Asymptotic stability of stationary Navier–Stokes flow in Besov spaces

Abstract

We discuss the asymptotic stability of stationary solutions to the incompressible Navier–Stokes equations on the whole space in Besov spaces. A critical estimate for the semigroup generated by the Laplacian with a perturbation is the main ingredient of the argument.

Keywords

Asymptotic stability incompressible Navier–Stokes equations Besov spaces

1. Introduction

In this article, we consider the asymptotic stability for stationary solutions of the incompressible Navier–Stokes equations: $\begin{array}{l} (N.S.) & \{\begin{matrix} \partial_{t} u - Δ u + (u \cdot \nabla) u + \nabla π = f, \\ div u = 0, \\ u (0) = a, \end{matrix} \end{array}$ in the whole space $R^{n}$ with $n ⩾ 3$ . Here, $a = a (x)$ and $f = f (x)$ are given initial data and external force, and u and π stand for the velocity and the pressure of the fluid, respectively. In this paper, since f is independent of time, we introduce the associate stationary solution U of the Navier–Stokes equations; $\begin{array}{l} (S) & \{\begin{matrix} - Δ U + (U \cdot \nabla) U + \nabla Π = f, \\ div U = 0 . \end{matrix} \end{array}$ The purpose of this paper is to investigate the asymptotic behavior of the non-stationary solution u for an initial data a close to a stationary solution U. More precisely, we prove that for such a u, $u (t) \to U$ as $t \to \infty$ in Besov spaces, provided that f and $a - U$ are sufficiently small. If u and U are solutions to (N.S.) and (S), respectively, $w : = u - U$ and $b : = a - U$ satisfy $\begin{array}{l} (E) & \{\begin{matrix} \partial_{t} w - Δ w + (w \cdot \nabla) U + (U \cdot \nabla) w + (w \cdot \nabla) w + \nabla ψ = 0, \\ div w = 0, \\ w (0) = b, \end{matrix} \end{array}$ with $ψ : = π - Π$ . Then the stability of U is reduced to the time decay problem $w (t) \to 0$ as $t \to \infty$ . For this purpose, following the idea from Kozono and Yamazaki [14], we introduce the semigroup generated by a perturbed Laplacian in the homogeneous Besov spaces. Using estimates established in this paper, we construct the solution w to the integral equation of (E) with the desired decay properties.

The stationary solutions in unbounded domains were first treated in [15]. We refer to [4] and [3] for further references.

In the whole space $R^{n}$ , there is an abundance of literature dedicated to the asymptotic stability problem for the incompressible Navier–Stokes equations. Gallagher, Iftimie and Planchon [9] proved the asymptotic stability of the trivial stationary solution $U \equiv 0$ in the Besov spaces ${\dot{B}}_{p, q}^{s (p)}$ with $q < \infty$ , where $s (p) : = - 1 + n / p$ , which is a scale invariant space under the scaling $u_{λ} (t, x) : = λ u (λ^{2} t, λ x)$ and $p_{λ} (t, x) : = λ^{2} p (λ^{2} t, λ x)$ . Auscher, Dubois and Tchamitchian [1] showed a similar result in ${BMO}^{- 1}$ for initial data from ${VMO}^{- 1}$ , which is the closure of $C_{0}^{\infty}$ in ${BMO}^{- 1}$ . For the non-trivial case $U ≢ 0$ , Kozono and Yamazaki [14] showed the stability of stationary solution belonging to Morrey spaces in the topology of Besov–Morrey spaces. Phan and Phuc [18] considered the stability problem in function spaces based on capacities. Bjorland, Brandolese, Iftimie and Schonbek [3] proved similar results in weak $L^{p}$ spaces, but with low integrability $p > n / 2$ in three dimensional case. With the use of Sobolev imbeddings, one can see that [14] gives the stability result in weak $L^{p}$ spaces with $p > n$ . In [3], they imposed a low integrability condition on the initial data and external force to address such small p.

The goal of this paper is to study the asymptotic stability in Besov spaces ${\dot{B}}_{p, \infty}^{s (p) - τ_{L}}$ for some $τ_{L} > 0$ and $p \in (n / 2, n)$ , with the restriction $s (p) - τ_{L} > 0$ coming from the product estimate in Lemma 2.2 below. The stability result, Theorem 1.2(ii) for Besov spaces also implies stability results for $L^{p, \infty}$ with some $p < n$ , see Remark 1.2. Thus, our estimate (3) can be regarded as a generalization of one in [3]. Our idea is to assume better behavior of low frequencies on the data.

Our result concerning with the existence of the steady state solution reads as follows.

Theorem 1.1.
Let $n ⩾ 3, n / 2 < p < n$ and $s (p) : = - 1 + n / p$ .

(i): If f is sufficiently small in ${\dot{B}}_{p, \infty}^{s (p) - 2}$ , then there uniquely exists $U \in {\dot{B}}_{p, \infty}^{s (p)}$ solving ( S ) in the sense that $\begin{array}{l} (1) & U = {(- Δ)}^{- 1} P f - {(- Δ)}^{- 1} P \nabla (U \otimes U) in {\dot{B}}_{p, \infty}^{s (p)}, \end{array}$ which satisfies $‖ U ‖_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}}$ .

(ii): Additionally, if we assume that $f \in {\dot{B}}_{p, \infty}^{s - 2}$ with $s \in (0, 1)$ and $‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p)}}$ is small, the solution above also satisfies $‖ U ‖_{{\dot{B}}_{p, \infty}^{s}} ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s - 2}}$ .
Remark 1.1.

Because all terms in (1) belong to ${\dot{B}}_{p, \infty}^{s (p)}$ , we can see that U solves the differential equation $\begin{array}{l} - Δ U + P \nabla (U \otimes U) = P f in {\dot{B}}_{p, \infty}^{s (p) - 2} \end{array}$ and all terms in this equation belong to ${\dot{B}}_{p, \infty}^{s (p) - 2}$ . For the definitions of the projection $P$ and ${(- Δ)}^{α / 2}$ on homogeneous Besov spaces, see Section 2.2.

After we finished writing this paper, we knew the same result was given by Kaneko, Kozono and Shimizu [10]. For the case, Bjorland, Brandolese, Iftimie and Schonbek [3] gave a negative result in $L^{p}$ spaces. Recently, Tsurumi [22] disproved the well-posedness of the stationary problem in the case $p ⩾ n$ .

Now we are in a position to state the stability of the stationary solutions constructed in Theorem 1.1 in ${\dot{B}}_{p, \infty}^{s (p) \pm τ}$ with some $τ > 0$ .
Theorem 1.2.
Let $n ⩾ 3$ and $n / 2 < p < n$ , and suppose that $a \in L^{n, \infty}$ satisfies $div a = 0$ in $S^{'}$ .

(i): Let $τ_{H} \in (0, 2 - n / p)$ . If $f \in {\dot{B}}_{p, \infty}^{s (p) - 2}$ is sufficiently small and $a - U$ is sufficiently small in ${\dot{B}}_{p, \infty}^{s (p)}$ , where $U \in {\dot{B}}_{p, \infty}^{s (p)}$ is the corresponding stationary solution with f constructed in (i) of Theorem 1.1 , then there uniquely exists $u \in B C ((0, \infty); {\dot{B}}_{p, \infty}^{s (p)})$ so that $u - U \in B C ((0, \infty); {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ and solves the differential equation $\begin{array}{l} (2) & \partial_{t} u (t) - Δ u (t) + P \nabla (u \otimes u) (t) = P f in {\dot{B}}_{p, \infty}^{s (p) - 2} \end{array}$ for $t > 0$ , with the initial condition $u (0) = a$ in the sense that for $α \in [0, 2]$ $\begin{array}{l} {‖ u (t) - a ‖}_{{\dot{B}}_{p, \infty}^{s (p) - α}} ≲ t^{α / 2} ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}} . \end{array}$ Moreover, it follows that $‖ u (t) - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}}$ and $\begin{array}{l} {‖ u (t) - U ‖}_{{\dot{B}}_{p, 1}^{s (p) + τ_{H}}} ≲ t^{- τ_{H} / 2} ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}} . \end{array}$ Further, $u (t) \to U$ in ${\dot{B}}_{p, \infty}^{s (p)}$ as $t \to \infty$ if and only if $e^{- t A} (a - U) \to 0$ in ${\dot{B}}_{p, \infty}^{s (p)}$ as $t \to \infty$ , see Section 3 for the definition of the semigroup $e^{- t A}$ .

(ii): Let $s \in (0, s (p))$ . If we additionally assume that $f \in {\dot{B}}_{p, \infty}^{s - 2}, a - U \in {\dot{B}}_{p, \infty}^{s}$ , and $‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}} + ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}}$ is sufficiently small, where $U \in {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, \infty}^{s}$ is the corresponding stationary solution with f constructed in (ii) of Theorem 1.1 , then the solution u above also satisfies $u \in B C ((0, \infty); {\dot{B}}_{p, \infty}^{s})$ and for all $τ_{L} \in (0, s (p) - s]$ $\begin{array}{l} (3) & {‖ u (t) - U ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} ≲ t^{- γ / 2} ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s}} where γ : = s (p) - τ_{L} - s . \end{array}$

In the theorem above, we note that the first part is almost similar to the result in [14] while the second part is new.
Remark 1.2.

