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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, Volume 32, Issue 2, Pages 278–297
DOI: https://doi.org/10.35634/vm220208
(Mi vuu811)
 

This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

Inverse image of precompact sets and regular solutions to the Navier-Stokes equations

A. A. Shlapunova, N. N. Tarkhanovb

a Siberian Federal University, pr. Svobodnyi, 79, Krasnoyarsk, 660041, Russia
b Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Str., 24/25, Potsdam (Golm), 14476, Germany
Full-text PDF (324 kB) Citations (1)
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Abstract: We consider the initial value problem for the Navier–Stokes equations over ${\mathbb R}^3 \times [0,T]$ with time $T>0$ in the spatially periodic setting. We prove that it induces open injective mappings ${\mathcal A}_s\colon B^{s}_1 \to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces of Bochner–Sobolev type parametrized with the smoothness index $s \in \mathbb N$. Finally, we prove that a map ${\mathcal A}_s$ is surjective if and only if the inverse image ${\mathcal A}_s ^{-1}(K)$ of any precompact set $K$ from the range of the map ${\mathcal A}_s$ is bounded in the Bochner space $L^{\mathfrak s} ([0,T], L^{{\mathfrak r}} ({\mathbb T}^3))$ with the Ladyzhenskaya–Prodi–Serrin numbers ${\mathfrak s}$, ${\mathfrak r}$.
Keywords: Navier–Stokes equations, regular solutions.
Funding agency Grant number
Foundation for the Development of Theoretical Physics and Mathematics BASIS
The first author was supported by a grant of the Foundation for the advancement of theoretical physics and mathematics “BASIS”.
Received: 21.01.2022
Accepted: 05.05.2022
Bibliographic databases:
Document Type: Article
UDC: 517
MSC: 76N10, 35Q30, 76D05
Language: English
Citation: A. A. Shlapunov, N. N. Tarkhanov, “Inverse image of precompact sets and regular solutions to the Navier-Stokes equations”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 32:2 (2022), 278–297
Citation in format AMSBIB
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\by A.~A.~Shlapunov, N.~N.~Tarkhanov
\paper Inverse image of precompact sets and regular solutions to the Navier-Stokes equations
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2022
\vol 32
\issue 2
\pages 278--297
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\crossref{https://doi.org/10.35634/vm220208}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4456920}
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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