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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
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.
Received: 21.01.2022 Accepted: 05.05.2022
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
Linking options:
https://www.mathnet.ru/eng/vuu811 https://www.mathnet.ru/eng/vuu/v32/i2/p278
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