Lemma 2.1 below tells us the following decay estimates; $\begin{array}{l} {‖ u (t) - U ‖}_{L^{n / (1 - τ_{H})}} ≲ {‖ u (t) - U ‖}_{{\dot{B}}_{p, 1}^{s (p) + τ_{H}}} ≲ t^{- τ_{H} / 2}, \\ {‖ u (t) - U ‖}_{L^{n / (1 + τ_{L}), \infty}} ≲ {‖ u (t) - U ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} ≲ t^{- γ / 2}, \end{array}$ where $\begin{array}{l} p < \frac{n p}{n - s p} ⩽ \frac{n}{1 + τ_{L}} < n < \frac{n}{1 - τ_{H}} < \frac{n p}{n - p} . \end{array}$

Since $s (p) - τ_{L} > 0$ in (ii), our result is independent to that of [3] where the asymptotic stability was discussed in weak $L^{p}$ spaces.

Here, we recall a merit of using Besov spaces instead of Lebesgue or weak Lebesgue spaces. Following Cannone [6], one can see that if $1 ⩽ r < p < \infty$ , then $\begin{array}{l} ‖ ω_{k} f ‖_{L^{r}} = ‖ f ‖_{L^{r}} and lim_{| k | \to \infty} ‖ ω_{k} f ‖_{{\dot{B}}_{p, \infty}^{- n (1 / r - 1 / p)}} = 0, \end{array}$ where $ω_{k} (x) : = e^{i k x}$ . Taking Lemma 2.1 below into account, our smallness condition on the external force f in ${\dot{B}}_{p, \infty}^{s (p) - 2}$ in Theorems 1.1 and 1.2 is satisfied when f sufficiently oscillates, even if the $L^{ℓ, \infty}$ norm of f with $ℓ : = n p / (n - (s (p) - 2) p) = n / 3$ is large. Note also that $L^{n / 3, \infty}$ is a scaling invariant space for external forces of the stationary equations (S). For example, in three dimensional case, Bjorland, Brandolese, Iftimie and Schonbek [3] assumed that $‖ Δ^{- 1} P f ‖_{L^{n, \infty}}$ is small to achieve stability results. This assumption is related to the smallness of $‖ f ‖_{L^{n / 3, \infty}}$ .

Applying the existence theorem of stationary solution in $L^{n, \infty}$ by Bjorland, Brandolese, Iftimie and Schonbek [3, Theorem 2.2], we can show the similar result. The proof is similar to that of Theorem 1.2 so we omit.
Theorem 1.3.
Let $n ⩾ 3$ and $n / 2 < p < n$ , and suppose that $a \in L^{n, \infty}$ satisfies $div a = 0$ in $S^{'}$ .

(i): Let $τ_{H} \in (0, 2 - n / p)$ . If ${(- Δ)}^{- 1} P f \in L^{n, \infty}$ is sufficiently small and $a - U$ is sufficiently small in ${\dot{B}}_{p, \infty}^{s (p)}$ , where $U \in L^{n, \infty}$ is the corresponding stationary solution with f constructed in [ 3 ], then there uniquely exists $u \in B C ((0, \infty); L^{n, \infty})$ so that $u - U \in B C ((0, \infty); {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ and solves the differential equation $\begin{array}{l} (4) & \partial_{t} u (t) - Δ u (t) + P \nabla (u \otimes u) (t) = P f \end{array}$ in ${\dot{W}}^{- 2, (n, \infty)} : = {(- Δ) g; g \in L^{n, \infty}}$ for $t > 0$ , with the initial condition $u (0) = a$ in the sense that for $α \in [0, 2]$ $\begin{array}{l} {‖ u (t) - a ‖}_{{\dot{B}}_{p, \infty}^{s (p) - α}} ≲ t^{α / 2} ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}} . \end{array}$ Moreover, it follows that $‖ u (t) - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}}$ and $\begin{array}{l} {‖ u (t) - U ‖}_{{\dot{B}}_{p, 1}^{s (p) + τ_{H}}} ≲ t^{- τ_{H} / 2} ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}} . \end{array}$ Further, $u (t) \to U$ in ${\dot{B}}_{p, \infty}^{s (p)}$ as $t \to \infty$ if and only if $e^{- t A} (a - U) \to 0$ in ${\dot{B}}_{p, \infty}^{s (p)}$ as $t \to \infty$ , see Section 3 for the definition of the semigroup $e^{- t A}$ .

(ii): Let $s \in (0, s (p))$ and $ℓ (s) : = n p / (n - s p) \in (p, \infty)$ . If we additionally assume that ${(- Δ)}^{- 1} P f \in L^{ℓ (s), \infty}, a - U \in {\dot{B}}_{p, \infty}^{s}$ , and $‖ {(- Δ)}^{- 1} P f ‖_{L^{n, \infty}} + ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}}$ is sufficiently small, where $U \in L^{n, \infty} \cap L^{ℓ (s), \infty}$ is the corresponding stationary solution with f constructed in [ 3 ], then the solution u above also satisfies $u \in B C ((0, \infty); L^{ℓ (s), \infty})$ and for all $τ_{L} \in (0, s (p) - s]$ $\begin{array}{l} (5) & {‖ u (t) - U ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} ≲ t^{- γ / 2} ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s}} where γ : = s (p) - τ_{L} - s . \end{array}$
Remark 1.3.

For the existence of stationary solutions in $L^{n, \infty}$ , used above, see Theorem 2.2 in [3].

From Lemma 2.1 below, we know that ${\dot{B}}_{p, \infty}^{s (p)} ↪ L^{n, \infty}$ and ${\dot{B}}_{p, \infty}^{s} ↪ L^{ℓ (s), \infty}$ .

Theorems 1.2 and 1.3 do not overlap. Although, the class of the external force in Theorem 1.2 is larger than that of Theorem 1.3, the class of non-stationary solution in Theorem 1.2 is smaller than that of Theorem 1.3.

This paper is organized as follows. In Section 2, we give embeddings between Besov spaces and weak $L^{p}$ spaces in Lemma 2.1. From this, we obtain a product estimate in Besov spaces. This is a corollary of Hölder inequality in weak $L^{p}$ spaces. After that, following Bourdaud [5], and also [2], we define Fourier multipliers including the Helmholtz projection and the resolvent operator for the Laplacian on the homogeneous Besov spaces. Section 3 is devoted to establish resolvent estimates for a perturbed Laplacian following Kozono and Yamazaki [14], and then give smoothing estimates for the semigroup generated by the Laplacian. A critical bound for the Duhamel term is also proved. In Section 4, we prove our main results Theorem 1.1 and Theorem 1.2.
2. Preliminaries

In this section, we recall definitions of function spaces and state some relevant inequalities which are the main tools for the rest of the paper.

We use the following notations. $S$ and $S^{'}$ denote the Schwartz spaces of rapidly decreasing smooth functions and tempered distributions, respectively. With reference [5] and [2], we make use of a subspace $S_{h}^{'}$ of $S^{'}$ in the definition of the homogeneous Besov spaces. This is a space of all tempered distributions f fulfilling $\begin{array}{l} ψ (λ D) f : = F^{- 1} [ψ (λ \cdot) \hat{f}] \to 0 in L^{\infty} as λ \to \infty, \end{array}$ for all $ψ \in S$ . The symbols ˆ and $F^{- 1}$ denote the Fourier transform and its inverse. $A ≲ B$ means $A ⩽ c B$ with a positive constant c. $A \approx B$ means $A ≲ B$ and $B ≲ A$ . Let $s (p) : = - 1 + n / p$ . For $p \in [1, \infty]$ , we denote $p^{'} \in [1, \infty]$ satisfying $1 / p + 1 / p^{'} = 1$ .

2.1. Function spaces

Let us recall the definition of Besov spaces. We fix $φ \in S (R^{n})$ satisfying supp $φ \subset {1 / 2 ⩽ | ξ | ⩽ 2}$ and $\sum_{j \in Z} φ (\frac{ξ}{2^{j}}) = 1$ for $ξ \in R^{n} ∖ {0}$ , and then $φ_{j} (D) f : = F^{- 1} [φ (\frac{\cdot}{2^{j}}) \hat{f}] \in S^{'}$ for $f \in S^{'}$ . The fact that for any $f \in S^{'}$ , $\sum_{j ⩾ 0} φ_{j} (D) f$ converges in the sense of $S^{'}$ is well-known. But, $\sum_{j ⩽ 0} φ_{j} (D) f$ may not converge in general. We can see this when f is a constant. It is known that $\sum_{j ⩽ 0} φ_{j} (D) f$ converges in $S^{'}$ and belongs to $S_{h}^{'}$ if $f \in S_{h}^{'}$ . Thus, it holds that $f = \sum_{j \in Z} φ_{j} (D) f \in S_{h}^{'}$ converges in $S^{'}$ for $f \in S_{h}^{'}$ . We refer to [5] and [2] for these facts.

For $p, q \in [1, \infty]$ and $s \in R$ , the homogeneous Besov space ${\dot{B}}_{p, q}^{s}$ is defined as $\begin{array}{l} {\dot{B}}_{p, q}^{s} : = {f \in S_{h}^{'}; ‖ f ‖_{{\dot{B}}_{p, q}^{s}} : = {‖ {2^{j s} {‖ φ_{j} (D) f ‖}_{L^{p}}}_{j \in Z} ‖}_{ℓ^{q}} < \infty} . \end{array}$ These spaces become Banach spaces, if $s < n / p$ when $q > 1$ or $s ⩽ n / p$ when $q = 1$ , see [2,5] and [8]. ${\dot{B}}_{p, q}^{s}$ is invariant with respect to the scaling for the Navier–Stokes equations when $s = s (p)$ . We refer to [2,5,8] and [19] for fundamental and important properties of Besov spaces.

