# Necessary and sufficient condition on initial data in the Besov space for solutions in the Serrin class of the Navier–Stokes equations

Hideo Kozono, Akira Okada, Senjo Shimizu*

*この研究の対応する著者

## 抄録

The Cauchy problem of the Navier–Stokes equations in Rn with the initial data a in the Besov space Bp,q-1+np(Rn) for n< p< ∞ and 1 ≤ q≤ ∞ is considered. We construct the local solution in Lα,q(0,T;Br,10(Rn)) for p≤ r< ∞ satisfying 2α+nr=1 with the initial data a∈Bp,q-1+np(Rn), where Lα,q denotes the Lorentz space. Conversely, if the solution belongs to Lα,q(0 , T; Lr(Rn)) with 2α+nr=1, then the initial data a necessarily belong to Br,q-1+nr(Rn). It implies that the initial data in the Besov space Bp,q-1+np(Rn) are a necessary and sufficient condition for the existence of solutions in the Serrin class.

本文言語 English 3015-3033 19 Journal of Evolution Equations 21 3 https://doi.org/10.1007/s00028-020-00614-w Published - 2021 9月

• 数学（その他）

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