TY  - INPR
A1  - Schulze, Bert-Wolfgang
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Green integrals on manifolds with cracks
N2  - We prove the existence of a limit in Hm(D) of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S ⊂ ∂D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchy problem in D with data on S, for an elliptic operator A of order m ≥ 1, whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of the double layer potential. A similar regularisation is constructed also for a mixed problem in D.
T3  - Preprint - (2000) 12 
Y1  - 2000
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25777
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  -