TY - JOUR

T1 - Removable time-dependent singularities of solutions to the Stokes equations

AU - Kozono, Hideo

AU - Ushikoshi, Erika

AU - Wakabayashi, Fumitaka

N1 - Funding Information:
The authors would like to express their thanks to the referee for his/her valuable comments.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2023/1/5

Y1 - 2023/1/5

N2 - Let Ω⊂RN and let ξ∈Cα([0,T];Ω) for [Formula presented]. We consider the situation that u=u(x,t) is a classical solution of the Stokes equations in ⋃0(Ω∖{ξ(t)})×{t}, that is, {ξ(t)}0 is regarded as the time-dependent singularities of u in Ω×(0,T). If u behaves around ξ(t) like |u(x,t)|=o(|x−ξ(t)|2−N+(1/α−2)) as x→ξ(t) uniformly in t∈(0,T), then {ξ(t)}0 is a family of removable singularities of u, which implies that u can be extended as a smooth solution in the whole space and time Ω×(0,T).

AB - Let Ω⊂RN and let ξ∈Cα([0,T];Ω) for [Formula presented]. We consider the situation that u=u(x,t) is a classical solution of the Stokes equations in ⋃0(Ω∖{ξ(t)})×{t}, that is, {ξ(t)}0 is regarded as the time-dependent singularities of u in Ω×(0,T). If u behaves around ξ(t) like |u(x,t)|=o(|x−ξ(t)|2−N+(1/α−2)) as x→ξ(t) uniformly in t∈(0,T), then {ξ(t)}0 is a family of removable singularities of u, which implies that u can be extended as a smooth solution in the whole space and time Ω×(0,T).

KW - Bogovskii lemma

KW - Moving singularity in time

KW - Removable singularity

KW - Stokes equations

KW - Uniqueness of weak solutions

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U2 - 10.1016/j.jde.2022.10.005

DO - 10.1016/j.jde.2022.10.005

M3 - Article

AN - SCOPUS:85140310546

SN - 0022-0396

VL - 342

SP - 472

EP - 489

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -