About some possible blow-up conditions for the 3-D Navier-Stokes equations

Abstract

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from \cite{Isab}, we prove that if one component of the velocity remains small enough in a sub-space of ${\dot{H}}^{\frac{1}{2}}$ "almost" scaling invariant, then the 3D Navier Stokes is globally wellposed. In a second time, we investigate the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form $L_T^p({\dot{B}}_{2,\infty }^{\alpha ,\frac{2}{p}-\frac{1}{2}-\alpha })$ or $L_T^p({\dot{B}}_{q,\infty }^{\frac{2}{p}+\frac{3}{q}-2})$.

References in corpus (5)

• Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor
• A certain necessary condition of potential blow up for Navier-Stokes equations
• Blow-up of critical Besov norms at a potential Navier-Stokes singularity
• A profile decomposition approach to the $L_t^{\infty }(L_x^3)$ Navier-Stokes regularity criterion
• Uniqueness result for the 3-D Navier-Stokes-Boussinesq Equations with Horizontal Dissipation

Cited by in corpus (1)

• On the global solvability of the axisymmetric Boussinesq system with critical regularity