# Notes on INS (8): Additional results and references (unsorted)

A list of other references and results (in no particular order)

1) A. Vasseur, Regularity criterion for 3d navier-stokes equations in terms of the direction of the velocity, Applications of Mathematics,Volume 54 (2009), Number 1, 47-52.

Abstract: In this short note, we give a link between the regularity of the solution u to the 3D Navier-Stokes equation, and the behavior of the direction of
the velocity u/|u|. It is shown that the control of div(u/|u|) in a suitable Lp_x (Lq_t ) norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. But in this case the condition is not on the vorticity, but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.

2) Peter Constantin and Charles Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J., 42(3):775–789, 1993.

3) Constantin, Some open problems and research directions in the mathematical study of fluid dynamics (a brief personal review of the problem), around 2000 ? (discusses the role played by vorticity; when $n=2$ there is an additional conservation law, which can be used to prove global regularity (?) )

4) Y. Zhou, M. Pokorny, On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component , J. Math. Phys. 50, 123514 (2009); and

Abstract: We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see “Regularity criteria for the three dimensional Navier–Stokes equations,” Indiana Univ. Math. J. 57, 2643 (2008) ] and Kukavica and Ziane [see “Navier-Stokes equations with regularity in one direction,” J. Math. Phys. 48, 065203 (2007) ]. In particular, for s ∊ [2,3], we get that the solution is regular if ∇u3Lt(0,T;Ls(3)), 2/t+3/s.

5) Y. Zhou, M. Pokorny, On the regularity of the solutions of the Navier–Stokes equations via one velocity component, Nonlinearity 23 (2010), No. 5.

We consider the regularity criteria for the incompressible Navier–Stokes equations connected with one velocity component. Based on the method from Cao and Titi (2008 Indiana Univ. Math. J. 57 2643–61) we prove that the weak solution is regular, provided , , or provided , if or if s (3, ∞]. As a corollary, we also improve the regularity criteria expressed by the regularity of or .

Abstract. In this paper we consider the three–dimensional Navier–Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one component of the velocity field, or alternatively in terms of one component of the pressure gradient, for the regularity of strong solutions to the three-dimensional Navier–Stokes equations.

Abstract. In this paper we consider the three–dimensional Navier–Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one direction derivative of the velocity field, namely, $u_z$, for the regularity of strong solutions to the three-dimensional Navier–Stokes equations.

Theorem (Cao). If $u$ is a weak solution such that $u_0 \in H^1$ and for some $T > 0$ we have $u_z \in L^\beta([0,T], L^\alpha(\Omega))$ with $\alpha > 27/16, \beta > 1$ and $3/\alpha + 2/\beta \leq 2$, then $u$ is a strong solution on the time interval $[0,T]$.

Remark. Philosophical question: why is it enough to control just one direction of the derivative ? The answer comes from interpolation  inequalities of the type

$\|\phi\|_{L^{3q}} \leq C (\|\phi_z\|_{L^q} + \|\phi\|_{L^q})^{1/3} \|\phi\|_{H^1}^{2/3}$