TY  - JOUR
A1  - Shlapunov, Alexander
A1  - Tarchanov, Nikolaj Nikolaevič
T1  - An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n
JF  - Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija
N2  - We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.
KW  - Navier-Stokes equations
KW  - de Rham complex
KW  - open mapping theorem
Y1  - 2021
U6  - https://doi.org/10.33048/semi.2021.18.108
SN  - 1813-3304
VL  - 18
IS  - 2
SP  - 1433
EP  - 1466
PB  - Institut Matematiki Imeni S. L. Soboleva
CY  - Novosibirsk
ER  - 
TY  - INPR
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - An open mapping theorem for the Navier-Stokes equations
N2  - We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016)10 
KW  - Navier-Stokes equations
KW  - weighted Hölder spaces
KW  - integral representation method
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-98687
SN  - 2193-6943
VL  - 5
IS  - 10
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Mera, Azal
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Navier-Stokes equations for elliptic complexes
N2  - We continue our study of invariant forms of the classical equations of mathematical physics,
such as the Maxwell equations or the Lamé 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)12 
KW  - Navier-Stokes equations
KW  - classical solution
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-85592
SN  - 2193-6943
VL  - 4
IS  - 12
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  -