TY - JOUR

T1 - Asymptotic Properties of Steady and Nonsteady Solutions to the 2D Navier-Stokes Equations with Finite Generalized Dirichlet Integral

AU - Kozono, Hideo

AU - Terasawa, Yutaka

AU - Wakasugi, Yuta

N1 - Funding Information:
Acknowledgements. The authors would like to thank Professor Yasunori Maekawa for helpful discussions. This work was supported by a JSPS Grant-in-Aid for Scientific Research(S) (grant no. JP16H06339).
Publisher Copyright:
© 2022 Department of Mathematics, Indiana University. All rights reserved.

PY - 2022

Y1 - 2022

N2 - We consider the stationary and non-stationary Navier-Stokes equations in the whole plane R2 and in the exterior domain outside of the large circle. The solution v is handled in the class with ∇v ∈ Lq for q ≥ 2. Since we deal with the case q ≥ 2, our class is larger in the sense of spatial decay at infinity than that of the finite Dirichlet integral, that is, for q = 2 where a number of results such as asymptotic behavior of solutions have been observed. For the stationary problem we shall show that ω(x) = o(|x|−(1/q+1/q2)) and ∇v(x) = o(|x|−(1/q+1/q2)log|x|) as |x| → ∞, where ω ≡ rotv. As an application, we prove the Liouville-type theorem under the assumption that ω ∈ Lq(R2). For the non-stationary problem, a generalized Lq-energy identity is clarified. We also apply it to the uniqueness of the Cauchy problem and the Liouville-type theorem for ancient solutions under the assumption that ω ∈ Lq(R2 × I).

AB - We consider the stationary and non-stationary Navier-Stokes equations in the whole plane R2 and in the exterior domain outside of the large circle. The solution v is handled in the class with ∇v ∈ Lq for q ≥ 2. Since we deal with the case q ≥ 2, our class is larger in the sense of spatial decay at infinity than that of the finite Dirichlet integral, that is, for q = 2 where a number of results such as asymptotic behavior of solutions have been observed. For the stationary problem we shall show that ω(x) = o(|x|−(1/q+1/q2)) and ∇v(x) = o(|x|−(1/q+1/q2)log|x|) as |x| → ∞, where ω ≡ rotv. As an application, we prove the Liouville-type theorem under the assumption that ω ∈ Lq(R2). For the non-stationary problem, a generalized Lq-energy identity is clarified. We also apply it to the uniqueness of the Cauchy problem and the Liouville-type theorem for ancient solutions under the assumption that ω ∈ Lq(R2 × I).

KW - Liouville-type theorems

KW - Navier-Stokes equations

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U2 - 10.1512/iumj.2022.71.8978

DO - 10.1512/iumj.2022.71.8978

M3 - Article

AN - SCOPUS:85136050645

SN - 0022-2518

VL - 71

SP - 1299

EP - 1316

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 5

ER -