis of the problem shows that the solution of theproblem exists globally, provided that the conductor with constantresistivity is connected. Furthermore, for some specialtemperature-resistivity relations, the unique stationary solution isshown to be global asymptotically stable. The results assert aphysical fact that the thermal produced by the Ohmic heating processwill runaway from the surfaces of the conductor, the temperature ofthe conductor remains bounded and solution of the system convergesasymptotically to the unique equilibrium.

THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR

We consider the linear Wigner–Fokker–Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker–Planck type operators in certain weighted L2-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate.

Efficient approximation of the solution of certain nonlinear reaction–diffusion equations with large absorption

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the absorption is large enough, compared with the flux in the boundary, there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the “continuous” equation. Furthermore, we exhibit an algorithm computing an ε-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.

Stationary free surface viscous flows without surface tension in three dimensions

We consider an incompressible, viscous, finite depth fluid flowing down a three-dimensional channel. In the absence of surface tension, we prove the existence of a unique stationary solution in weighted Sobolev spaces. The result is based on a thorough study of the linearized problem, particularly the pseudodifferential operator relating the normal velocity of the fluid and the normal component of the associated stress tensor along the free surface, and requires the use of the Nash-Moser implicit function theorem.

Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case

Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of W concentrating at the origin.

Tail behaviour of stationary solutions of random difference equations: the case of regular matrices

Given a sequence of i.i.d. random variables with generic copy such that M is a regular matrix and Q takes values in , we consider the random difference equation Under suitable assumptions stated below, this equation has a unique stationary solution R such that for some and some finite positive and continuous function K on , holds true. A rather long proof of this result, originally stated by Kesten [Acta Math. 131 (1973), pp. 207–248] at the end of his famous article, was given by Le Page [Séminaires de probabilités Rennes 1983, University of Rennes I, Rennes, 1983, p. 116]. The purpose of this article is to show how regeneration methods can be used to provide a much shorter argument (particularly for the positivity of K). It is based on a multidimensional extension of Goldie's implicit renewal theory developed in Goldie [Ann. Appl. Probab. 1 (1991), pp. 126–166].

Convergence to Stable Laws for Multidimensional Stochastic Recursions: The Case of Regular Matrices

Given a sequence (M n , Q n ) n ≥ 1 of i.i.d. random variables with generic copy (M, Q) ∈ GL(d, ℝ) ×ℝd , we consider the random difference equation (RDE) $$ R_{n}=M_{n}R_{n-1}+Q_{n}, $$ n ≥ 1, and assume the existence of κ > 0 such that $$ \lim_{n \to \infty} \left(\mathbb{E}\ensuremath{\left\| {M_1 \cdots M_n}^\kappa \right\|}\right)^{\frac{1}{n}} = 1 . $$ We prove, under suitable assumptions, that the sequence S n = R 1 + ... + R n , appropriately normalized, converges in law to a multidimensional stable distribution with index κ. As a by-product, we show that the unique stationary solution R of the RDE is regularly varying with index κ, and give a precise description of its tail measure.

Asymptotic behavior for an Ohmic heating model in thermal electricity

In this paper, we consider the asymptotic behavior for a nonlocal parabolic problem with Dirichlet boundary condition, which is raised from the thermal-electricity and is so-called an Ohmic heating model. It models the system of the temperature of a conductor in the device, which is connected in series with another conductor with constant. The electrical resistivity of the one of the conductors depends on the temperature and the other one remains constant. An analysis of the nonlinear problem shows that the solution exists global and the unique stationary solution is globally asymptotically stable. The results assert that the temperature of the conductor remains bounded and the system converges asymptotically to the unique equilibrium.

The ‘Farmerian’ Approach to Ending a Finance‐Induced Recession: Notes on Stability and Dynamics

In this paper, I explore the out‐of‐equilibrium dynamics of Farmer's ME‐NA model. Maintaining the assumption that all the variables continuously have the same time reference, I build a model that describes the adjustments of the value of output and the interest rate under the hypothesis that public debt and the stock market value are given. Within this framework, I show that the economy has a unique stationary solution whose dynamics are stable. Moreover, simulating the model under its baseline calibration, I show that adjustments towards the steady‐state occur through convergent oscillations while the most promising way out from a finance‐induced recession is a policy mix that combines a mild fiscal expansion with interventions aimed at altering the trade‐off between holding risky and safe assets.

Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space

We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝn admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted L p energy method.

Approximating travelling waves by equilibria of non-local equations

We consider an evolution equation of parabolic type in R having a travelling wave solution. We study the effects on the dynamics of an appropriate change of variables which transforms the equation into a non-local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. This procedure allows us to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples. We analyze the relation of the new equation with the original one in the entire real line. We also analyze the behavior of the non-local problem in a bounded interval with appropriate boundary conditions. We show that it has a unique stationary solution which approaches the traveling wave as the interval gets larger and larger and that is asymptotically stable for large enough intervals.

