On a family of results concerning direction of vorticity and regularity for the Navier–Stokes equations

Abstract

These notes concern existence, and suitable formulation, of meaningful conditions on the direction of the vorticity which guarantee the regularity of the solutions to the evolution Navier–Stokes equations. A main concern here is to compare the different situations which appear in considering slip and no-slip boundary conditions. The paper reviews mainly results obtained in some of the references cited.

Appendix

As remarked at the end of the introduction, we present here some reflections upon the global structure and significance of our results, taken as a whole.

As the reader has verified, the statements presented above are split into two families of sufficient conditions for regularity, namely, $\,\beta \ge \, \frac{1}{2}\,$ and $\,\beta \le \, \frac{1}{2}\,$. In Theorem 1.2, the advantage of assuming $\,\beta >\, \frac{1}{2}\,$ is counterbalanced by replacing in Eq. (5) the constant $\,c\,$ by a function $g\in L^a(0,T;L^b(\Omega ))\,$. On the other hand, in Theorem 1.3, we mitigate the penalizing situation $\,\beta <\,\frac{1}{2}\,$ by assuming (8). This situation may give the wrong idea that the two families of results are relatively independent. On the contrary, the above formal separation is not substantial. In fact, the two families glue perfectly at the intersection point $\,\beta =\,\frac{1}{2}\,$ since the conclusion (namely, “condition (6) implies regularity”) is the same in both cases. On the other hand, a step by step analysis of the proofs given for each of the two above theorems, shows that, inside each class, the results have the same “strength”, independently of the value of the parameter $\,\beta$. Since the two families “glue” at point $\,\frac{1}{2}\,,$ we conclude that we have just one family of strictly connected results, all having an equivalent “strength”.

We may also show the “equal strength” of the above sufficient conditions for regularity by appealing to scaling techniques. Let us illustrate this possibility by showing that the sufficient conditions for regularity $\quad \sin \theta (x,y,t)\le g(t,x)|x-y|^\beta \,,$ as $\,\beta \,$ goes from $\frac{1}{2}$ to $1\,,$ enjoy the same strength.

Assume that $\,(\,(u(x,\,t),\,p(x,\,t)\,)\,$ is a solution to the Navier–Stokes equations in $\,(\,0,\,+\,\infty \,) \times \,R^3\,$. Then

$$\begin{aligned} (\,(u_{\lambda }(x,\,t),\,p_{\lambda }(x,\,t)\,)\equiv \, (\,(\,{\lambda }\,u(\,{\lambda }x,\,{\lambda }^2 t),\,{\lambda }^2 p(\,{\lambda }x,\,{\lambda }^2 t)\,) \end{aligned}$$

is a solution in the same domain. In particular

$$\begin{aligned} {\omega }_{\lambda }(x,\,t)\equiv \,\text {curl}\,\,u_{\lambda }(x,\,t)=\,{\lambda }^2\,{\omega }(\,{\lambda }x,\,{\lambda }^2 t). \end{aligned}$$

Set

$$\begin{aligned} \theta _{\lambda }(x,y,t)\overset{def}{=}\angle (\omega _{\lambda }(x,t), \omega _{\lambda }(y,t)). \end{aligned}$$

Then, by appealing to

$$\begin{aligned} \sin \theta (x,y,t)= \frac{\,|\,\omega (x,t) \times \, \omega (y,t)\,|\,}{\,|\omega (x,t)|\,|\omega (y,t)|\,}\,, \end{aligned}$$

it follows that

$$\begin{aligned} \sin \theta _{\lambda }(x,y,t)=\,\sin \theta ({\lambda }\,x,{\lambda }\,y,{\lambda }^2\,t). \end{aligned}$$

Assume now that the solution $u(x,t)$ satisfies

$$\begin{aligned} \sin \theta (x,y,t)\le g(t,x)|x-y|^\beta \,, \end{aligned}$$

for some $\,\beta \in \,[\,\frac{1}{2},\,1\,]$, where $g\in L^a(0,\,+\infty \,;L^b(R^3))\,,$ and the exponents are defined by Eq. (4). It follows that

$$\begin{aligned} \sin \theta _{\lambda }(x,y,t)\le g_{\lambda }(x,\,t)|x-y|^\beta \,, \end{aligned}$$

where the function $\,g_{\lambda }\,$ is given by

$$\begin{aligned} g_{\lambda }(x,\,t)\overset{def}{=} {\lambda }^{\beta } \,g(\,{\lambda }x,\,{\lambda }^2 t). \end{aligned}$$

It follows that

$$\begin{aligned} \Vert \,g_{\lambda }\,\Vert _{L^a(\,0,\,+\infty ;\,L^b(R^3)\,)} =\,{\lambda }^\frac{1}{2} \,\Vert \,g\,\Vert _{L^a(\,0,\,+\infty ;\,L^b(R^3)\,)}, \end{aligned}$$

for all $\,\beta \in \,[\,\frac{1}{2},\,1\,]$. The equivalence of the “strength” of the different sufficient conditions for regularity follows from the independence of the exponent $\frac{1}{2}\,$ with respect to $\,\beta \,$. The reader may verify that weaker (resp. stronger) sufficient conditions for regularity lead to larger (resp. smaller) exponents.

Finally we show that the above common strength is at the same level as a classical “Prodi-Serrin” integrability conditions for regularity. In fact, for $\,\beta =\,0\,$, condition (7) is superfluous, since it holds automatically. Furthermore, condition (8) simply reads $\,\omega \in L^2(0,T; L^3(\Omega ))\,$. This means $\,u\in L^2(0,T; H^{1,\,3}(\Omega ))\,$, which is a class of regularity, see [1]. This class is formally equivalent, in an obvious sense, to the classical “Prodi-Serrin” condition $\,u\in L^2(0,T;L^{\infty }(\Omega ))\,$.

The above argument lead us to call all the above family of $\,\beta -$dependent results, sharp results. Note that, in Theorem 1.2, the weak regularity allowed by (4) to the coefficients $\,g(t,\,x)\,$ is fundamental to obtain sharp results. This is the reason why proving the “minimal regularity” for the coefficients $\,g(t,\,x)\,,$ is taken here into considerable attention. A similar remark applies in relation to (9) and Theorem 1.3.