TY - JOUR

T1 - Necessary and sufficient condition on initial data in the Besov space for solutions in the Serrin class of the Navier–Stokes equations

AU - Kozono, Hideo

AU - Okada, Akira

AU - Shimizu, Senjo

N1 - Funding Information:
The authors would like to thank the anonymous referee for providing valuable suggestions, especially Remark after Lemma 2.5. The research of H. Kozono was partially supported by JSPS Grant-in-Aid for Scientific Research (S) 16H06339. The research of S. Shimizu was partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03945.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2021/9

Y1 - 2021/9

N2 - 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.

AB - 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.

KW - Inhomogeneous Besov space

KW - Navier–Stokes equations

KW - Serrin class

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U2 - 10.1007/s00028-020-00614-w

DO - 10.1007/s00028-020-00614-w

M3 - Article

AN - SCOPUS:85090221080

SN - 1424-3199

VL - 21

SP - 3015

EP - 3033

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 3

ER -