A Variational Approach to the Stationary Solutions of the Burgers Equation

We consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by Bertini et al. [Comm. Pure Appl. Math., 64 (2011), pp. 649–696], we analyze a Lyapunov functional for such an equation which gives the large deviations asymptotic of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functional, whose minimizer is given by the unique stationary solution, as the length of the interval diverges. In particular, we focus on boundary data corresponding to a standing wave solution to the Burgers equation in the whole line. In this case, the limiting functional has a one-parameter family of minimizers and we compute the sharp asymptotic cost corresponding to a given shift of the stationary solution.

Mathematics of the NFAT Signaling Pathway

This paper is a mathematical study of some aspects of the signaling pathway leading to the activation of the transcription factor NFAT (nuclear factor of activated T cells). Activation takes place by dephosphorylation at multiple sites. This has been modeled by Salazar and Höfer using a large system of ordinary differential equations depending on many parameters. With the help of chemical reaction network theory we show that for any choice of the parameters this system has a unique stationary solution for each value of the conserved quantity given by the total amount of NFAT and that all solutions converge to this stationary solution at late times. The dephosphorylation is carried out by calcineurin, which in turn is activated by a rise in calcium concentration. We study the way in which the dynamics of the calcium concentration influences NFAT activation, an issue also considered by Salazar and Höfer with the help of a model arising from work of Somogyi and Stucki. Criteria are obtained for convergence to equilibrium of solutions of the model for the calcium concentration.

Solving Intrinsic Parameters of Camera Calibrated from Controlled Motion Sequences and Homography

A camera calibration based on controlled motion sequences and homography is adopted. First condition of unique solution for constrained equation set is proofed. Then two sets three orthogonal motions are carried out for camera in manipulator, thus 6 homographic matrixes of plane are obtained at 7 view points, and thus the intrinsic parameters of camera are achieved. The experiments on real images validate our technology based on homography.

Late-time behaviour of the Einstein-Vlasov system with Bianchi I symmetry

The late-time behaviour of the Einstein-dust system is well understood for homogeneous spacetimes. For the case of Bianchi I we have been able to show that the late-time behaviour of the Einstein-Vlasov system is well approximated by the Einstein-dust system assuming that one is close to the unique stationary solution which is the attractor of the Einstein-dust system.

A Criterion for Stationary Solutions of Retarded Linear Equations with Additive Noise

In this work, we shall consider stationary (mild) solutions for a class of retarded functional linear differential equations with additive noise in Hilbert spaces. We first introduce a family of Green operators for the stochastic systems and establish stability results which will play an important role in the investigation of stationary solutions. A criterion imposed on the Green operators is presented to identify a unique stationary solution for the systems considered. Under strong quasi-Feller property, it is shown that this criterion is a sufficient and necessary condition to guarantee a unique stationary solution, based on a method having its origins in optimal control theory.

Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

Nous démontrons que les équations différentielles stochastiques (EDS) conduites par des mouvements Browniens fractionnaires à paramètre de Hurst H>½ ont des propriétés ergodiques similaires aux EDS usuelles conduites par des mouvements Browniens. L’intérêt principal du présent article est de pouvoir traîter également des systèmes hypoelliptiques satisfaisant la condition de Hörmander. Nous montrons qu’une version adéquate de la propriété de Feller forte est vérifiée par de tels systèmes et nous en déduisons que, sous une propriété de controllabilité usuelle, ils admettent une unique solution stationnaire qui ait un sens physique. L’ingrédient principal de notre analyse est une borne supérieure sur les moments inverses de la matrice de Malliavin associée, conditionnée au passé du bruit.

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions.Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.

Thresholds, news impact surfaces and dynamic asymmetric multivariate GARCH

Dynamic Asymmetric Multivariate GARCH (DAMGARCH) is a new model that extends the Vector ARMA‐GARCH (VARMA‐GARCH) model of Ling and McAleer (2003) by introducing multiple thresholds and time‐dependent structure in the asymmetry of the conditional variances. Analytical expressions for the news impact surface implied by the new model are also presented. DAMGARCH models the shocks affecting the conditional variances on the basis of an underlying multivariate distribution. It is possible to model explicitly asset‐specific shocks and common innovations by partitioning the multivariate density support. This article presents the model structure, describes the implementation issues, and provides the conditions for the existence of a unique stationary solution, and for consistency and asymptotic normality of the quasi‐maximum likelihood estimators. The article also presents an empirical example to highlight the usefulness of the new model.