L 2 decay for the Navier-Stokes flow in halfspaces

Abstract

We use the following notations: U = L'(Dn), 1 < r < oo, denotes the usual Lebesgue space of scalar, as well as vector, functions defined on the halfspace D n. The norm of

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

“…$||e^{-tA}a||_{2}\leq C(1+t)^{-\frac{n+2}{4}}$ , so the existence of a weak solution with decay property (3.14) is deduced in exactly the same way as in [1], The assumption implies $a\in L_{\sigma}^{q}$ for alll $<q\leq 2$ ;so Proposition 3.2 and (3.13) together imply $||e^{-tA}a||_{q}\leq C(1+t)^{-\frac{1}{2}-\frac{n}{2}(1-\frac{1}{q})}$ for all $1<q\leq 2$ . (3.15) By using this, we can deduce assertion (3.15) in astandard manner.…”

“…Consider the Helmholtz decomposition ( We know (see [1]) that $-A_{r}$ generates abounded analytic semigroup $\{e^{-tA}\}_{t\geq 0}$ in $L_{\sigma}^{r}$ so that for each $a\in L_{\sigma^{\backslash }}^{r}$ the filnction $v(t)=(\uparrow'"\iota^{n}')=e^{-tA}a$ gives aunique so lution of (S) in $L_{\sigma}^{r}$ .…”

“…We first deal with the weak solutions, which are known (see [1], [10]) to exist globally in time for all $a\in L_{\sigma}^{2}$ , satisfying the identity : $\langle u(t), \varphi\rangle=\langle e^{-tA}a, \varphi\rangle+\int_{0}^{t}\langle u\otimes u, \nabla e^{-(t-s)A}\varphi\rangle ds$ given in (i) is constructed via approximate solutions $\{u_{N}\}$ as given in [1], [7], [13], which satisfy $\lim_{Narrow\infty}\int_{0}^{\infty}||u_{N}(t)-u(t)||_{2}^{2}dt=0$ .…”

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

“…For the case d = ∞ the existence of the Helmholtz projection in (i) for instance is proved in [40]; assertion (ii) for the case of d = ∞ is obtained in [35]. We also refer to [8] Now we are in position to establish the Helmholtz decomposition of the space FM 0 (R n−1 , X p ). To be precise, we will show that…”

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]).…”