English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The first goal of our paper is to give a new type of regularity criterion forsolutions $u$ to Navier-Stokes equation in terms of some supercritical functionspace condition $u \in L^{\infty}(L^{\alpha ,*})$ (with$\frac{3}{4}(17^{\frac{1}{2}} -1)< \alpha < 3 $) and some exponential controlon the growth rate of $\dv (\frac{u}{|u|})$ along the streamlines of u. Thisregularity criterion greatly improves the previous one in\cite{smoothnesscriteria}. The proof leading to the regularity criterion of ourpaper basically follows the one in \cite{smoothnesscriteria}. However, we alsopoint out that totally new idea which involves the use of the new supercriticalfunction space condition is necessary for the success of our new regularitycriterion in this paper. The second goal of our paper is to construct adivergence free vector field $u$ within a flow-invariant tubular region withincreasing twisting of streamlines towards one end of a bundle of streamlines.The increasing twisting of streamlines is controlled in such a way that theassociated quantities $\|u\|_{L^{\alpha}}$ and $\|\dv(\frac{u}{|u|})\|_{L^{6}}$blow up while preserving the finite energy property $u\in L^{2}$ at the sametime. The purpose of such a construction is to demonstrate the necessity to gobeyond the scope covered by some previous regularity criteria such as\cite{Vasseur2} or \cite{Esca}.

On possible isolated blow-up phenomena of the 3D-Navier-Stokes equation  and a regularity criterion in terms of supercritical function space condition  and smoothness condition along the streamlines