On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier\u2013Stokes equations perturbed by various random forces of low dimension

Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing

We study controllability issues for the 2D Euler and Navier- Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional torus T^2. We assume the system to be controlled by a degenerate forcing applied to fixed number of modes. In our previous work [3, 5, 4] we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity (\nu > 0). Methods of dfferential geometric/Lie algebraic control theory have been used for that study. In [3] criteria for global controllability of nite-dimensional Galerkin approximations of 2D and 3D NS systems have been established. It is almost immediate to see that these criteria are also valid for the Galerkin approximations of the Euler systems. In [5, 4] we established a much more intricate suf- cient criteria for global controllability in finite-dimensional observed component and for L2-approximate controllability for 2D NS system. The justication of these criteria was based on a Lyapunov-Schmidt reduction to a finite-dimensional system. Possibility of such a reduction rested upon the dissipativity of NS system, and hence the previous approach can not be adapted for Euler system. In the present contribution we improve and extend the controllability results in several aspects: 1) we obtain a stronger sufficient condition for controllability of 2D NS system in an observed component and for L2- approximate controllability; 2) we prove that these criteria are valid for the case of ideal incompressible uid (\nu = 0); 3) we study solid controllability in projection on any finite-dimensional subspace and establish a sufficient criterion for such controllability

Navier--Stokes equations: controllability by means of low modes forcing

We study controllability issues for 2D and 3D Navier-Stokes (NS) sys- tems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of differential geometric/Lie algebraic control theory are used to establish global control- lability of finite-dimensional Galerkin approximations of 2D and 3D NS and Euler systems, global controllability in finite-dimensional projection of 2D NS system and L2-approximate controllability for 2D NS system. Beyond these main goals we obtain results on boundedness and contin- uous dependence of trajectories of 2D NS system on degenerate forcing, when the space of forcings is endowed with so called relaxation metric

Sub-Riemannian metrics: minimality of abnormal geodesics versus subanalyticity

We study sub-Riemannian (Carnot-Caratheodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and the lack of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Annales IHP. Analyse nonlinéaire 13 , p. 635-690] we describe in this paper two classes of the germs of distributions (called 2-generating and medium fat) such that the corresponding sub-Riemannian metrics are subanalytic. To characterize these classes of distributions we determine the dimensions of the manifolds on which generic germs of distributions of given rank are respectively 2-generating or medium fat

Symplectic geometry of constrained optimization

In this paper, we discuss geometric structures related to the Lagrange multipliers rule. The practical goal is to explain how to compute or estimate the Morse index of the second variation. Symplectic geometry allows one to effectively do it even for very degenerate problems with complicated constraints. The main geometric and analytic tool is an appropriately rearranged Maslov index. We try to emphasize the geometric framework and omit analytic routine. Proofs are often replaced with informal explanations, but a well-trained mathematician will easily rewrite them in a conventional way. We believe that Vladimir Arnold would approve of such an attitude

Ensemble controllability by Lie algebraic methods

We study possibilities to control an ensemble (a parameterized family) of nonlinear control systems by a single parameter-independent control. Proceeding by Lie algebraic methods we establish genericity of exact controllability property for finite ensembles, prove sufficient approximate control- lability condition for a model problem in R3 , and provide a variant of Rashevsky-Chow theorem for approximate controllability of control-linear ensembles

Experimental and modeling studies of desensitization of P2X3 receptors

The function of ATP-activated P2X3 receptors involved in pain sensation is modulated by desensitization, a phenomenon poorly understood. The present study used patch-clamp recording from cultured rat or mouse sensory neurons and kinetic modeling to clarify the properties of P2X3 receptor desensitization. Two types of desensitization were observed, a fast process (t1/2 = 50 ms; 10 \u3bcM ATP) following the inward current evoked by micromolar agonist concentrations, and a slow process (t1/2 = 35 s; 10 nM ATP) that inhibited receptors without activating them. We termed the latter high-affinity desensitization (HAD). Recovery from fast desensitization or HAD was slow and agonist-dependent. When comparing several agonists, there was analogous ranking order for agonist potency, rate of desensitization and HAD effectiveness, with 2-methylthioadenosine triphosphate the strongest and \u3b2,\u3b3-methylene-ATP the weakest. HAD was less developed with recombinant (ATP IC50 = 390 nM) than native P2X 3 receptors (IC50 = 2.3 nM). HAD could also be induced by nanomolar ATP when receptors seemed to be nondesensitized, indicating that resting receptors could express high-affinity binding sites. Desensitization properties were well accounted for by a cyclic model in which receptors could be desensitized from either open or closed states. Recovery was assumed to be a multistate process with distinct kinetics dependent on the agonist-dependent dissociation rate from desensitized receptors. Thus, the combination of agonist-specific mechanisms such as desensitization onset, HAD, and resensitization could shape responsiveness of sensory neurons to P2X3 receptor agonists. By using subthreshold concentrations of an HAD-potent agonist, it might be possible to generate sustained inhibition of P2X 3 receptors for controlling chronic pain. Copyright \ua9 2006 The American Society for Pharmacology and Experimental Therapeutics