Asymptotic profiles of nonstationary incompressible Navier–Stokes flows in the half-space

On L 1 -summability and asymptotic profiles for smooth solutions to Navier?Stokes equations in a 3D exterior domain

“…In the case of the Cauchy problem, [8] and [23] proved that the weak and strong solutions admit various types of asymptotic expansions, in terms of the space-time derivatives of Gaussianlike functions, provided that the initial data satisfy appropriate moment conditions. Similar results are given in [9] and [10] for solutions in the half-space. In this paper we first derive asymptotic expansions for u and ∇p, both of which contain a term that is not in L 1 .…”

“…As a corollary, we can prove the existence of a lower bound of rates of time-decay of the L 1 -solutions, as is done in the case of the Cauchy problem ( [8]) and the problem in the half-space ( [9]). The paper is organized as follows: In section 2 we introduce necessary notation and then state the main results.…”

“…Since strong solutions are required to be in BC([0, ∞) : L 3 ( )), it follows that the solutions treated in Theorem 3 belong to BC([0, ∞) : L q ( )) for all 1 ≤ q ≤ 3. For solutions satisfying (2.6), we then deduce the space-time asymptotic profiles, which are analogous to those obtained in [8], [23] for the Cauchy problem and in [9], [10] for the problem in the half-space. Theorem 4.…”

Existence of Strong Mild Solution of the Navier-Stokes Equations in the Half Space With Nondecaying Initial Data

“…Stokes solutions in R n + have also been derived in [17]. This solution formula have been used in the L q framework, mainly for 1 < q < ∞ (see [4,6], see also [2,3,8,15] for the L 1 or L ∞ estimates of the Stokes flow or its gradient). The solution formula in [16] has been used mainly for L ∞ framework (see [5,13,16]).…”

Moment estimates for weak solutions to the Navier–Stokes equations in half‐space

“…It should be noted that, in our results, the solution u and initial velocity u 0 belong to the same weighted space. (2) Fujigaki and Miyakawa [13] showed that the weak solution u can decay as …”