Abstract
We consider the nonstationary Stokes and Navier-Stokes equations in an aperture domain Ω ⊂ R n , n ≥ 2 . For this purpose, we prove L p - L q type estimate of the Stokes semigroup in the aperture domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin. We apply them to the Navier-Stokes initial value problem in the aperture domain. As a result, we can prove the global existence of a unique solution to the Navier-Stokes problem with the vanishing flux condition and some decay properties as t → ∞ , when the initial velocity is sufficiently small in the L n space. Moreover we can prove the time-local existence of a unique solution to the Navier-Stokes problem with the non-trivial flux condition.
Highlights
An aperture domain Ω ⊂ Rn (n ≥ 2) is an unbounded domain with noncompact boundary ∂Ω
When Ω is a perturbed half-space, as was mentioned, Kubo and Shibata [30] proved (1.1) for 1 ≤ p ≤ q ≤ ∞ (p = ∞, q = 1) and (1.2) for 1 ≤ p ≤ q < ∞(q = 1) when n ≥ 2. It is well-known that we can prove the global existence of the solution to the NavierStokes problem with small Ln data as an application of the Lp-Lq estimate of the Stokes semigroup
After notation is fixed, we present the statement of our main results: Theorem 2.1 on the local energy decay estimates of the Stokes semigroup, Theorem 2.2 on the Lp-Lq estimates of the Stokes semigroup, Theorem 2.3 on the global existence and decay properties of the Navier-Stokes flow with φ(u) ≡ 0, Theorem 2.4 on some further asymptotic behaviors of the obtained flow under additional summability assumption on the initial data and Theorem 2.6 on
Summary
An aperture domain Ω ⊂ Rn (n ≥ 2) is an unbounded domain with noncompact boundary ∂Ω. When Ω is a perturbed half-space, as was mentioned, Kubo and Shibata [30] proved (1.1) for 1 ≤ p ≤ q ≤ ∞ (p = ∞, q = 1) and (1.2) for 1 ≤ p ≤ q < ∞(q = 1) when n ≥ 2 It is well-known that we can prove the global existence of the solution to the NavierStokes problem with small Ln data as an application of the Lp-Lq estimate of the Stokes semigroup. The time-global existence was proved by many authors in the following domain cases: Giga and Miyakawa [21] for bounded domains, Kato [25] for the whole space, Ukai [39] and Kozono [26] for the half-space, Iwashita [24] and Wiegner [40] for the exterior domain, Abe and Shibata [1] for the infinite layer, Kubo and Shibata [30] for the perturbed half-space and Hishida [22] for the aperture domain. In a similar way to Kato [25] with the aid of an auxiliary function (flux carrier) of Heywood [23], we prove that the time-local existence of the unique strong solution to (NS) when the flux is non-trivial
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
