Rapid Time-Decay Phenomenon of the Incompressible Navier—Stokes Flow in Exterior Domains

Abstract

This paper focuses on the rapid time-decay phenomenon of the 3D incompressible Navier—Stokes flow in exterior domains. By using the representation of the flow in exterior domains, together with the estimates of the Gaussian kernel, the tensor kernel, and the Stokes semigroup, we prove that under the assumption $\int_0^\infty {\int_{\partial \Omega } {T\left[ {u,p} \right]\left( {y,t} \right) \cdot \nu d{S_y}dt = 0} }$ for the body pressure tensor T[u,p], if ${u_0} \in {L^1}\left( \Omega \right) \cap L_\sigma ^3\left( \Omega \right) \cap {W^{{2 \over 5},{5 \over 4}}}\left( \Omega \right)$ with ∥u₀∥ ≤ η for some sufficiently small number η > 0, then rapid time-decay phenomenon of the Navier—Stokes flow appears. If additionally ∣x∣^αu₀ ∈ L^r0 (Ω) for some 0 < α < 1 and $1 < {r_0} < {\left({1 - {\alpha \over 3}} \right)^{ - 1}}$ or α = 1 and r₀ = 1, then the flow exhibits higher decay rates as t → ∞.