Next, we recall weak $L^{p}$ spaces. Let f be a measurable function and $p \in [1, \infty)$ . We say f belongs to $L^{p, \infty}$ if $\begin{array}{l} ‖ f ‖_{L^{p, \infty}} : = sup_{λ > 0} λ {| {x \in R^{n}; | f (x) | > λ} |}^{1 / p} < \infty . \end{array}$ When $p \in (1, \infty)$ , this space can be characterized by means of real interpolation; $\begin{array}{l} L^{p, \infty} = {(L^{p_{0}}, L^{p_{1}})}_{θ, \infty} \end{array}$ where $θ \in (0, 1), 1 / p = (1 - θ) / p_{0} + θ / p_{1}$ and $1 < p_{0} < p < p_{1} ⩽ \infty$ . The latter space is a Banach space equipped with the norm $\begin{array}{l} sup_{λ > 0} λ^{- θ} K (λ, f; L^{p_{0}}, L^{p_{1}}) . \end{array}$ The K-functional is defined by $\begin{array}{l} K (λ, f; L^{p_{0}}, L^{p_{1}}) : = inf_{\begin{array}{c} f = f_{0} + f_{1} \\ f_{0} \in L^{p_{0}}, f_{1} \in L^{p_{1}} \end{array}} (‖ f_{0} ‖_{L^{p_{0}}} + λ ‖ f_{1} ‖_{L^{p_{1}}}) . \end{array}$ The space $L^{n, \infty}$ is also scaling invariant space for the Naver–Stokes equations, and we know the following inclusions, $\begin{array}{l} {\dot{B}}_{1, \infty}^{n - 1} ↪ {\dot{B}}_{p, \infty}^{s (p)} ↪ L^{n, \infty} ↪ {\dot{B}}_{q, \infty}^{s (q)} ↪ {BMO}^{- 1}, \end{array}$ where $1 ⩽ p < n < q < \infty$ , and the last space is treated by Koch and Tataru [11]. Kozono and Yamazaki [13] constructed small global solutions in the frame work of Besov–Morrey spaces.

We make use of next lemma to establish product estimates in Besov spaces in Lemma 2.2 below.

Lemma 2.1.
Let $1 < p < \infty, - n / p^{'} ⩽ s < n / p$ and $ℓ : = n p / (n - s p) \in [1, \infty)$ .

(i): If $s > 0$ , then ${\dot{B}}_{p, \infty}^{s} ↪ L^{ℓ, \infty}$ .

(ii): If $s < 0$ , then $L^{ℓ, \infty} ↪ {\dot{B}}_{p, \infty}^{s}$ .
Proof.
Since $s < 0$ implies $ℓ < p$ , (ii) is immediately shown by Bernstein’s inequality.

(i): We take $ℓ_{0}, ℓ_{1} ⩾ 1$ so that $p < ℓ_{0} < ℓ < ℓ_{1} < \infty$ and $2 / ℓ = 1 / ℓ_{0} + 1 / ℓ_{1}$ . Bernstein’s inequality yields that $\begin{array}{l} {‖ \sum_{j ⩾ λ_{}} φ_{j} (D) f ‖}_{L^{ℓ_{0}}} ≲ 2^{α λ_{}} ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} and {‖ \sum_{j < λ_{}} φ_{j} (D) f ‖}_{L^{ℓ_{1}}} ≲ 2^{β λ_{}} ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}}, \end{array}$ where $α = n (1 / p - 1 / ℓ_{0}) - s < 0$ and $β = n (1 / p - 1 / ℓ_{1}) - s > 0$ . Therefore, one has $K (λ, f; L^{ℓ_{0}}, L^{ℓ_{1}}) ≲ (2^{α λ_{}} + λ 2^{β λ_{}}) ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}}$ . Optimizing $λ_{} \in Z$ , we obtain $\begin{array}{l} ‖ f ‖_{{(L^{ℓ_{0}}, L^{ℓ_{1}})}_{1 / 2, \infty}} ⩽ sup_{λ > 0} λ^{- 1 / 2} ({‖ \sum_{j ⩾ λ_{}} φ_{j} (D) f ‖}_{L^{ℓ_{0}}} + λ {‖ \sum_{j < λ_{*}} φ_{j} (D) f ‖}_{L^{ℓ_{1}}}) ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$ Thus, from the characterization of $L^{ℓ, \infty}$ by real interpolation, the desired inequality $‖ f ‖_{L^{ℓ, \infty}} ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}}$ is proved. □

2.2. Product estimates

The following estimate is applied to control the convection term $(g \cdot \nabla) h = \nabla \cdot (g \otimes h)$ under the condition div $g = 0$ . Instead of the paraproduct argument, we use embeddings between Besov spaces and weak $L^{p}$ spaces in Lemma 2.1. As we see, the inequality is a consequence of the Hölder inequality in weak $L^{p}$ spaces.

Lemma 2.2.
Let $n ⩾ 1, 0 < s < 1$ and $n / (n - (1 - s)) < p < n / s$ . Then, for $g \in L^{n, \infty}$ and $h \in {\dot{B}}_{p, \infty}^{s}$ , $\begin{array}{l} ‖ g h ‖_{{\dot{B}}_{p, \infty}^{s - 1}} ≲ ‖ g ‖_{L^{n, \infty}} ‖ h ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$
Proof.
From (ii) in Lemma 2.1, we see that $L^{ℓ, \infty} ↪ {\dot{B}}_{p, \infty}^{s - 1}$ with $ℓ = n p / (n - (s - 1) p) \in (1, \infty)$ . One has $\begin{array}{l} ‖ g h ‖_{{\dot{B}}_{p, \infty}^{s - 1}} & ≲ ‖ g h ‖_{L^{ℓ, \infty}} \\ ≲ ‖ g ‖_{L^{n, \infty}} ‖ h ‖_{L^{r, \infty}} \\ ≲ ‖ g ‖_{L^{n, \infty}} ‖ h ‖_{{\dot{B}}_{p, \infty}^{s}}, \end{array}$ where $r : = n p / (n - s p) > 1$ and (i) of Lemma 2.1 has been used in the third inequality. □

2.3. Operators on homogeneous Besov spaces

We make clear definitions of the projection $P$ , the fractional derivative and integral operator ${(- Δ)}^{α / 2}$ on homogeneous Besov spaces. Moreover, we define ${(λ + Δ)}^{- b / 2}$ for $b > 0$ and λ lying in a sector in $C$ . Estimates for ${(λ + Δ)}^{- 1}$ are used in the next section. Remark that our definition includes the end-point cases $p = 1$ and ∞, although $P$ is a singular integral operator.

Let $p, q \in [1, \infty], s \in R, f \in {\dot{B}}_{p, q}^{s}$ and $m \in C^{\infty} (R^{n} ∖ {0})$ . We consider the convergence of the sum $\begin{array}{l} (6) & m (D) f : = \sum_{j \in Z} F^{- 1} [m φ (\cdot / 2^{j}) \hat{f}] . \end{array}$ Proposition 2.14 in [2] is useful to show that the sum above converges in $S^{'}$ and also $m (D) f \in S_{h}^{'}$ . To see these properties, it is enough to prove the following: $\begin{array}{l} (7) & sup_{j \in N} 2^{- j N} {‖ F^{- 1} [m φ (\cdot / 2^{j}) \hat{f}] ‖}_{L^{\infty}} < \infty with some N \in Z \\ (8) & {‖ \sum_{j = - m_{2}}^{- m_{1}} F^{- 1} [m φ (\cdot / 2^{j}) \hat{f}] ‖}_{L^{\infty}} \to 0 as m_{2} > m_{1} \to \infty \end{array}$ Moreover, once we prove (7) and (8) with any $\tilde{φ} \in C_{0}^{\infty} (R^{n})$ with the support of $\tilde{φ}$ being a compact subset of $R^{n} ∖ {0}$ instead of φ (this is a function as in the beginning of Section 2.1 and the property $\sum_{j \in Z} \tilde{φ} (ξ / 2^{j}) = 1$ for $ξ \in R^{n} ∖ {0}$ , we can see that $m (D) f$ is independent of φ. Indeed, for such $\tilde{φ}$ and any $ψ \in S$ , if the Fourier support of ψ is a compact subset of $R^{n} ∖ {0}$ , then it holds that $\begin{array}{l} ⟨ \sum_{j \in Z} F^{- 1} [m (φ (\cdot / 2^{j}) - \tilde{φ} (\cdot / 2^{j})) \hat{f}], ψ ⟩ = ⟨ \hat{f}, \sum_{j \in Z} m \hat{ψ} [φ (\cdot / 2^{j}) - \tilde{φ} (\cdot / 2^{j})] ⟩ = 0 . \end{array}$ Here, we remark that the last series is a finite one. Since the space consisting of such ψ is dense in $L^{2}$ , it turns out that $m (D) f$ is independent of the choice of the cut-off function φ.

2.3.1. Helmholtz projection and fractional Laplacian

Let $f \in {\dot{B}}_{p, q}^{s}$ with $1 ⩽ p, q ⩽ \infty, s \in R$ and $a \in R$ . Then, we assume that $m \in C^{\infty} (R^{n} ∖ {0})$ is a-homogeneous, $m (λ ξ) = λ^{a} m (ξ)$ for $λ > 0$ , and satisfies $| \partial^{α} m (ξ) | ≲ | ξ |^{a - | α |}$ for all $α \in {(N \cup {0})}^{n}$ and $ξ \neq 0$ . We can see that $m (D) f \in {\dot{B}}_{p, q}^{s - a}$ , provided that $\begin{array}{l} (9) & a ⩾ s - n / p when q = 1 or a > s - n / p when q \in (1, \infty] . \end{array}$ To see this, we observe that $‖ F^{- 1} [m φ (\cdot / 2^{j})] ‖_{L^{p^{'}}} ≲ 2^{j (a + n / p)}$ . This and (9) ensure that (7) with $N = a + n / p - s$ and (8) hold. Hence, we see from Lemma 2.23 in [2] that $m (D) f \in {\dot{B}}_{p, q}^{s - a}$ and $\begin{array}{l} {‖ m (D) f ‖}_{{\dot{B}}_{p, q}^{s - a}} ≲ {‖ {2^{j (s - a)} {‖ F^{- 1} [m φ (\cdot / 2^{j}) \hat{f}] ‖}_{L^{p}}}_{j \in Z} ‖}_{ℓ^{q}} ≲ ‖ f ‖_{{\dot{B}}_{p, q}^{s}} . \end{array}$ Applying this argument, we can define the Helmholtz projection $P f : = m (D) f$ with $m (ξ) : = {(δ_{i, j} + ξ_{i} ξ_{j} / | ξ |^{2})}_{1 ⩽ i, j ⩽ n}$ and obtain the boundedness $\begin{array}{l} ‖ P f ‖_{{\dot{B}}_{p, q}^{s}} ≲ ‖ f ‖_{{\dot{B}}_{p, q}^{s}} \end{array}$ when $\begin{array}{l} (10) & s ⩽ n / p when q = 1 or s < n / p when q \in (1, \infty] . \end{array}$ We remark that this inequality holds even if $p = 1, \infty$ . Similarly, let us denote ${(- Δ)}^{a / 2} f : = m_{a} (D) f$ with $m_{a} (ξ) : = | ξ |^{a}$ . Under the condition (9), ${(- Δ)}^{a / 2}$ is an operator from ${\dot{B}}_{p, q}^{s}$ to ${\dot{B}}_{p, q}^{s - a}$ with the estimate $\begin{array}{l} {‖ {(- Δ)}^{a / 2} f ‖}_{{\dot{B}}_{p, q}^{s - a}} ≲ ‖ f ‖_{{\dot{B}}_{p, q}^{s}} . \end{array}$

2.3.2. Resolvent operator

Let $f \in {\dot{B}}_{p, q}^{s}$ with $1 ⩽ p, q ⩽ \infty$ and $s \in R$ . Let $b ⩾ 0, ω \in (0, π / 2)$ and $\begin{array}{l} S_{ω} : = {z \in C ∖ {0}; | arg (z) | ⩾ ω} . \end{array}$ For $λ \in S_{ω}$ , we denote $\tilde{m_{b}} (ξ) : = {(λ - | ξ |^{2})}^{- b / 2}$ and observe that $| \partial^{α} \tilde{m_{b}} (ξ) | ≲ min (| λ |^{- b / 2}, | ξ |^{- b}) | ξ |^{- | α |}$ for $ξ \neq 0$ . Thus it holds that $‖ F^{- 1} [\tilde{m_{b}} (\cdot) φ (\cdot / 2^{j})] ‖_{L^{1}} = ‖ F^{- 1} [\tilde{m_{b}} (2^{j} \cdot) φ (\cdot)] ‖_{L^{1}} ≲ min (| λ |^{- b / 2}, 2^{- j b})$ . Hence under the condition (10), it turns out that (7) with $N = n / p - s$ and (8) hold. Therefore, we see that ${(λ + Δ)}^{- b / 2} f : = \tilde{m_{b}} (D) f = \sum_{j \in Z} F^{- 1} [\tilde{m_{b}} φ (\cdot / 2^{j}) \hat{f}]$ converges in $S^{'}$ and belongs to $S_{h}^{'}$ . Moreover, Lemma 2.23 in [2] gives $\begin{array}{l} (11) & {‖ {(λ + Δ)}^{- b / 2} f ‖}_{{\dot{B}}_{p, q}^{s}} ≲ {‖ {2^{j s} {‖ F^{- 1} [\tilde{m_{b}} φ (\cdot / 2^{j}) \hat{f}] ‖}_{L^{p}}}_{j \in Z} ‖}_{ℓ^{q}} ≲ | λ |^{- b / 2} ‖ f ‖_{{\dot{B}}_{p, q}^{s}}, \\ (12) & {‖ {(λ + Δ)}^{- b / 2} f ‖}_{{\dot{B}}_{p, q}^{s + b}} ≲ {‖ {2^{j (s + b)} {‖ F^{- 1} [\tilde{m_{b}} φ (\cdot / 2^{j}) \hat{f}] ‖}_{L^{p}}}_{j \in Z} ‖}_{ℓ^{q}} ≲ ‖ f ‖_{{\dot{B}}_{p, q}^{s}} . \end{array}$ The implicit constants above are independent of $λ \in S_{ω}$ . Especially, we define the resolvent operator $R_{- Δ} (λ) : = {(λ + Δ)}^{- 1}$ . If (10) holds, then $(λ + Δ)$ is an injection from ${\dot{B}}_{p, q}^{s} \cap {\dot{B}}_{p, q}^{s + 2} \subset {\dot{B}}_{p, q}^{s}$ to ${\dot{B}}_{p, q}^{s}$ and also $I = R_{- Δ} (λ) (λ + Δ) = (λ + Δ) R_{- Δ} (λ)$ on ${\dot{B}}_{p, q}^{s}$ .

Next, we shall consider ${\dot{B}}_{p, q}^{s}$ - ${\dot{B}}_{p_{0}, q}^{s}$ estimates with $p_{0} > p$ for ${(λ + Δ)}^{- b / 2}$ when $\begin{array}{l} (13) & b ⩾ n / p . \end{array}$ To do that, we observe the estimate $‖ F^{- 1} [\tilde{m_{b}} (\cdot) φ (\cdot / 2^{j})] ‖_{L^{r}} ≲ min (| λ |^{- b / 2}, 2^{- j b}) 2^{j n (1 - 1 / r)}$ for $r \in (1, \infty)$ by interpolating the $L^{1}$ -estimate above and the $L^{\infty}$ -estimate, $\begin{array}{l} {‖ F^{- 1} [\tilde{m_{b}} φ (\cdot / 2^{j})] ‖}_{L^{\infty}} ≲ {‖ \tilde{m_{b}} φ (\cdot / 2^{j}) ‖}_{L^{1}} ≲ min (| λ |^{- b / 2}, 2^{- j b}) 2^{j n} . \end{array}$ Therefore, we obtain $\begin{array}{l} {‖ {(λ + Δ)}^{- b / 2} f ‖}_{{\dot{B}}_{p_{0}, q}^{s}} ≲ {‖ {2^{j s} {‖ F^{- 1} [\tilde{m_{b}} φ (\cdot / 2^{j}) \hat{f}] ‖}_{L^{p_{0}}}}_{j \in Z} ‖}_{ℓ^{q}} ≲ | λ |^{- (b - n (1 / p - 1 / p_{0})) / 2} ‖ f ‖_{{\dot{B}}_{p, q}^{s}} . \end{array}$

2.3.3. Composition operator

Let $f \in {\dot{B}}_{p, q}^{s}$ with $1 ⩽ p, q ⩽ \infty$ and $s \in R$ . Let $0 ⩽ a ⩽ b$ . We consider the composition operator $m_{a, b} (D)$ with $m_{a, b} (ξ) : = | ξ |^{a} {(λ - | ξ |^{2})}^{- b / 2}$ where $λ \in S_{ω}$ . It is not hard to see, from the previous sections, that if $\begin{array}{l} (14) & a - b ⩾ s - n / p when q = 1 or a - b > s - n / p when q \in (1, \infty], \end{array}$ then $\sum_{j \in Z} F^{- 1} [m_{a, b} φ (\cdot / 2^{j}) \hat{f}]$ converges in $S^{'}, m_{a, b} (D) f \in S_{h}^{'}$ and $\begin{array}{l} {‖ m_{a, b} (D) f ‖}_{{\dot{B}}_{p, q}^{s}} ≲ | λ |^{- b / 2} ‖ f ‖_{{\dot{B}}_{p, q}^{s}} and {‖ m_{a, b} (D) f ‖}_{{\dot{B}}_{p, q}^{s + b}} ≲ ‖ f ‖_{{\dot{B}}_{p, q}^{s}} . \end{array}$ Further, we can see that $m_{a, b} (D) = m_{a} (D) \circ \tilde{m_{b}} (D) = \tilde{m_{b}} (D) \circ m_{a} (D)$ . In fact, it holds that $\begin{array}{l} m_{a} (D) \circ \tilde{m_{b}} (D) f & = \sum_{j \in Z} F^{- 1} [m_{a} φ (\cdot / 2^{j}) \hat{\tilde{m_{b}} (D) f}] = \sum_{j \in Z} F^{- 1} [m_{a} \tilde{m_{b}} φ (\cdot / 2^{j}) \tilde{φ} (\cdot / 2^{j}) \hat{f}] \\ = m_{a, b} (D) f, \end{array}$ with a cut-off function $\tilde{φ}$ being chosen so that $φ (ξ / 2^{j}) \tilde{φ} (ξ / 2^{j}) = φ (ξ / 2^{j})$ , and we have used the fact that the definitions of the operators are independent of test functions. Similarly, we have $\tilde{m_{b}} (D) \circ m_{a} (D) = m_{a, b} (D)$ .

3. Resolvent estimates and a critical estimate for the semigroup

Let $n ⩾ 3$ and $n / 2 < p < n$ . These restrictions are due to Lemma 2.2 with $s = s (p) = - 1 + n / p$ .

Here, we discuss resolvent estimates for the Laplacian with the perturbation $B$ ; $\begin{array}{l} B [w] : = P \nabla (U \otimes w + w \otimes U) \\ A [w] : = - Δ w + B [w] \end{array}$ where $U \in L^{n, \infty}$ . Although we construct stationary solutions $U \in {\dot{B}}_{p, \infty}^{s (p)} \subset L^{n, \infty}$ in Theorem 1.1, the weaker condition $U \in L^{n, \infty}$ is enough for this section. We consider estimates for $A$ with the domain $D (A) : = {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, \infty}^{s (p) + 2}$ . It is known that this is a Banach space equipped the norm $‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p)}} + ‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p) + 2}}$ , although the latter space is not so, see [2]. It seems that $D (A)$ is not a dense subspace of ${\dot{B}}_{p, \infty}^{s (p)}$ , because ${\dot{B}}_{p, \infty}^{s (p)}$ is a kind of $ℓ^{\infty}$ space in the sense of real interpolation.

3.1. Resolvent estimates for the perturbed operator $A$

We remark that $b = 2$ fulfills the condition (13), and then start with the estimates for $A$ and $B$ .

Lemma 3.1.
It follows that for $s \in (0, 1)$ $\begin{array}{l} {‖ B [w] ‖}_{{\dot{B}}_{p, \infty}^{s - 2}} ≲ ‖ U ‖_{L^{n, \infty}} ‖ w ‖_{{\dot{B}}_{p, \infty}^{s}} and {‖ A [w] ‖}_{{\dot{B}}_{p, \infty}^{s - 2}} ≲ (1 + ‖ U ‖_{L^{n, \infty}}) ‖ w ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$
Proof.
This is done by the argument in Section 2.3 and Lemma 2.2. □

Let $s \in (0, 1)$ and $λ \in S_{ω}$ with $ω \in (0, π / 2)$ . Combining the estimate for $B$ above with (12), we see that if $‖ U ‖_{L^{n, \infty}}$ is sufficiently small, then $(1 - R_{- Δ} (λ) B)$ is invertible on ${\dot{B}}_{p, \infty}^{s}$ with $\begin{array}{l} {(1 - R_{- Δ} (λ) B)}^{- 1} = \sum_{ℓ = 0}^{\infty} {[R_{- Δ} (λ) B]}^{ℓ} and {‖ {(1 - R_{- Δ} (λ) B)}^{- 1} ‖}_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s}} ⩽ 2 . \end{array}$ Because it holds that $(λ - A) = (λ + Δ) (I - R_{- Δ} (λ) B)$ as a map from $D (A)$ to ${\dot{B}}_{p, \infty}^{s}$ , we see that $\begin{array}{l} R_{A} (λ) : = {(1 - R_{- Δ} (λ) B)}^{- 1} R_{- Δ} (λ) \end{array}$ is an operator from ${\dot{B}}_{p, \infty}^{s}$ to itself, and satisfies $(λ - A) R_{A} (λ) = R_{A} (λ) (λ - A) = I$ on ${\dot{B}}_{p, \infty}^{s}$ with the estimate $\begin{array}{l} {‖ R_{A} (λ) ‖}_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s}} ≲ | λ |^{- 1} . \end{array}$ Moreover, we establish smoothing estimates. To see these, we make use of an auxiliary operator by Kozono and Yamazaki [14] $\begin{array}{l} C_{θ} : = {(- Δ)}^{- (2 - θ) / 2} B {(- Δ)}^{- θ / 2} \end{array}$ with $θ \in [0, 2]$ . This operator has the following properties.
Lemma 3.2.
Let $s < n / p$ and $θ \in [0, 2]$ with $0 < s + θ < 1$ . Assume that U is sufficiently small in $L^{n, \infty}$ .

(i): $\begin{array}{l} ‖ C_{θ} ‖_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s}} ≲ ‖ U ‖_{L^{n, \infty}} . \end{array}$

(ii): $\begin{array}{l} \sum_{j = 0}^{\infty} {[R_{- Δ} (λ) B]}^{j} = {(- Δ)}^{- θ / 2} \sum_{j = 0}^{\infty} {[R_{- Δ} (λ) (- Δ) C_{θ}]}^{j} {(- Δ)}^{θ / 2} . \end{array}$

(iii): $\begin{array}{l} {‖ \sum_{j = 1}^{\infty} {[R_{- Δ} (λ) (- Δ) C_{θ}]}^{j} ‖}_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s}} ≲ ‖ U ‖_{L^{n, \infty}} . \end{array}$
Proof.
(i): This can be seen from the mapping properties of the negative powers of $(- Δ)$ in Section 2.3 and Lemma 3.1.

(ii): Using the commutativity of $R_{- Δ} (λ)$ and ${(- Δ)}^{- θ / 2}$ , we have $\begin{array}{l} {[R_{- Δ} (λ) B]}^{2} = {(- Δ)}^{- θ / 2} {[R_{- Δ} (λ) (- Δ) C_{θ}]}^{2} {(- Δ)}^{θ / 2} . \end{array}$ Thus, the desired equality holds.

(iii): Since $\begin{array}{l} R_{- Δ} (λ) (- Δ) g = \sum_{j \in Z} F^{- 1} [m (ξ) φ (ξ / 2^{j}) \hat{g}] \end{array}$ with $m (ξ) : = {(λ + | ξ |^{2})}^{- 1} | ξ |^{2}$ and $| \partial^{α} m (ξ) | ≲ min (| λ |^{- 1}, | ξ |^{- 2}) | ξ |^{2 - | α |} ⩽ | ξ |^{- | α |}$ for all $ξ \neq 0$ , one has the boundedness $‖ R_{- Δ} (λ) (- Δ) g ‖_{{\dot{B}}_{p, \infty}^{s}} ≲ ‖ g ‖_{{\dot{B}}_{p, \infty}^{s}}$ . Combining this with (i) and the smallness of $‖ U ‖_{L^{n, \infty}}$ , one has the absolute convergence of the sum. □
Lemma 3.3.
Let $- 2 < s < 1, - 2 ⩽ τ ⩽ 2, - 2 < s + τ < 1$ and $λ \in S_{ω}$ with $ω \in (0, π / 2)$ . Assume that $‖ U ‖_{L^{n, \infty}}$ is sufficiently small.

(i): $\begin{array}{l} {‖ R_{- Δ} (λ) B R_{A} (λ) ‖}_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s + τ}} ≲ | λ |^{- (2 - τ) / 2} ‖ U ‖_{L^{n, \infty}} . \end{array}$

(ii): If $τ ⩾ 0$ , then $\begin{array}{l} {‖ R_{A} (λ) ‖}_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s + τ}} ≲ | λ |^{- (2 - τ) / 2} (1 + ‖ U ‖_{L^{n, \infty}}) . \end{array}$
Proof.
(i): The conditions on exponents ensure the existence of $η \in [0, 1] \cap (- s / 2, (1 - s) / 2) \cap (τ / 2, (τ + 2) / 2)$ . Thus, $\tilde{η} : = (τ + 2) / 2 - η \in [0, 1]$ enjoys $τ + 2 = 2 (η + \tilde{η})$ . Then, we take $θ \in [0, 2]$ so that $s - n / p + 2 η < θ$ .

Following Kozono and Yamazaki [14], we rewrite $\begin{array}{l} R_{- Δ} (λ) B R_{A} (λ) = R_{- Δ} (λ) {(- Δ)}^{(2 - θ) / 2} C_{θ} \sum_{j = 0}^{\infty} {[R_{- Δ} (λ) (- Δ) C_{θ}]}^{j} {(- Δ)}^{θ / 2} R_{- Δ} (λ), \end{array}$ and then we can obtain $\begin{array}{l} \{\begin{matrix} ‖ R_{- Δ} (λ) {(- Δ)}^{(2 - θ) / 2} ‖_{{\dot{B}}_{p, \infty}^{s + τ + (2 - θ) - 2 \tilde{η}} \to {\dot{B}}_{p, \infty}^{s + τ}} ≲ | λ |^{- (1 - \tilde{η})}, \\ ‖ C_{θ} \sum_{j = 0}^{\infty} {[R_{- Δ} (λ) (- Δ) C_{θ}]}^{j} ‖_{{\dot{B}}_{p, \infty}^{s + τ + (2 - θ) - 2 \tilde{η}} \to {\dot{B}}_{p, \infty}^{s + τ + (2 - θ) - 2 \tilde{η}}} ≲ ‖ U ‖_{L^{n, \infty}}, \\ ‖ {(- Δ)}^{θ / 2} R_{- Δ} (λ) ‖_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s - θ + 2 η}} ≲ | λ |^{- (1 - η)} . \end{matrix} \end{array}$ Here, we remark that $s + τ + (2 - θ) - 2 \tilde{η} = s - θ + 2 η$ . The first and last inequalities follow from the argument in Section 2.3. The second one is a combination of (i) and (iii) in Lemma 3.2. Thus, the desired inequality is obtained.

(ii): Interpolating (11) and (12), we have $\begin{array}{l} {‖ R_{- Δ} (λ) f ‖}_{{\dot{B}}_{p, \infty}^{s + τ}} ≲ | λ |^{- (2 - τ) / 2} ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$ Since $\begin{array}{l} (15) & R_{A} (λ) = R_{- Δ} (λ) + R_{- Δ} (λ) B R_{A} (λ), \end{array}$ the estimate for $R_{A} (λ)$ is shown by this and (i). □

Now we are position to define the semigroup with respect to $A$ by the Dunford integral $\begin{array}{l} e^{- t A} : = \frac{1}{2 π i} \int_{Γ} e^{- t λ} R_{A} (λ) d λ (t > 0), \end{array}$ where $Γ : = Γ_{-} \cup Γ_{0} \cup Γ_{+}$ , $\begin{array}{l} Γ_{0} : = {z \in C; z = e^{i ψ}, | ψ | ⩽ θ}, Γ_{\pm} : = {z \in C; z = r e^{\pm i θ}, r ⩾ 1}, \end{array}$ and $Γ_{\pm}$ are oriented upwards and $Γ_{0}$ is counterclockwise, with some $θ \in (ω, π / 2)$ . We refer to [20] and [15] for the standard properties of the semigroup.

Next estimates for the semigroup are used in the proof of Theorem 1.2. Similar estimates for the heat semigroup were shown by Kozono, Ogawa and Taniuchi [12].
Lemma 3.4.
Let $- 2 < s < 1$ . If $‖ U ‖_{L^{n, \infty}}$ is sufficiently small, then the following estimates hold for $t > 0$ .

(i): $\begin{array}{l} {‖ e^{- t A} f ‖}_{{\dot{B}}_{p, \infty}^{s}} ≲ (1 + ‖ U ‖_{L^{n, \infty}}) ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$ (ii): For $τ \in (0, 1 - s)$ , $\begin{array}{l} {‖ e^{- t A} f ‖}_{{\dot{B}}_{p, 1}^{s + τ}} ≲ t^{- τ / 2} (1 + ‖ U ‖_{L^{n, \infty}}) ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$ (iii): For $τ \in [0, 2]$ with $τ ⩽ 2 + s$ , $\begin{array}{l} {‖ e^{- t A} f - f ‖}_{{\dot{B}}_{p, \infty}^{s - τ}} ≲ t^{τ / 2} ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$
Remark 3.1.

Especially, (ii) with large τ is important in Lemma 3.6 below. In there, we use (ii) with $τ > 2$ .

Remark that we assume that $U \in L^{n, \infty}$ rather than $U \in L^{n}$ or $U \in {\dot{B}}_{p, \infty}^{s (p)}$ .

It seems that these estimates cannot be deduced from similar results in Besov–Morrey spaces by Kozono and Yamazaki [14].

Proof.
(i): This is an easy consequence of Lemma 3.3.

(ii): This inequality with ${\dot{B}}_{p, \infty}^{s + τ}$ in the place of ${\dot{B}}_{p, 1}^{s + τ}$ immediately follows from the same reason as (i). Applying an interpolation inequality by Machihara and Ozawa [16], we deduce the desired inequality as follows; if $0 < τ_{1} < τ < τ_{2} < 1 - s, τ = (1 - θ) τ_{1} + θ τ_{2}$ with $θ \in (0, 1)$ $\begin{array}{l} {‖ e^{- t A} f ‖}_{{\dot{B}}_{p, 1}^{s + τ}} & ≲ {‖ e^{- t A} f ‖}_{{\dot{B}}_{p, \infty}^{s + τ_{1}}}^{1 - θ} {‖ e^{- t A} f ‖}_{{\dot{B}}_{p, \infty}^{s + τ_{2}}}^{θ} \\ ≲ t^{- τ / 2} (1 + ‖ U ‖_{L^{n, \infty}}) ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$

(iii): To show this, we use the identity (15) and thus $\begin{array}{l} e^{- t A} f - f = (e^{t Δ} f - f) + \frac{1}{2 π i} \int_{Γ} e^{- t λ} R_{- Δ} (λ) B R_{A} (λ) f d λ . \end{array}$ Because $m (ξ) : = e^{- | ξ |^{2} / 4} - 1$ fulfills $| \partial^{α} m (ξ) | ≲ | ξ |^{2 - | α |}$ , the operator norm of the first term is controlled by a constant multiplied by $t^{τ / 2}$ . On the other hand, (i) in Lemma 3.3 yields $\begin{array}{l} {‖ \int_{Γ} e^{- t λ} R_{- Δ} (λ) B R_{A} (λ) d λ ‖}_{{\dot{B}}_{p, \infty}^{s} \to {\dot{B}}_{p, \infty}^{s - τ}} ≲ t^{τ / 2} ‖ U ‖_{L^{n, \infty}} . \end{array}$ Thus, the proof is completed. □

In next lemma, we see the differentiability of $e^{- t A} f$ with respect to t. Lemma 3.5.
Let $0 < s < 1$ and $0 ⩽ τ ⩽ 2$ with $τ < s$ . Suppose that $‖ U ‖_{L^{n, \infty}}$ is sufficiently small. Then, $\begin{array}{l} {‖ \frac{e^{- t A} f - f}{t} + A f ‖}_{{\dot{B}}_{p, \infty}^{s - 2 - τ}} ≲ t^{τ / 2} ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$
Proof.
Similarly as in [14] and [18], we write $\begin{array}{l} \frac{e^{- t A} f - f}{t} + A f & = \frac{1}{2 π i t} \int_{Γ} e^{- t λ} [R_{A} (λ) - λ^{- 1}] f d λ + A f \\ = \frac{1}{2 π i t} \int_{Γ^{'}} \frac{e^{- λ}}{λ} R_{A} (λ / t) A f d λ - \frac{1}{2 π i} \int_{Γ^{'}} \frac{e^{- λ}}{λ^{2}} A f d λ \\ = \frac{1}{2 π i} \int_{Γ^{'}} \frac{e^{- λ}}{λ^{2}} A R_{A} (λ / t) A f d λ, \end{array}$ where $Γ^{'} : = {t λ; λ \in Γ}$ . We have used the identities $R_{A} (λ) - λ^{- 1} = λ^{- 1} R_{A} (λ) A = λ^{- 1} A R_{A} (λ)$ in the last two equalities. Using Lemmas 3.1 and 3.3, we conclude as follows; $\begin{array}{l} {‖ \frac{e^{- t A} f - f}{t} + A f ‖}_{{\dot{B}}_{p, \infty}^{s - 2 - τ}} ≲ t^{τ / 2} {(1 + ‖ U ‖_{L^{n, \infty}})}^{2} \int_{Γ^{'}} e^{- | Re λ |} | Re λ |^{- (2 + τ / 2)} d λ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$ □

3.2. A critical estimate

We end this section with a critical estimate for the semigroup. This type of estimate was first proved by Meyer [17] in $L^{n, \infty}$ . Yamazaki [24] independently proved it, on domains, applying estimates in the dual spaces. Our estimate is analogous to that of [21] and [23], where the estimate for the heat semigroup was considered.

Lemma 3.6.
Let $0 < s < 1$ . It holds that $\begin{array}{l} {‖ \int_{t_{0}}^{t} e^{- (t - σ) A} P g (σ) d σ ‖}_{{\dot{B}}_{p, \infty}^{s}} ≲ sup_{t_{0} < σ < t} {‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s - 2}} \end{array}$ for $- \infty ⩽ t_{0} < t < \infty$ .
Remark 3.2.

Taking the norm inside the integral, one has from Lemma 3.4, $‖ e^{- (t - τ) A} P g (τ) ‖_{{\dot{B}}_{p, \infty}^{s}} ≲ {(t - τ)}^{- 1} ‖ g (τ) ‖_{{\dot{B}}_{p, \infty}^{s - 2}}$ , which makes the integral diverges at $σ = t$ . This difficulty can be overcome by the characterization ${\dot{B}}_{p, \infty}^{s} = {({\dot{B}}_{p, \infty}^{s_{0}}, {\dot{B}}_{p, \infty}^{s_{1}})}_{θ, \infty}$ with $s = (1 - θ) s_{0} + θ s_{1}$ .

This inequality is related to the $L^{\infty}$ -maximal regularity. Indeed, ignoring $P$ and combining Lemma 3.6 and Lemma 3.1, we obtain the following. $\begin{array}{l} {‖ A \int_{0}^{t} e^{- (t - σ) A} f (σ) d σ ‖}_{{\dot{B}}_{p, \infty}^{s (p) - 2}} & ≲ {‖ \int_{0}^{t} e^{- (t - σ) A} f (σ) d σ ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} \\ ≲ sup_{0 < σ < t} {‖ f (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - 2}} . \end{array}$ In other words, if $u (t) : = \int_{0}^{t} e^{- (t - σ) A} f (σ) d σ$ , then $\begin{array}{l} ‖ A u ‖_{L^{\infty} ((0, \infty); {\dot{B}}_{p, \infty}^{s (p) - 2})} ≲ ‖ f ‖_{L^{\infty} ((0, \infty); {\dot{B}}_{p, \infty}^{s (p) - 2})} . \end{array}$ Further, if u solves the differential equation $\partial_{t} u + A u = f$ , then it follows that $\begin{array}{l} ‖ \partial_{t} u ‖_{L^{\infty} ((0, \infty); {\dot{B}}_{p, \infty}^{s (p) - 2})} + ‖ A u ‖_{L^{\infty} ((0, \infty); {\dot{B}}_{p, \infty}^{s (p) - 2})} ≲ ‖ f ‖_{L^{\infty} ((0, \infty); {\dot{B}}_{p, \infty}^{s (p) - 2})} . \end{array}$ It is known that the $L^{\infty}$ -maximal regularity cannot be deduced from $L^{p}$ -maximal regularity with some $p \in (1, \infty)$ through extrapolation.

