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Open AccessArticle

The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions

Institute Physics, Mathematics and Computer Sciences, Baltic Federal University, Kaliningrad 236016, Russia
Axioms 2019, 8(2), 41; https://doi.org/10.3390/axioms8020041
Received: 31 January 2019 / Revised: 7 March 2019 / Accepted: 1 April 2019 / Published: 7 April 2019
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
In this article, I consider local solutions of the 3D Navier–Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters that are invariant with respect to the scaling procedure. Since in spaces of even dimensions the scaling procedure is a conformal mapping on the Heisenberg group, then an application of invariant parameters can be considered as the application of conformal invariants. It gives the possibility to prove the sufficient and necessary conditions for existence of a global regular solution. This is the main result and one among some new statements. With some compliments, the rest improves well-known classical results. View Full-Text
Keywords: Navier–Stokes equations; global solutions; regular solutions; a priori estimates; weak solutions; kinetic energy; dissipation Navier–Stokes equations; global solutions; regular solutions; a priori estimates; weak solutions; kinetic energy; dissipation
MDPI and ACS Style

Semenov, V.I. The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions. Axioms 2019, 8, 41.

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