TY  - JOUR
A1  - Mera, Azal
A1  - Shlapunov, Alexander A.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Navier-Stokes Equations for Elliptic Complexes
JF  - Journal of Siberian Federal University. Mathematics & Physics
N2  - We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
KW  - Navier-Stokes equations
KW  - classical solution
Y1  - 2019
U6  - https://doi.org/10.17516/1997-1397-2019-12-1-3-27
SN  - 1997-1397
SN  - 2313-6022
VL  - 12
IS  - 1
SP  - 3
EP  - 27
PB  - Sibirskij Federalʹnyj Universitet
CY  - Krasnojarsk
ER  - 
TY  - JOUR
A1  - Shlapunov, Alexander  A.
A1  - Tarchanov, Nikolaj Nikolaevič
T1  - Inverse image of precompact sets and regular solutions to the Navier-Stokes equations
JF  - Vestnik Udmurtskogo Universiteta. Matematika, mechanika, kompʹjuternye nauki
N2  - We consider the initial value problem for the Navier-Stokes equations over R-3 x [0, T] with time T > 0 in the spatially periodic setting. 
We prove that it induces open injective mappings A(s): B-1(s) -> B-2(s-1) where B-1(s), B-2(s-1) are elements from scales of specially constructed function spaces of Bochner-Sobolev typeparametrized with the smoothness index s is an element of N. 
Finally, we prove that a map Asis surjective if and only if the inverse image A(s)(- 1) (K) of any pre compact set K from the range of the map Asis bounded in the Bochner space L-s([0, T], L-r(T-3))with the Ladyzhenskaya-Prodi-Serrin numbers s, r.
KW  - Navier-Stokes equations
KW  - regular solutions
Y1  - 2022
U6  - https://doi.org/10.35634/vm220208
SN  - 1994-9197
SN  - 2076-5959
VL  - 32
IS  - 2
SP  - 278
EP  - 297
PB  - Udmurtskij gosudarstvennyj universitet
CY  - Iževsk
ER  -