This manuscript derives an evolution equation for the symmetric part of the gradient of the velocity (the strain tensor) in the incompressible Navier-Stokes equation on R 3 , and proves the existence of L 2 mild solutions to this equation. We use this equation to obtain a simplified identity for the growth of enstrophy for mild solutions that depends only on the strain tensor, not on the nonlocal interaction of the strain tensor with the vorticity. The resulting identity allows us to prove a new family of scale-critical, necessary and sufficient conditions for the blow-up of a solution at some finite time T max < +∞, which depend only on the history of the positive part of the second eigenvalue of the strain matrix. Since this matrix is trace-free, this severely restricts the geometry of any finite-time blow-up. This regularity criterion provides analytic evidence of the numerically observed tendency of the vorticity to align with the eigenvector corresponding to the middle eigenvalue of the strain matrix. This regularity criterion also allows us to prove as a corollary a new scale critical, one component type, regularity criterion for a range of exponents for which there were previously no known critical, one component type regularity criteria. Furthermore, our analysis permits us to extend the known time of existence of smooth solutions with fixed initial enstrophy E 0 = 1 2 ∇ ⊗ u 0 2 L 2 by a factor of 4,920.75-although the previous constant in the literature was not expected to be close to optimal, so this improvement is less drastic than it sounds, especially compared with numerical results. Finally, we will prove the existence and stability of blow-up for a toy model ODE for the strain equation.
In this paper, we will introduce the inviscid vortex stretching equation, which is a model equation for the 3D Euler equation where the advection of vorticity is neglected. We will show that there are smooth solutions of this equation which blowup in finite-time, even when restricting to axisymmetric, swirl-free solutions. This provides further evidence of the role of advection in depleting nonlinear vortex stretching for solutions of the 3D Euler equation.
In this paper, we will prove a regularity criterion that guarantees solutions of the Navier-Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space L 4 t L 2 x , where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier-Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe's regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli's regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier-Stokes equation must be fully three dimensional at all length scales.
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