TY - JOUR

T1 - Characterization of initial data in the homogeneous 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 research of H.K. was partially supported by JSPS Grant-in-Aid for Scientific Research (S) Grant Number JP16H06339.The research of S.S. was partially supported by JSPS Grant-in-Aid for Scientific Research (B) Grant Number JP16H03945.

PY - 2020/3/15

Y1 - 2020/3/15

N2 - Consider the Cauchy problem of the Navier-Stokes equations in Rn with initial data a in the homogeneous Besov space [Formula presented] for ntΔa can be controlled in Lα,q(0,∞;B˙r,1 0(Rn)) for [Formula presented] with p≦r<∞, where Lα,q denotes the Lorentz space. As an application, the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than Serrin's. Conversely, if the global solution belongs to the usual Serrin class Lα,q(0,∞;Lr(Rn)) for r and α as above with 1r,q −1+nr(Rn). Moreover, we prove that such solutions are analytic in the space variables. Our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in Lp(Rn) with Hölder continuity in time.

AB - Consider the Cauchy problem of the Navier-Stokes equations in Rn with initial data a in the homogeneous Besov space [Formula presented] for ntΔa can be controlled in Lα,q(0,∞;B˙r,1 0(Rn)) for [Formula presented] with p≦r<∞, where Lα,q denotes the Lorentz space. As an application, the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than Serrin's. Conversely, if the global solution belongs to the usual Serrin class Lα,q(0,∞;Lr(Rn)) for r and α as above with 1r,q −1+nr(Rn). Moreover, we prove that such solutions are analytic in the space variables. Our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in Lp(Rn) with Hölder continuity in time.

KW - Analyticity

KW - Homogeneous Besov space

KW - Navier-Stokes equations

KW - Serrin class

UR - http://www.scopus.com/inward/record.url?scp=85075988838&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075988838&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2019.108390

DO - 10.1016/j.jfa.2019.108390

M3 - Article

AN - SCOPUS:85075988838

VL - 278

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 5

M1 - 108390

ER -