Introduction and results

Consider the nonstationary Navier-Stokes system in ${\mbox{\mb R}}^n$, $n\geq 2$, in the form of the integral equation:

$$u(t)=e^{-tA}a-\int_0^te^{-(t-s)A}P\nabla\cdot(u\otimes u)(s)ds. \leqno {\rm (IE)}$$

Here $\{e^{-tA}\}_{t\geq 0}$ is the heat semigroup, $P$ is the bounded projetion from the space ${\mbox{\boldmath$L$}}^q$, $1<q<\infty$, of vector fields to the subspace of ${\mbox{\boldmath$L$}}^q$ consisting of all divergence-free vector fields. As is shown in [4], the operator $e^{-tA}P\nabla\cdot$ has the kernel function $F=(F_{\ell,jk})_{j,k,\ell=1}^n$ with

$$F_{\ell,jk}(x,t)=\partial _{\ell}E_t(x)\delta_{jk} +\int_0^{... ...{\ell}E_{s+t}(x)ds, \qquad \partial _j=\partial /\partial x_j,$$

where $x=(x_1,\ldots,x_n)\in{\mbox{\mb R}}^n$, $t>0$, and $E_t(x)=(4\pi t)^{-\frac{n}{2}}e^{-\vert x\vert^2/4t}$.

Inspired by Takahashi [9], we proved in [4] the following

Theorem 1. Fix $1\leq\gamma\leq n+1$ and let $a$ be divergence-free, satisfying

$$\vert(e^{-tA}a)(x)\vert\leq c(1+\vert x\vert)^{-\gamma},\qqu... ...e^{-tA}a)(x)\vert\leq c(1+t)^{-\frac{\gamma}{2}}. \leqno (1.1)$$

If $c>0$ is small, then (IE) possesses a solution $u$ such that, with another $c>0$,

$$\vert u(x,t)\vert\leq c(1+\vert x\vert)^{-\gamma},\qquad \vert u(x,t)\vert\leq c(1+t)^{-\frac{\gamma}{2}}.$$

However, in proving Theorem 1, the Hardy space theory was used to find relevant decay estimates in $t$. In this paper we show that Theorem 1 can be deduced by a more elementary method, i.e., without using the Hardy space theory.

In [4] we also gave a class of initial data $a$ for which $e^{-tA}a$ satisfies (1.1). But, the assumption on $a$ imposed in [4] was too complicated and restrictive. So, we here give a simpler version of the assumption on $a$ that ensures the validity of (1.1). More specifically, we shall show the following

Theorem 2. Let $a$ be divergence-free and satisfy

$$c_0=\sup_y(1+\vert y\vert)^{\gamma}\vert a(y)\vert<\infty\qquad\mbox{for some $0<\gamma\leq n+1$}.$$

Suppose further that

$$c_1=\int\vert a(y)\vert dy<\infty\quad\mbox{when $\gamma=n$}... ... y\vert\vert a(y)\vert dy<\infty\quad\mbox{when $\gamma=n+1$}.$$

Then

$$\vert(e^{-tA}a)(x)\vert\leq cd(1+\vert x\vert)^{-\gamma},\qq... ...^{-tA}a)(x)\vert\leq cd(1+t)^{-\frac{\gamma}{2}}. \leqno (1.2)$$

Here, $d=c_0+c_1$ when $\gamma=n$ or $\gamma=n+1$; and $d=c_0$ otherwise.

Starting from Theorems 1 and 2, we deduced in [3] a space-time asymptotic expansion of solutions $u$ in the case $\gamma=n+1$, which was then applied in [8] to find a lower bound estimate of rates of energy decay for weak solutions of (IE).

Finally, we supplement the recent result of Brandolese [2] on the existence of solutions which decay more rapidly than those treated in Theorem 1. Consider the divergence-free vector fields $a$ such that

$(a)$ $a_j$ is odd in $x_j$ and is even in each of the other variables.

$(b)$ $a$ is cyclically symmetric in the sense that

$$a_1(x_1,\ldots,x_n)=a_2(x_n,x_1,\ldots,x_{n-1})=\cdots=a_n(x_2,\ldots,x_n,x_1).$$

Brandolese [2] shows the existence of a solution $u$ satisfying $(a)$ and $(b)$ above, with the estimates

$$\vert u(x,t)\vert\leq c(1+\vert x\vert)^{-n-2},\qquad \vert u(x,t)\vert\leq c(1+t)^{-\frac{n+2}{2}}. \leqno (1.3)$$

Observe that the solutions above decay more rapidly than those treated in Theorem 1. But, estimate (1.3) seems not optimal. Indeed, we shall prove

Theorem 3. Suppose $a$ satisfies $(a)$ and $(b)$, and

$$c_0=\sup(1+\vert y\vert)^{n+3}\vert a(y)\vert<\infty,\qquad c_1=\int\vert y\vert^3\vert a(y)\vert dy<\infty. \leqno (1.4)$$

(i) The function $x\mapsto e^{-tA}a(x)$ satisfies $(a)$ and $(b)$, and there hold the estimates

$$\vert(e^{-tA}a)(x)\vert\leq c(c_0+c_1)(1+\vert x\vert)^{-n-3... ... \vert(e^{-tA}a)(x)\vert\leq c(c_0+c_1)(1+t)^{-\frac{n+3}{2}}.$$

(ii) If $c_0+c_1$ is small, then (IE) has a strong solution $u$ satisfying $(a)$, $(b)$, and

$$\vert u(x,t)\vert\leq c(1+\vert x\vert)^{-n-3},\qquad\vert u(x,t)\vert\leq c(1+t)^{-\frac{n+3}{2}}. \leqno (1.5)$$

Brandolese [2] proves the existence of solutions which satisfy (1.3), assuming

$$c'_0=\sup(1+\vert y\vert)^{n+2}\vert a(y)\vert<\infty,\qquad c'_1=\int\vert y\vert^2\vert a(y)\vert dy<\infty. \leqno (1.6)$$

This is the reason why (1.3) is deduced in [2] instead of (1.5). However, the result of [3] implies that the corresponding solution $u$ has to satisfy

$$\Vert u(t)\Vert _{\infty}=o(t^{-\frac{n+2}{2}})\qquad\mbox{as $t\to\infty$}$$

whenever $a$ satisfies $(a)$, $(b)$ and (1.6). Thus, (1.3) is not optimal even under the assumption (1.6). On the other hand, in view of the result given in [7], estimate (1.5) seems to be optimal for general solutions satisfying $(a)$ and $(b)$. Namely, one can reasonably expect the existence of a solution $u$ satisfying $(a)$ and $(b)$ such that

$$\Vert u(t)\Vert _{\infty}\geq ct^{-\frac{n+3}{2}}\qquad \mbox{for large\ \ $t>0$},$$

even when $a$ is in $\mathcal{S}$ and satisfies $(a)$, $(b)$ and $\int y^{\alpha}a(y)dy=0$ for every multi-index $\alpha$.