Proof.
Firstly, we rewrite the integral as follows; $\begin{array}{l} \int_{t_{0}}^{t} e^{- (t - σ) A} P g (σ) d σ = \int_{0}^{\infty} e^{- σ A} P \tilde{g} (σ) d σ \end{array}$ with $\tilde{g} (σ) : = g (t - σ) χ_{(t_{0}, t)} (σ)$ . Here, we take $ε \in (0, 1 - s)$ . From the characterization ${\dot{B}}_{p, \infty}^{s} = {({\dot{B}}_{p, \infty}^{s - ε}, {\dot{B}}_{p, \infty}^{s + ε})}_{1 / 2, \infty}$ , we bound $\begin{array}{l} {‖ \int_{0}^{\infty} e^{- σ A} P \tilde{g} (σ) d σ ‖}_{{\dot{B}}_{p, \infty}^{s}} ≲ sup_{λ > 0} λ^{- 1 / 2} (‖ I ‖_{{\dot{B}}_{p, \infty}^{s - ε}} + λ ‖ II ‖_{{\dot{B}}_{p, \infty}^{s + ε}}), \end{array}$ where $I : = \int_{0}^{λ_{ε}} e^{- σ A} P \tilde{g} (σ) d σ$ and $II : = \int_{λ_{ε}}^{\infty} e^{- σ A} P \tilde{g} (σ) d σ$ with $λ_{ε} > 0$ . Using the smoothing estimates for the semigroup and the boundedness of $P$ , one can see $\begin{array}{l} ‖ I ‖_{{\dot{B}}_{p, \infty}^{s - ε}} ≲ λ_{ε}^{ε / 2} sup_{σ > 0} {‖ \tilde{g} (σ) ‖}_{{\dot{B}}_{p, \infty}^{s - 2}} and ‖ II ‖_{{\dot{B}}_{p, \infty}^{s + ε}} ≲ λ_{ε}^{- ε / 2} sup_{σ > 0} {‖ \tilde{g} (σ) ‖}_{{\dot{B}}_{p, \infty}^{s - 2}} . \end{array}$ Optimizing $λ_{ε} > 0$ , that is $λ_{ε} = λ^{1 / ε}$ , we obtain the desired inequality. □

4. Proof of Theorems 1.1 and 1.2

We prove Theorem 1.1 in Section 4.1 and Theorem 1.2 in Section 4.2, respectively.

4.1. Construction of stationary solutions

We construct stationary solutions U by using successive approximations; $\begin{array}{l} U_{0} : = {(- Δ)}^{- 1} P f and U_{m + 1} : = U_{0} - {(- Δ)}^{- 1} P \nabla (U_{m} \otimes U_{m}), (m = 0, 1, 2, \dots) . \end{array}$ The boundedness of the projection $P$ on ${\dot{B}}_{p, \infty}^{s (p) - 2}$ and Lemma 2.2 give us $\begin{array}{l} ‖ U_{m + 1} ‖_{{\dot{B}}_{p, \infty}^{s (p)}} & ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}} + ‖ U_{m} \otimes U_{m} ‖_{{\dot{B}}_{p, \infty}^{s (p) - 1}} \\ ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}} + ‖ U_{m} ‖_{L^{n, \infty}} ‖ U_{m} ‖_{{\dot{B}}_{p, \infty}^{s (p)}} \\ ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}} + ‖ U_{m} ‖_{{\dot{B}}_{p, \infty}^{s (p)}}^{2}, \end{array}$ which is enough to complete the proof of (i).

Since one has $\begin{array}{l} ‖ U_{m + 1} ‖_{{\dot{B}}_{p, \infty}^{s}} & ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s - 2}} + ‖ U_{m} \otimes U_{m} ‖_{{\dot{B}}_{p, \infty}^{s - 1}} \\ ≲ ‖ f ‖_{{\dot{B}}_{p, \infty}^{s - 2}} + ‖ U_{m} ‖_{{\dot{B}}_{p, \infty}^{s (p)}} ‖ U_{m} ‖_{{\dot{B}}_{p, \infty}^{s}}, \end{array}$ the proof of (ii) is completed.

4.2. Construction of non-stationary solutions and their asymptotic stability

Let us define $ε : = ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s (p)}}$ . We consider the equation $\begin{array}{l} (E) & \{\begin{matrix} \partial_{t} w + A [w] + P (w \cdot \nabla) w = 0, \\ w (0) = b : = a - U . \end{matrix} \end{array}$ For simplicity, we denote $\begin{array}{l} ‖ w ‖_{X (t)} : = sup_{σ ⩽ t} {‖ w (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} and ‖ w ‖_{X_{τ} (t)} : = sup_{σ ⩽ t} σ^{τ / 2} {‖ w (σ) ‖}_{{\dot{B}}_{p, 1}^{s (p) + τ}} . \end{array}$ Define $\begin{array}{l} w_{0} (t) : = e^{- t A} b and w_{m + 1} (t) : = w_{0} (t) - B (w_{m}, w_{m}) (t), \end{array}$ where $\begin{array}{l} B (g, h) (t) : = \int_{0}^{t} e^{- (t - σ) A} P \nabla (g \otimes h) (σ) d σ . \end{array}$ Lemmas 3.4 and 3.6 give us the following: $\begin{array}{l} ‖ w_{0} ‖_{X (t)} ≲ ε \\ ‖ w_{0} ‖_{X_{τ_{H}} (t)} ≲ ε \\ {‖ B (g, h) ‖}_{X (t)} ≲ ‖ g ‖_{X (t)} ‖ h ‖_{X (t)} \\ {‖ B (g, h) ‖}_{X_{τ_{H}} (t)} ≲ (‖ g ‖_{X (t)} + ‖ g ‖_{X_{τ_{H}} (t)}) (‖ h ‖_{X (t)} + ‖ h ‖_{X_{τ_{H}} (t)}) . \end{array}$ These imply that ${w_{m}}_{m = 0}^{\infty}$ is a Cauchy sequence in both $B C ((0, \infty); {\dot{B}}_{p, \infty}^{s (p)})$ and $B C ((0, \infty); {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ . Let us give a proof of the last inequality. Fix $τ_{0} \in (τ_{H} / 2, min (τ_{H}, 1 - n / (2 p)))$ . Firstly, we use (ii) of Lemma 2.1 to see $L^{r, \infty} ↪ {\dot{B}}_{p, \infty}^{s (p) + 2 τ_{0} - 1}$ with $r = n / (1 - τ_{0})$ . Thus, from Hölder’s inequality and (i) of Lemma 2.1, we have $\begin{array}{l} ‖ g h ‖_{{\dot{B}}_{p, \infty}^{s (p) + 2 τ_{0} - 1}} ≲ ‖ g ‖_{{\dot{B}}_{p, \infty}^{s (p) + τ_{0}}} ‖ h ‖_{{\dot{B}}_{p, \infty}^{s (p) + τ_{0}}} . \end{array}$ Combining this, (ii) of Lemma 3.4 and the mapping properties established in Section 2.3.1, one obtains $\begin{array}{l} {‖ B (g, h) ‖}_{X_{τ_{H}} (t)} ≲ ‖ g ‖_{X_{τ_{0}} (t)} ‖ h ‖_{X_{τ_{0}} (t)} . \end{array}$ Finally, the interpolation inequality $‖ g ‖_{X_{τ_{0}} (t)} ≲ ‖ g ‖_{X (t)}^{1 - τ_{0} / τ_{H}} ‖ g ‖_{X_{τ_{H}} (t)}^{τ_{0} / τ_{H}}$ and Young’s inequality yield the desired estimates.

We shall show that $w_{0} \in B C ((0, \infty); {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ . Fix $t \in (0, \infty)$ and take $ε \in (0, t / 2)$ . From (iii) of Lemma 3.4, it hold that for small $θ > 0$ $\begin{array}{l} {‖ w_{0} (t + ε) - w_{0} (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ ε^{θ / 2} {‖ e^{- t A} b ‖}_{{\dot{B}}_{p, \infty}^{s (p) + θ}} \end{array}$ and $\begin{array}{l} {‖ w_{0} (t) - w_{0} (t - ε) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ ε^{θ / 2} {‖ e^{- (t - ε)} b ‖}_{{\dot{B}}_{p, \infty}^{s (p) + θ}} ≲ ε^{θ / 2} {‖ e^{- t / 2 A} b ‖}_{{\dot{B}}_{p, \infty}^{s (p) + θ}} . \end{array}$ Using these estimates and (ii) of Lemma 3.4, we can show also the continuity $w_{0} \in B C ((0, \infty); {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ . On the other hand, since it holds that $\begin{array}{l} B (w_{0}, w_{0}) (t) - B (w_{0}, w_{0}) (t \pm ε) \\ = \int_{0}^{min (t, t \pm ε)} e^{- σ A} P \nabla [(w_{0} \otimes w_{0}) (t - σ) - (w_{0} \otimes w_{0}) (t \pm ε - σ)] d σ \\ + \int_{min (t, t \pm ε)}^{max (t, t \pm ε)} e^{- σ A} P \nabla (w_{0} \otimes w_{0}) (max (t, t \pm ε) - σ) d σ, \end{array}$ we see $w_{1} \in B C ((0, \infty), {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ from Lebesgue’s convergence theorem. Repeating this argument gives us $w_{m} \in B C ((0, \infty), {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ . Therefore, there exists $w \in B C ((0, \infty), {\dot{B}}_{p, \infty}^{s (p)} \cap {\dot{B}}_{p, 1}^{s (p) + τ_{H}})$ solving the integral equation $\begin{array}{l} w (t) = e^{- t A} b - B (w, w) (t) . \end{array}$

