Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

Abstract

This paper is concerned with two dual aspects of the regularity question for the Navier–Stokes equations. First, we prove a local-in-time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space $L^3$, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and Šverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if $(0, T^*)$ is a singular point, then

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{3}(B_{R}(0))}\geqq \gamma _{univ},\qquad R=O(\sqrt{T^*-t}). \end{aligned}$$

This result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely $L^{3,\infty }$ and the Besov space ${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }$, $p\in (3,\infty )$.

Appendices

Appendix A: Auxiliary Results

The first result is the classical existence of mild solutions for critical initial data $a\in L^3_\sigma ({\mathbb {R}}^3)$.

Proposition 15

([17]). There exists universal constants $\gamma ,\, K_0\in (0,\infty )$ such that the following holds true. For for all $u_{0,a}\in L^3_\sigma ({\mathbb {R}}^3)$ with $\Vert u_{0,a}\Vert _{L^3}\leqq \gamma$, there exists a unique smooth mild solution $a\in C([0,\infty );L^3)\cap L^\infty ((0,\infty );L^3)$ such that $a(\cdot ,0)=u_{0,a}$, $a\in C((0,\infty );W^{1,3}\cap BUC_\sigma )$ and

$$\begin{aligned}&\sup _{t\in (0,\infty )}\big (\Vert a(\cdot ,t)\Vert _{L^3}+t^{\frac{1}{8}}\Vert a(\cdot ,t)\Vert _{L^{4}}+t^\frac{1}{5}\Vert a(\cdot ,t)\Vert _{L^5}+t^\frac{1}{2}\Vert a(\cdot ,t)\Vert _{L^\infty }\nonumber \\&\quad +t^\frac{1}{2}\Vert \nabla a(\cdot ,t)\Vert _{L^3}\big ) \leqq K_0\Vert u_{0,a}\Vert _{L^3}. \end{aligned}$$

(96)

Moreover $a\in L^5({\mathbb {R}}^3\times (0,\infty ))$ and

$$\begin{aligned} \Vert a\Vert _{L^5({\mathbb {R}}^3\times (0,\infty ))}\leqq K_0\Vert u_{0,a}\Vert _{L^3}. \end{aligned}$$

(97)

The second result is an estimate for local energy solutions, so-called Lemarié-Rieusset solutions [33, Chapter 32 and 33]. Before stating this result, we give the definition of such solutions. See also [28] and [24, Definition 3.1].

Definition 16

(Local energy solutions). A pair (u, p) is called a local energy solution to (1) in ${\mathbb {R}}^3\times (0,\infty )$ with the initial data $u_0\in L^2_{uloc,\sigma }({\mathbb {R}}^3)$, $\sup _{|{{\bar{x}}}|\geqq R}\Vert u_0\Vert _{L^2(B_1({{\bar{x}}}))}{\mathop {\longrightarrow }\limits ^{R\rightarrow \infty }} 0$, if (u, p) satisfies the following conditions:

• (i) We have $u\in L^\infty _{loc}([0,T); L^2_{uloc,\sigma } ({\mathbb {R}}^3))$ and $p\in L^\frac{3}{2}_{loc} ({\mathbb {R}}^3\times (0,\infty ))$, and

  $$\begin{aligned} \sup _{{{\bar{x}}}\in {\mathbb {R}}^3} \int \nolimits _0^{T} \Vert \nabla u \Vert _{L^2 (B_1({{\bar{x}}})\cap {\mathbb {R}}^3)}^2 d t <\infty , \end{aligned}$$

  for all finite $T\in (0, \infty )$.

• (ii) The pair (u, p) is a solution to (1) in the sense of distributions.

• (iii) The function $t\mapsto \langle u(\cdot ,t), \varphi \rangle _{L^2({\mathbb {R}}^3)}$ belongs to C([0, T)) for any compactly supported $\varphi \in L^2({\mathbb {R}}^3)$. Moreover, for any compact set $K\subset {\mathbb {R}}^3$,

  $$\begin{aligned} \lim _{t\rightarrow 0} \Vert u(\cdot ,t) -u_0 \Vert _{L^2(K)} =0. \end{aligned}$$

  (98)

• (iv) The pair (u, p) satisfies the local energy inequality: for any $0\leqq \phi \in C^\infty _c({\mathbb {R}}^3\times (0,\infty ))$, for all $t\in (0,\infty )$,

  $$\begin{aligned} \begin{aligned}&\int \nolimits _{B_1(0)}|u(x,t)|^2\phi (x,t) \,\text {d}x+2\int \nolimits _{0}^t\int \nolimits _{{\mathbb {R}}^3}|\nabla u|^2\phi \,\text {d}x\,\text {d}s\\&\quad \leqq \ \int \nolimits _{0}^t\int \nolimits _{{\mathbb {R}}^3}|u|^2(\partial _t\phi +\Delta \phi )\,\text {d}x\,\text {d}s+\int \nolimits _{0}^t\int \nolimits _{{\mathbb {R}}^3}(|u|^2+2p)u\cdot \nabla \phi \,\text {d}x\,\text {d}s. \end{aligned} \end{aligned}$$

Notice that (98) enables one to transfer the mild decay of the initial data to the solution u.

