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

This paper introduces an infinite linear hierarchy for the homogeneous,incompressible three-dimensional Navier-Stokes equation. The Cauchy problem ofthe hierarchy with a factorized divergence-free initial datum is shown to beequivalent to that of the incompressible Navier-Stokes equation in$\mathcal{H}^1.$ This allows us to present an explicit formula for solutions tothe incompressible Navier-Stokes equation under consideration. The obtainedformula is an expansion in terms of binary trees encoding the collisionhistories of the "particles" in a concise form. Precisely, each term in thesummation of $n$ "particles" collision is expressed by a $n$-parameter singularintegral operator with an explicit kernel in Fourier space, describing a kindof processes of two-body interaction of $n$ "particles". Therefore, thisformula is a physical expression for the solutions of the incompressibleNavier-Stokes equation.

An infinite linear hierarchy for the incompressible Navier-Stokes equation and application