Measurable solutions for stochastic evolution equations without uniqueness

A method for obtaining measurable solutions to stochastic evolution equations in which there is no uniqueness for the corresponding non-stochastic equation is presented. It involves a technique based on a measurable selection theorem for set-valued functions. No assumptions are needed on the underlying probability space. An application is given to the stochastic Navier–Stokes problem in arbitrary dimensions. We also show the existence of measurable solutions to stochastic ordinary differential equations in which there is no uniqueness. A finite-dimensional generalization is given to adapted solutions in the case of a normal filtration and path uniqueness.     

Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients

We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed.     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

On the existence and long-time behavior of solutions to stochastic three-dimensional Navier–Stokes–Voigt equations

We consider the 3D stochastic Navier–Stokes–Voigt equations in bounded domains with the homogeneous Dirichlet boundary condition and infinite-dimensional Wiener process. First, we prove the existence and uniqueness of solutions to the problem. Then we investigate the mean square exponential stability and the almost sure exponential stability of the stationary solutions.     

Global Mild Solution of Stochastic Generalized Navier–Stokes Equations with Coriolis Force

In this paper, we apply Littlewood–Paley theory and Itô integral to get the global existence of stochastic Navier–Stokes equations with Coriolis force in Fourier–Besov spaces. As a comparison, we also give corresponding results of the deterministic Navier–Stokes equations with Coriolis force.     

Optimal control for 3D stochastic Navier–Stokes equations

Loeb space methods are used to prove existence of an optimal control for general 3D stochastic Navier–Stokes equations with multiplicative noise. The possible non-uniqueness of the solutions mean that it is necessary to utilize the notion of a non-standard approximate solution developed in the paper by N.J. Cutland and Keisler H.J. 2004, Global attractors for 3-dimensional stochastic Navier–Stokes equations, Journal of Dynamics and Differential Equations, pp. 16205–16266, for the study of attractors.     

Stochastic 3D tamed Navier–Stokes equations: Existence,uniqueness and small time large deviation principles

In this paper, we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3D tamed Navier–Stokes equations in case of Dirichlet boundary conditions for the Stokes–Laplacian. Then we prove a small time large deviation principle for the solutions.     

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.     

Global existence of weak and strong solutions to Cucker–Smale–Navier–Stokes equations in R2

We study the existence theory for the Cucker–Smale–Navier–Stokes (in short, CS–NS) equations in two dimensions. The CS–NS equations consist of Cucker–Smale flocking particles described by a Vlasov-type equation and incompressible Navier–Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS–NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.     

Existence and uniqueness of solutions to a model describing miscible liquids

The existence and the uniqueness of solutions to a problem of miscible liquids are investigated in this note. The model consists of Navier–Stokes equations with Korteweg stress terms coupled with the reaction–diffusion equation for the concentration. We assume that the fluid is incompressible and the Boussinesq approximation is adopted. The global existence and uniqueness of solutions is established for some optimal conditions on the reaction source term and the external force functions.     

Global classical solutions of viscous liquid–gas two‐phase flow model

In this paper, we consider the global existence and uniqueness of the classical solutions for the three‐dimensional where the existence of global classical solutions to the compressible Navier–Stokes equations was obtained by using the continuity methods under the assumption that the initial energy is sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

Ergodicity for the 3D stochastic Navier–Stokes equations

We consider the Kolmogorov equation associated with the stochastic Navier–Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u_m of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier–Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Construction of weak solutions of a certain stochastic Navier–Stokes equation

We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.     

A second‐order backward difference time‐stepping scheme for penalized Navier–Stokes equations modeling filtration through porous media

We investigate the stability and convergence of a fully implicit, linearly extrapolated second‐order backward difference time‐stepping scheme for the penalized Navier–Stokes equations modeling filtration through porous media. In the penalization approach, an extended Navier–Stokes equation is used in the entire computational domain with suitable resistance terms to mimic the presence of porous medium. It is widely used as an alternative to the heterogeneous approach in which different types of partial differential equations (PDEs) are used in fluid and porous subregions along with suitable continuity conditions at the interface. However, the introduction of extra resistance terms makes the penalized Navier–Stokes equations more nonlinear. We prove that the linearly extrapolated scheme is unconditionally stable and derive optimal order error estimates without any stability condition. To show feasibility and applicability of the approach, it is used to numerically solve a passive control problem in which flow around a solid body is controlled by adding porous layers on the surface. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 681–705, 2016     

Analysis of nonlocal model of compressible fluid in 1‐D

We study a nonlocal modification of the compressible Navier–Stokes equations in mono‐dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view, our system is an intermediate between the Euler and Navier–Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and global existence of solutions. Particularly, we show the uniform in time bound on the density of fluid. Copyright © 2010 John Wiley & Sons, Ltd.     

Feedback control for nonlinear evolutionary equations with applications

In the paper we are concerned with the feedback control system governed by nonlinear evolutionary equations involving weakly continuous operators. By using the Rothe method and a surjectivity result for weakly continuous operators, we first present the solvability for the evolutionary equation. Then we show the existence of solutions to the feedback control system. We also consider an existence result for an optimal control problem. Moreover, we apply the main results to a class of differential variational inequalities, evolutionary hemivariational inequalities and the non-stationary Navier–Stokes–Voigt equation with a subgradient inclusion condition.     

Numerical analysis of the stochastic Stokes equations of Wick type

We propose a finite element method for the numerical solution of the stochastic Stokes equations of the Wick type. We give existence and uniqueness results for the continuous problem and its approximation. Optimal error estimates are derived and algorithmic aspects of the method are discussed. Our method will reduce the problem of solving stochastic Stokes equations to solving a set of deterministic ones. Moreover, one can reconstruct particular realizations of the solution directly from Wiener chaos expansions once the coefficients are available. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.