About the behaviour of regular Navier-Stokes solutions near the blow-up

Eugénie Poulon (1)

1 LJLL - Laboratoire Jacques-Louis Lions

In this paper, we present some results about blow up of regular solutions to the homogeneous incompressible Navier-Stokes system, in the case of data in the Sobolev space $\dot{H}^{s}(\mathbb{R}^3)$, where $\displaystyle{\frac{1}{2} < s < \frac{3}{2}} \cdotp$ Firstly, we will introduce the notion of minimal blow up Navier-Stokes solutions and show that the set of such solutions is not only nonempty but also compact in a certain sense. Secondly, we will state an uniform blow up rate for minimal Navier-Stokes solutions. The key tool is profile theory as established by P. Gérard \cite{PG}.

Mots clés: Incompressible Navier-Stokes equation; blow-up; profile decomposition

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Eugénie Poulon. About the behaviour of regular Navier-Stokes solutions near the blow-up. 2014. ⟨hal-01010898v2⟩