Next, we shall show that w, in fact, fulfills the differential equation in (E) in ${\dot{B}}_{p, \infty}^{s (p) - 2}$ . We take $t > 0$ and $t_{1}, t_{2} > 0$ such that $t_{1} < t < t_{2}$ . We decompose $\begin{array}{l} \frac{w (t_{2}) - w (t_{1})}{t_{2} - t_{1}} = \frac{e^{- (t_{2} - t_{1}) A} - 1}{t_{2} - t_{1}} w (t) - P \nabla (w \otimes w) (t) - (I + II + III), \end{array}$ where $\begin{array}{l} I : = \frac{e^{- (t_{2} - t_{1}) A} - 1}{t_{2} - t_{1}} (w (t) - w (t_{1})) \\ II : = \frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} e^{- (t_{2} - σ) A} P \nabla ((w \otimes w) (σ) - (w \otimes w) (t)) d σ \\ III : = \frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} (e^{- (t_{2} - σ) A} - 1) P \nabla (w \otimes w) (t) d σ . \end{array}$ It follows from Lemma 3.5 that $\begin{array}{l} {‖ \frac{e^{- (t_{2} - t_{1}) A} - 1}{t_{2} - t_{1}} w (t) + A w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - 2}} ≲ {(t_{2} - t_{1})}^{τ / 2} {‖ w (t) ‖}_{{\dot{B}}_{p, 1}^{s (p) + τ_{H}}} \to 0 as t_{2}, t_{1} \to t . \end{array}$ (iii) of Lemma 3.4 yields $\begin{array}{l} ‖ I ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}} ≲ {‖ w (t) - w (t_{1}) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} \to 0 as t_{2}, t_{1} \to t . \end{array}$ We can see the same convergence for the second part from mapping properties established above; $\begin{array}{l} ‖ II ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}} & ≲ \frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} {‖ w \otimes w (σ) - w \otimes w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - 1}} d σ \\ ≲ ‖ w ‖_{X (t_{2})} sup_{t_{1} ⩽ σ ⩽ t_{2}} {‖ w (σ) - w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} \to 0 \end{array}$ as $t_{2}, t_{1} \to t$ . Since $\begin{array}{l} ‖ III ‖_{{\dot{B}}_{p, \infty}^{s (p) - 2}} & ≲ \frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} {(t_{2} - σ)}^{τ_{H} / 2} {‖ P \nabla (w \otimes w) (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p) + τ_{H} - 2}} d σ \\ ≲ {(t_{2} - t_{1})}^{τ_{H} / 2} {‖ w \otimes w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p) + τ_{H} - 1}} \\ ≲ {(t_{2} - t_{1})}^{τ_{H} / 2} {‖ w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} {‖ w (t) ‖}_{{\dot{B}}_{p, 1}^{s (p) + τ_{H}}} \to 0 \end{array}$ as $t_{2}, t_{1} \to t$ , we obtain that $\begin{array}{l} \partial_{t} w (t) + A w (t) + P \nabla (w \otimes w) (t) = 0 in {\dot{B}}_{p, \infty}^{s (p) - 2} . \end{array}$ Now, let us define $u : = w + U \in B C ((0, \infty), {\dot{B}}_{p, \infty}^{s (p)})$ . Thus, u satisfies the differential equation $\begin{array}{l} \partial_{t} u (t) - Δ u (t) + P \nabla (u \otimes u) (t) = P f in {\dot{B}}_{p, \infty}^{s (p) - 2}, \end{array}$ and also fulfills $‖ u (t) - U ‖_{{\dot{B}}_{p, 1}^{s (p) + τ_{H}}} ≲ ε t^{- τ_{H} / 2} \to 0$ as $t \to \infty$ .

To end the proof of this part, we shall show the equivalence of convergences. We borrow an argument from Cannone and Karch [7]. Suppose that $u (t) \to U$ in ${\dot{B}}_{p, \infty}^{s (p)}$ as $t \to \infty$ . It is sufficient to verify that $B (w, w) (t)$ converges to zero. Dividing the integral into two parts, we obtain that for $δ \in (0, 1)$ , $\begin{array}{l} (16) & {‖ B (w, w) (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ \int_{0}^{δ} {(1 - σ)}^{- 1} {‖ w (t σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}}^{2} d σ + sup_{δ t ⩽ σ ⩽ t} {‖ w (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}}^{2} \to 0 \end{array}$ as $t \to \infty$ . Next, we assume that ${lim}_{t \to \infty} ‖ e^{- t A} b ‖_{{\dot{B}}_{p, \infty}^{s (p)}} = 0$ . It is enough to see that $B (w, w)$ tends to zero, again. Applying the inequality in (16), one can see that $\begin{array}{l} \underset{t \to \infty}{lim sup} {‖ B (w, w) (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ ε (1 + log \frac{1}{1 - δ}) \underset{t \to \infty}{lim sup} {‖ w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} . \end{array}$ As a consequence, $\begin{array}{l} \underset{t \to \infty}{lim sup} {‖ w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} ≲ ε (1 + log \frac{1}{1 - δ}) \underset{t \to \infty}{lim sup} {‖ w (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} . \end{array}$ Taking sufficiently small δ, we conclude that $u (t) \to U$ in ${\dot{B}}_{p, \infty}^{s (p)}$ as $t \to \infty$ .

(ii) Let us denote $M : = ‖ a - U ‖_{{\dot{B}}_{p, \infty}^{s}}$ and $\begin{array}{l} ‖ w ‖_{Y (t)} : = sup_{σ ⩽ t} {‖ w (σ) ‖}_{{\dot{B}}_{p, \infty}^{s}} . \end{array}$ Similarly as the estimate on ${\dot{B}}_{p, \infty}^{s (p)}$ , it follows from Lemmas 2.2, 3.4 and 3.6 that $\begin{array}{l} ‖ w_{0} ‖_{Y (t)} ≲ M and {‖ B (g, h) ‖}_{Y (t)} ≲ min (‖ g ‖_{X (t)} ‖ h ‖_{Y (t)}, ‖ g ‖_{Y (t)} ‖ h ‖_{X (t)}) . \end{array}$ Further, the continuity $w_{m} \in B C ((0, \infty), {\dot{B}}_{p, \infty}^{s})$ is also easily seen. Thus, we find that $w \in B C ((0, \infty), {\dot{B}}_{p, \infty}^{s})$ with $‖ w ‖_{Y (t)} ≲ M$ .

We claim that $\begin{array}{l} {‖ B (g, h) (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} \\ ≲ t^{- γ / 2} sup_{0 < σ ⩽ t / 2} min ({‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s}}, {‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}}) \\ (17) & + sup_{t / 2 ⩽ σ ⩽ t} min ({‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}}, {‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}}) . \end{array}$ We show this inequality after completing the proof of Theorem 1.2.

We shall complete the proof, assuming (17). Following Bjorland, Brandolese, Iftimie and Schonbek [3], we have from (17) $\begin{array}{l} W_{m + 1} (t) & : = sup_{T ⩾ t} {‖ w_{m + 1} (T) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} \\ ≲ sup_{T ⩾ t} (T^{- γ / 2} M + ε M T^{- γ / 2} + ε sup_{T / 2 ⩽ σ ⩽ T} {‖ w_{m} (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}}) \\ ≲ t^{- γ / 2} M + ε sup_{t / 2 ⩽ σ} {‖ w_{m} (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} . \end{array}$ Hence, it turns out that $W_{m + 1} (t) ≲ t^{- γ / 2} M + ε W_{m} (t / 2)$ for all $t \in (0, \infty)$ , and then one has that ${lim sup}_{m \to \infty} W_{m} (t) ≲ t^{- γ / 2} M$ for all $t \in (0, \infty)$ . For any $θ \in S$ , $\begin{array}{l} | ⟨ w (t), θ ⟩ | ⩽ \underset{m \to \infty}{lim sup} W_{m} (t) ‖ θ ‖_{{\dot{B}}_{p^{'}, 1}^{- (s (p) - τ_{L})}} . \end{array}$ By duality, we see that $‖ w (t) ‖_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} ≲ t^{- γ / 2} M$ .

We show the inequality (17). Following the argument in [3], we decompose $B (g, h) (t) = e^{- t A / 2} B (g, h) (t / 2) + B^{'} (g, h) (t)$ , where $\begin{array}{l} B^{'} (g, h) (t) : = \int_{t / 2}^{t} e^{- (t - σ) A} P \nabla (g \otimes h) (σ) d σ . \end{array}$ The first part is controlled by Lemmas 3.4 and 3.6; $\begin{array}{l} {‖ e^{- t A / 2} B (g, h) (t / 2) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} \\ ≲ t^{- γ / 2} {‖ B (g, h) (t / 2) ‖}_{{\dot{B}}_{p, \infty}^{s}} \\ ≲ t^{- γ / 2} sup_{0 < σ ⩽ t / 2} min ({‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s}}, {‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}}) . \end{array}$ For the latter part, Lemma 3.6 also gives $\begin{array}{l} {‖ B^{'} (g, h) (t) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} & ≲ sup_{t / 2 ⩽ σ ⩽ t} {‖ (g \otimes h) (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L} - 1}} \\ ≲ sup_{t / 2 ⩽ σ} min ({‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}}, {‖ g (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p) - τ_{L}}} {‖ h (σ) ‖}_{{\dot{B}}_{p, \infty}^{s (p)}}) . \end{array}$ Applying Lemma 2.2, we complete the proof of the claim (17).

Footnotes

Acknowledgements

The third author would like to thank Professor Toshiaki Hishida for several comments. The authors would like to thank Professors Kenta Kaneko, Hideo Kozono and Senjo Shimizu for showing us their preprint [10]. The authors would like to thank Professors Yoshio Tsutsumi and Yasunori Maekawa for letting us know about the second part of Remark . We would like to thank the referees for their careful reading of our manuscript as well as for her/his valuable suggestions. The works of the first and third authors were partially supported by JSPS, through “Program to Disseminate Tenure Tracking System”. The work of the second author was partially supported by JSPS, through Gand-in-Aid for Young Scientists (B) 17K14215. The work of the third author was partially supported by JSPS, through Grand-in-Aid for Young Scientists (B) 15K20919.

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Asymptotic stability of stationary Navier–Stokes flow in Besov spaces

Abstract

Keywords

1. Introduction

2.1. Function spaces

2.3.1. Helmholtz projection and fractional Laplacian

2.3.2. Resolvent operator

2.3.3. Composition operator

3. Resolvent estimates and a critical estimate for the semigroup

3.1. Resolvent estimates for the perturbed operator A

4.1. Construction of stationary solutions

4.2. Construction of non-stationary solutions and their asymptotic stability

Footnotes

Acknowledgements

References

3.1. Resolvent estimates for the perturbed operator $A$