Symmetry reduction and exact solutions of the Navier-Stokes equations. II

Abstract

6 Symmetry properties and exact solutions of system (3.12)As was mentioned in Sec. 3, ansatzes (3.4)-(3.7) reduce the NSEs (1.1) to the systems of PDEs of a similar structure that have the general form (see (3.12)):

Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations

“…[12], [37] and the review [40]). But, at best of our knowledge, no examples of solutions in R 3 with fast decay of the energy have been constructed so far.…”

On the normalizers of subalgebras in an infinite Lie algebra

“…If the basis of S does not depend on arbitrary functions (all functions in (3) are specified), then the coordinates of operator X and constants b l i will be the only unknown quantities in Eq. (4). Note that these equations may be both algebraic and differential.…”

Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations

“…Classification of infinite-dimensional subalgebras of this algebra was studied in [6]. 3 Short reviews devoted to invariant solutions of the Navier-Stokes equations can be found in [14][15][16]. 4 Firstly the approach of partially invariant solutions to the NavierStokes equations was applied in [17].…”