Proposition 17

([24, 33, Lemma 3.1]). There exist two universal constants $c_1,\ K_1\in (0,\infty )$ such that for all $u_0\in L^2_{uloc}({\mathbb {R}}^3)$, for all local energy solution u to (1) with initial data $u_0$, we have

$$\begin{aligned}&\sup _{s\in (0,S_{lews})}\sup _{{{\bar{x}}}\in {\mathbb {R}}^3}\int \nolimits _{B_1({{\bar{x}}})}\frac{|u(x,s)|^2}{2}\, \,\text {d}x\nonumber \\&\qquad +\sup _{{{\bar{x}}}\in {\mathbb {R}}^3}\int \nolimits _0^{S_{lews}}\int \nolimits _{B_1({{\bar{x}}})}|\nabla u(x,s)|^2\, \,\text {d}x\, \,\text {d}s\leqq K_1M^2 \end{aligned}$$

(99)

where

$$\begin{aligned} S_{lews}:=c_1^2\min (M^{-4},1)\quad \text{ and }\quad M:=\Bigg (\sup _{{{\bar{x}}}\in {\mathbb {R}}^3}\int \nolimits _{B_1({{\bar{x}}})}|u_0(x)|^2\, \,\text {d}x\Bigg )^\frac{1}{2}<\infty . \end{aligned}$$

Following [23, 28, 33], if u is a local energy weak solution to (1) in the sense of Definition 16 and a is a mild solution to (1), then $v-a$ solves

$$\begin{aligned} \partial _{t}v-\Delta v+v\cdot \nabla v+ a\cdot \nabla v+v\cdot \nabla a+ \nabla q=0,\,\,\,\,\nabla \cdot v=0,\quad x\in {\mathbb {R}}^3,\ t>0 \end{aligned}$$

in the sense of distributions, and we have the following global representation formula for the pressure: for all ${{\bar{x}}}\in {\mathbb {R}}^3$, for all $(x,t)\in B_\frac{3}{2}({{\bar{x}}})\times (0,T)$,

$$\begin{aligned} \begin{aligned} q(x,t)&=\ -\frac{1}{3}|v(x,t)|^2-\int \nolimits _{B_2({{\bar{x}}})}\nabla ^2N(x-y):(v\otimes v+a\otimes v+v\otimes a)\,\hbox {d}y\\&\quad -\int \nolimits _{{\mathbb {R}}^3\setminus B_2({{\bar{x}}})}(\nabla ^2N(x-y)-\nabla ^2N({{\bar{x}}}-y)):(v\otimes v+a\otimes v+v\otimes a)\,\hbox {d}y. \end{aligned} \end{aligned}$$

(100)

Here $N(x)=-\frac{1}{4\pi |x|}$. Notice that (100) provides a proof of estimate (84) in the case $a=0$. With ${{\bar{x}}}=0$, we have for all $(x,t)\in B_\frac{3}{2}(0)\times (0,S_{lews})$,

$$\begin{aligned} q(x,t):=-\frac{1}{3}|v(x,t)|^2+q_{loc}(x,t)+q_{nonloc}(x,t), \end{aligned}$$

with

$$\begin{aligned}&\Vert q_{loc}(\cdot ,t)\Vert _{L^\frac{5}{3}_{x,t}(B_\frac{3}{2}(0)\times S_{lews})}\leqq \ C\Vert v^2\Vert _{L^\frac{5}{3}_{t,x}}\leqq C(1+M^2),\\&\Vert q_{nonloc}(\cdot ,t)\Vert _{L^\infty (B_\frac{3}{2}(0))}\leqq \ C\sum _{\xi \in {\mathbb {Z}}^3}\frac{1}{1+|\xi |^4}\int \nolimits _{B_1(\xi )}|v|^2\leqq C(1+M^2), \end{aligned}$$

where we used Calderón-Zygmund estimates for the first bound.

We conclude this appendix by a useful local representation formula for the pressure. The following lemma is well known, see for instance Caffarelli, Kohn and Nirenberg’s paper [9] (specifically p.782 of [9]).

