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## Archive of Issues

Germany; Russia Krasnoyarsk; Potsdam
Year
2022
Volume
32
Issue
2
Pages
278-297
 Section Mathematics Title Inverse image of precompact sets and regular solutions to the Navier–Stokes equations Author(-s) Shlapunov A.A.a, Tarkhanov N.N.b Affiliations Siberian Federal Universitya, University of Potsdamb 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 уравнения Навье–Стокса, регулярные решения UDC 517 MSC 76N10, 35Q30, 76D05 DOI 10.35634/vm220208 Received 21 January 2022 Language English Citation Shlapunov A.A., Tarkhanov N.N. Inverse image of precompact sets and regular solutions to the Navier–Stokes equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 2, pp. 278-297. References Adams R.A. Sobolev spaces, Academic Press, 2003. Agranovich M.S. 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