Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

In this paper, we establish the global fast dynamics for the time‐dependent Ginzburg–Landau equations of superconductivity. We show the squeezing property and the existence of finite‐dimensional exponential attractors for the system. In addition we prove the existence of the global attractor in L² × L² for the Ginzburg–Landau equations in two spatial dimensions. Copyright © 1999 John Wiley & Sons, Ltd.     

Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise

In this paper, we consider the ergodicity of invariant measures for the stochastic Ginzburg-Landau equation with degenerate random forcing. First, we show the existence and pathwise uniqueness of strong solutions with H¹-initial data, and then the existence of an invariant measure for the Feller semigroup by the Krylov-Bogoliubov method. Then in the case of degenerate additive noise, using the notion of asymptotically strong Feller property, we prove the uniqueness of invariant measures for the transition semigroup.     

Ergodicity of stochastic 2D Navier–Stokes equation with Lévy noise

In this paper we deal with the 2D Navier-Stokes equation perturbed by a Lévy noise force whose white noise part is non-degenerate and that the intensity measure of Poisson measure is σ-finite. Existence and uniqueness of invariant measure for this equation is obtained, two main properties of the Markov semigroup associated with this equation are proved. In other words, strong Feller property and irreducibility hold in the same space.     

Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993–1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.     

On the global existence and small dispersion limit for a class of complex Ginzburg–Landau equations

In this paper we consider a class of complex Ginzburg–Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial‐value problem in d‐dimensional torus ??^d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

Invariant measures and regularity properties of perturbed Ornstein-Uhlenbeck semigroups

This paper deals with perturbations of the Ornstein-Uhlenbeck operator on L²-spaces with respect to a Gaussian measure μ. We perturb the generator of the Ornstein-Uhlenbeck semigroup by a certain unbounded, non-linear drift, and show various properties of the perturbed semigroup such as compactness and positivity. Strong Feller property, existence and uniqueness of an invariant measure are discussed as well.     

Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Convergence analysis of a linearized Crank–Nicolson scheme for the two‐dimensional complex Ginzburg–Landau equation

A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L² ‐norm by the discrete energy method. The convergence order is \begin{align*}\mathcal{O}(\tau^2+h_1^2+h^2_2)\end{align*}, where τ is the temporal grid size and h₁,h₂ are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A linearized Crank–Nicolson–Galerkin FEM for the time‐dependent Ginzburg–Landau equations under the temporal gauge

We propose a decoupled and linearized fully discrete finite element method (FEM) for the time‐dependent Ginzburg–Landau equations under the temporal gauge, where a Crank–Nicolson scheme is used for the time discretization. By carefully designing the time‐discretization scheme, we manage to prove the convergence rate , where τ is the time‐step size and r is the degree of the finite element space. Due to the degeneracy of the problem, the convergence rate in the spatial direction is one order lower than the optimal convergence rate of FEMs for parabolic equations. Numerical tests are provided to support our error analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1279–1290, 2014     

Weak solutions to the Ginzburg–Landau model in superconductivity with the Coulomb gauge

We first prove the uniqueness of weak solutions (ψ,A) to the 3‐D Ginzburg–Landau model in superconductivity with zero magnetic diffusivity and the Coulomb gauge if , which is a critical space for some positive constant T. We also prove the global existence of solutions when and A₀∈L³. Copyright © 2016 John Wiley & Sons, Ltd.     

Perfectness of Conelike *–Semigroups in ℚk

Let S be an abelian *–semigroup in ℚ^k. We give a sufficient condition for every positive definite function on S to have a unique representing measure on the dual semigroup of S (i.e. S is perfect). To characterize perfectness for any abelian *–semigroupis a challenging, but not yet generally solved problem. In this paper, we characterize the structure of involutions on an abelian *–semigroup which is a subset of ℚ^k, and show that any conelike *–semigroups in ℚ^k are perfect.     

Non-random Invariant Sets for Some Systems of Parabolic Stochastic Partial Differential Equations

Abstract

In this article we investigate a class of non-autonomous, semilinear, parabolic systems of stochastic partial differential equations defined on a smooth, bounded domain 𝒪 ? ? ⁿ and driven by an infinite-dimensional noise defined from an L ²(𝒪)-valued Wiener process; in the general case the noise can be colored relative to the space variable and white relative to the time variable. We first prove the existence and the uniqueness of a solution under very general hypotheses, and then establish the existence of invariant sets along with the validity of comparison principles under more restrictive conditions; the main ingredients in the proofs of these results consist of a new proposition concerning Wong–Zakaï approximations and of the adaptation of the theory of invariant sets developed for deterministic systems. We also illustrate our results by means of several examples such as certain stochastic systems of Lotka–Volterra and Landau–Ginzburg equations that fall naturally within the scope of our theory.     

Existence and Uniqueness of Invariant Measures for SPDEs with Two Reflecting Walls

In this article, we study stochastic partial differential equations with two reflecting walls h ¹ and h ², driven by space-time white noise with non-constant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. The strong Feller property is also obtained.     

Uniqueness of Invariant Measures of Infinite Dimensional Stochastic Differential Equations Driven by Lévy Noises

In this paper, we are attempting to study the uniqueness of invariant measures of a stochastic differential equation driven by a Lévy type noise in a real separable Hilbert space. To investigate this problem, we study the strong Feller property and irreducibility of the corresponding Markov transition semigroup respectively. To show the strong Feller property, we generalize a Bismut–Elworthy–Li type formula to our Markov transition semigroup under a non-degeneracy condition of the coefficient of the Wiener process.     

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

Applications of Feller–Reuter–Riley Transition Functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.     

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

In this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing.     

Log-Harnack inequality for stochastic Burgers equations and applications

By proving an L²-gradient estimate for the corresponding Galerkin approximations, the log-Harnack inequality is established for the semigroup associated to a class of stochastic Burgers equations. As applications, we derive the strong Feller property of the semigroup, the irreducibility of the solution, the entropy-cost inequality for the adjoint semigroup, and entropy upper bounds of the transition density.