Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

Solvability of the Navier—Stokes System with L 2 Boundary Data

We prove the existence of the very weak solution of the Dirichlet problem for the Navier—Stokes system with L ² boundary data. Under the small data assumption we also prove the uniqueness. We use the penalization method to study the linearized problem and then apply Banach's fixed point theorem for the nonlinear problem with small boundary data. We extend our result to the case with no small data assumption by splitting the data on a large regular and small irregular part. Accepted 15 March 1999     

Shape Optimization in Viscous Compressible Fluids

Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid.     

Shape Optimization in Viscous Compressible Fluids

   Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid.     

The Effects of Flexion and Torsion on a Fluid Flow Through a Curved Pipe

We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow in powers of ɛ is computed. The first term in the expansion depends only on the tangential injection along the central curve γ of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the effects of the curvature (flexion) of γ in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to γ . The boundary layers at the ends of the pipe are studied. The error estimate is proved. Accepted 21 March 2001. Online publication 9 August 2001.     

On the Two-Dimensional Navier—Stokes Equations with the Free Boundary Condition

In this article we consider the two-dimensional Navier—Stokes equations with free boundary condition (open surface), and derive a number of different results: a new orthogonal property for the nonlinear term, improved a priori estimates on the solution, an upper bound on the dimension of the attractor which agrees with the conventional theory of turbulence; finally, for elongated rectangular domains, an improved Lieb—Thirring (collective Sobolev) inequality leads to an upper bound on the dimension of the attractor which might be optimal. Accepted 11 July 1996     

Dynamics for Linear Feedback Controlled Two-Dimensional Bénard Equations with Distributed Controls

The long-time behavior of solutions for some feedback distributed control problems associated with the Bénard equations is studied. Some linear feedback solutions for the Bénard equations are constructed. Then we prove that these feedback solutions possess the decay (in time) properties. Accepted 5 March 2001. Online publication 20 June 2001.     

On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Deterministic and Stochastic Control of Navier—Stokes Equation with Linear, Monotone, and Hyperviscosities

This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models. Accepted 7 June 1999     

Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise

We prove that any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.