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.     

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists.     

Lévy noise with memory

In this article we propose a characteristic functional for Lévy stable noise with temporal correlations. The non-Markovian characteristic functional is inferred from a physical model that describes a particle interacting with an inhomogeneous environment.     

Finite difference approximations and dynamics simulations for the Lévy Fractional Klein‐Kramers equation

The Klein‐Kramers equation describes position and velocity distribution of Langevin dynamics, the diffusion equation and Fokker‐Planck equation are its special cases for characterizing position distribution and velocity distribution, respectively. Incorporating the mechanisms of Lévy flights into the Klein‐Kramers formalism leads to the Lévy fractional Klein‐Kramers equation, which can effectively describe Lévy flights in the presence of an external force field in the phase space. For numerically solving the Lévy fractional Klein‐Kramers equation, this article presents the explicit and implicit finite difference schemes. The discrete maximum principle is generalized, using this result the detailed stability and convergence analyses of the schemes are given. And the extrapolation and some other possible techniques for improving the convergent rate or making the schemes efficient in more general cases are also discussed. The extensive numerical experiments are performed to confirm the effectiveness of the numerical schemes or simulate the superdiffusion processes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.     

A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators

We prove the equivalence of the three different definitions of the viscosity solution for the integro-differential equation with the Lévy operator. The two of the definitions are known in the preceding works of the author and the others, and the last one is new. A construction of a sequence of the approximating test functions to the subsolution (or the supersolution) is indispensable for the proof, and it is done explicitly in the paper.     

A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation

In this paper, we consider the stabilization of the nonstationary incompressible Navier–Stokes equations around a stationary solution by a boundary linear feedback control. The feedback operator is obtained from the solution of the algebraic Bernoulli equation associated with the penalized linearized Navier–Stokes equations around an unstable stationary solution and is used to locally stabilize the original nonlinear equations. We give the explicit factorized form of the stabilizing solution of the algebraic Bernoulli equation. The numerical effectiveness of this approach is demonstrated by stabilizing the vortex shedding behind a circular obstacle. Copyright © 2011 John Wiley & Sons, Ltd.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

The quintessential option pricing formula under Lévy processes

The basic digital method for option pricing developed in Ingersoll [J. Ingersoll, Digital contracts: Simple tools for pricing complex derivatives, Journal of Business 73 (1) (2000) 67–88] and Buchen and Skipper [P. Buchen, M. Skipper, The quintessential option pricing formula, School of Mathematics and Statistics, University of Sydney, 2003, pp. 1–31] is generalized to a Lévy environment. The approach is combined with the mathematical methodology of Boyarchenko and Levendorski [S.I. Boyarchenko, S.Z. Levendorski, Non-Gaussian Merton–Black–Scholes theory, World Scientific, 2002] that employs pseudo-differential operators whose symbol is expressed in terms of the characteristic exponent of the underlying Lévy process. Some new valuation formulas are obtained.     

On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{?}^j{u(t,x)|=O(|x|}^{1?n?j})$ and $|{?}^j{p(t,x)|=O(|x|}^{?n?j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.     

An optimal dividends problem with transaction costs for spectrally negative Lévy processes

We consider an optimal dividends problem with transaction costs where the reserves are modeled by a spectrally negative Lévy process. We make the connection with the classical de Finetti problem and show in particular that when the Lévy measure has a log-convex density, then an optimal strategy is given by paying out a dividend in such a way that the reserves are reduced to a certain level c₁ whenever they are above another level c₂. Further we describe a method to numerically find the optimal values of c₁ and c₂.     

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

We show that the L^p spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general L^p space when the initial velocity is in , 1<rq<∞ (1<r<q=∞).     

Mild solutions of stochastic Navier‐Stokes equation with jump noise in ‐spaces

In this paper, we study solvability of the local mild solution of stochastic Navier‐Stokes equation with jump noise in ‐spaces. This research work has mainly carried out by exploiting some interesting works of Tosio Kato.     

A note on integration of the Navier–Stokes equations

In this study the 2D Navier–Stokes equations are used to obtain a new self-similar equation. The latter equation, subject to appropriate boundary conditions and volume discharge, describes the pressure distribution and velocity field of a plane free jet.     

The large time behavior of solutions to 3D Navier–Stokes equations with nonlinear damping

This paper studies the Cauchy problem of the 3D Navier–Stokes equations with nonlinear damping term | u | ^β?1u (β ≥ 1). For β ≥ 3, we derive a decay rate of the L²‐norm of the solutions. Then, the large time behavior is given by comparing the equation with the classic 3D Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.     

A risk model driven by Lévy processes

We present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a Lévy process into the model. This seeks to account for the discrete nature of claims and asset prices. We give several explicit examples of Lévy processes that can be used to drive a risk model. This allows us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modeling. We also extend classical results on ruin probabilities to this model. Copyright © 2003 John Wiley & Sons, Ltd.     

Vanishing viscosity limit of Navier–Stokes Equations in Gevrey class

In this paper, we consider the inviscid limit for the periodic solutions to Navier–Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier–Stokes equation is independent of viscosity, and that the solutions of the Navier–Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover, the convergence rate in Gevrey class is presented. Copyright © 2017 John Wiley & Sons, Ltd.     

Ergodicity of 2D stochastic Ginzburg–Landau–Newell equations driven by degenerate noise

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate random forcing. The existence and pathwise uniqueness of strong solutions with H¹‐initial data is established, and then the existence of an invariant measure for the Feller semigroup is shown by Krylov–Bogoliubov theorem. Because of the coupled items in the stochastic Ginzburg–Landau–Newell equations, the higher order momentum estimates can be only obtained in the L²‐norm. We show the ergodicity of invariant measure for the transition semigroup by asymptotically strong Feller property and the support property. Copyright © 2017 John Wiley & Sons, Ltd.     

Wellposedness for the Navier–Stokes flow in the exterior of a rotating obstacle

In this paper we study the Navier–Stokes boundary‐initial value problem in the exterior of a rotating obstacle, in two and three spatial dimensions. We prove the local in time existence and uniqueness of strong solutions. Moreover, we show that the solutions are global in time, in two spatial dimensions. Copyright © 2005 John Wiley & Sons, Ltd.     

Pullback attractors of 2D Navier–Stokes equations with weak damping,distributed delay,and continuous delay

In this present paper, the existence of pullback attractors for the 2D Navier–Stokes equation with weak damping, distributed delay, and continuous delay has been considered, by virtue of classical Galerkin's method, we derived the existence and uniqueness of global weak and strong solutions. Using the Aubin–Lions lemma and some energy estimate in the Banach space with delay, we obtained the uniform bounded and existence of uniform pullback absorbing ball for the solution semi‐processes; we concluded the pullback attractors via verifying the pullback asymptotical compactness by the generalized Arzelà–Ascoli theorem. Copyright © 2016 John Wiley & Sons, Ltd.