Critical strong Feller regularity for Markov solutions to the Navier-Stokes equations (Q638465)

The author considers the Navier-Stokes equation on a three-dimensional torus, \[ \dot u + (u\cdot\nabla)u+\nabla p = \nu\Delta u +\dot\eta,\quad \mathrm{div}\; u = 0, \] driven by a Gaussian noise \(\dot\eta\). Using an abstract selection principle for Markov families and the short time coupling with a smooth process, it is shown that every Markov solution to the Navier-Stokes equation has the strong Feller property in the topology of the domain of \(A^\alpha\) for every \(\alpha>\frac12\), where \(A\) is the Stokes operator.