Apparaît dans la collection : Vorticity, rotation and symmetry (III) - approaching limiting cases of fluid flows / Vorticité, rotation et symétrie (III) – analyse des situations limites en théorie des fluides

The purpose of the talk will be the proof of the following result for the homogeneous incompressible Navier-Stokes system in dimension three: given an initial data $v_0$ with vorticity $\Omega_0= \nabla \times v_0$ in $L^{\tfrac{3}{2}}$ (which implies that $v_0$ belongs to the Sobolev space $H^{\tfrac{1}{2}}$ ), we prove that the solution $v$ given by the classical Fujita-Kato theorem blows up in a finite time $T^²$ only if, for any $p$ in ]4,6[ and any unit vector $e$ in $\mathbb{R}^3$ ; there holds $\int_{0}^{T^²}\left | v(t)\cdot e\right |^p_{\frac{1}{2}+\frac{2}{p}}dt=\infty $. We remark that all these quantities are scaling invariant under the scaling transformation of Navier-Stokes system.