# Stochastic vorticity equation: existence, uniqueness and regularity results in the flat torus and in the whole plane

Università degli Studi Trento

38123 Povo (TN)

Tel +39 04 61/281508-1625-1701-3786

dept.math [at] unitn.it

**Luogo: **Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari "-1"**Ore**: 11:45

### Relatrice:

**Margherita Zanella**(Università di Pavia)

### Abstract:

In this talk we deal with the two-dimensional stochastic Navier-Stokes equations in their vorticity form. We consider at first the equations on the flat torus, with a stochastic forcing term given by a Gaussian noise, white in time and colored in space. We prove existence and uniqueness of a weak solution process in the martingale measure approach (Walsh notion of solution). Moreover, we prove the space-time continuity of the solution process and we study its regularity in the Malliavin sense.

Then we consider the Navier-Stokes equations in vorticity form in the whole plane R2 with a white noise forcing term of multiplicative type, whose spatial covariance is not regular enough to apply the Itô calculus in Lq spaces, q>1. We prove the existence of a unique strong (in the probability sense) solution.

The talk is based on two joint works with B. Ferrario.

**Referente: **Stefano Bonaccorsi