The internal stabilization by noise of the linearized Navier-Stokes equation

Abstract

One shows that the linearized Navier-Stokes equation in , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller , , where are independent Brownian motions in a probability space and is a system of functions on with support in an arbitrary open subset . The stochastic control input is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution.