Section
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Mathematics
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Title
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Inverse image of precompact sets and regular solutions to the Navier–Stokes equations
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Author(-s)
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Shlapunov A.A.a,
Tarkhanov N.N.b
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Affiliations
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Siberian Federal Universitya,
University of Potsdamb
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Abstract
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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 ${\mathfrak s}$, ${\mathfrak r}$.
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Keywords
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уравнения Навье–Стокса, регулярные решения
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UDC
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517
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MSC
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76N10, 35Q30, 76D05
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DOI
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10.35634/vm220208
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Received
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21 January 2022
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Language
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English
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Citation
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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.
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