Lemma 18

Suppose that $p\in L^{1}(B_{2}(0))$ and $V_{ij}\in L^{q}(B_{2}(0))$ for some $q>1$. Furthermore, suppose

$$\begin{aligned} -\Delta p= \partial _{i}\partial _{j} V_{ij} \end{aligned}$$

(101)

in $B_{2}(0)$ in a distributional sense. Then for $\varphi \in C_{0}^{\infty }(B_{1}(0))$ we have

$$\begin{aligned} \varphi p= & {} {\mathcal {R}}_{i}{\mathcal {R}}_{j}(\varphi V_{ij}) \nonumber \\&-N*((\partial _{i}\partial _{j}\varphi )V_{ij})-2\partial _{j}N*((\partial _{i}\varphi )V_{ij})-N*(p\Delta \varphi )-2\partial _{j}N*((\partial _{j}\varphi )p).\nonumber \\ \end{aligned}$$

(102)

Appendix B: The Case of $\mathbf{L }^{{\varvec{3,\infty }}}$ Initial Data

1.1 B.1. Preliminary Material

Given a measurable subset $\Omega \subseteq {\mathbb {R}}^{d}$, let us define the Lorentz spaces. For a measurable function $f:\Omega \rightarrow {\mathbb {R}}$ define:

$$\begin{aligned} d_{f,\Omega }(\alpha ):=\mu (\{x\in \Omega : |f(x)|>\alpha \}), \end{aligned}$$

(103)

where $\mu$ denotes the Lebesgue measure. The Lorentz space $L^{p,q}(\Omega )$, with $p\in [1,\infty [$, $q\in [1,\infty ]$, is the set of all measurable functions g on $\Omega$ such that the quasinorm $\Vert g\Vert _{L^{p,q}(\Omega )}$ is finite. Here,

$$\begin{aligned} \Vert g\Vert _{L^{p,q}(\Omega )}:= & {} \Big (p\int \nolimits _{0}^{\infty }\alpha ^{q}d_{g,\Omega }(\alpha )^{\frac{q}{p}}\frac{d\alpha }{\alpha }\Big )^{\frac{1}{q}}, \end{aligned}$$

(104)

$$\begin{aligned} \Vert g\Vert _{L^{p,\infty }(\Omega )}:= & {} \sup _{\alpha >0}\alpha d_{g,\Omega }(\alpha )^{\frac{1}{p}}. \end{aligned}$$

(105)

It is known there exists a norm, which is equivalent to the quasinorm defined above, for which $L^{p,q}(\Omega )$ is a Banach space. For $p\in [1,\infty )$ and $1\leqq q_{1}< q_{2}\leqq \infty$, we have the following continuous embeddings:

$$\begin{aligned} L^{p,q_1}(\Omega ) \hookrightarrow L^{p,q_2}(\Omega ), \end{aligned}$$

(106)

and the inclusion is known to be strict.

Let X be a Banach space with norm $\Vert \cdot \Vert _{X}$, $a<b$, $p\in [1,\infty )$ and $q\in [1,\infty ]$. Then $L^{p,q}(a,b;X)$ will denote the space of strongly measurable X-valued functions f(t) on (a, b) such that

$$\begin{aligned} \Vert f\Vert _{L^{p,q}(a,b; X)}:= \big \Vert \Vert f(t)\Vert _{X}\big \Vert _{L^{p,q}(a,b)}<\infty . \end{aligned}$$

(107)

In particular, if $1\leqq q_{1}< q_{2}\leqq \infty$, we have the following continuous embeddings:

$$\begin{aligned} L^{p,q_1}(a,b; X) \hookrightarrow L^{p,q_2}(a,b; X), \end{aligned}$$

(108)

and the inclusion is known to be strict.

Let us recall a known proposition known as ‘O’Neil’s convolution inequality’ (Theorem 2.6 of O’Neil’s paper [43]).

Proposition 19

Suppose $1< p_{1}, p_{2}, r<\infty$ and $1\leqq q_{1}, q_{2}, s\leqq \infty$ are such that

$$\begin{aligned} \frac{1}{r}+1=\frac{1}{p_1}+\frac{1}{p_{2}} \end{aligned}$$

(109)

and

$$\begin{aligned} \frac{1}{q_1}+\frac{1}{q_{2}}\geqq \frac{1}{s}. \end{aligned}$$

(110)

Suppose that

$$\begin{aligned} f\in L^{p_1,q_1}({\mathbb {R}}^{d})\,\,\text {and}\,\,g\in L^{p_2,q_2}({\mathbb {R}}^{d}). \end{aligned}$$

(111)

Then

$$\begin{aligned}&f*g \in L^{r,s}({\mathbb {R}}^d)\,\,\mathrm{with} \end{aligned}$$

(112)

$$\begin{aligned}&\Vert f*g \Vert _{L^{r,s}({\mathbb {R}}^d)}\leqq 3r \Vert f\Vert _{L^{p_1,q_1}({\mathbb {R}}^d)} \Vert g\Vert _{L^{p_2,q_2}({\mathbb {R}}^d)}. \end{aligned}$$

(113)

We will use an inequality that we will refer to as ‘Hunt’s inequality’. The statement below and proof can be found in Hunt’s paper [22] (Theorem 4.5, p. 271 of [22]).

Proposition 20

Suppose that $0<p,q,r\leqq \infty$ and $0<s_1,s_2\leqq \infty$. Furthermore, suppose that p, q, r, $s_1$ and $s_2$ satisfy the following relations:

$$\begin{aligned} \frac{1}{p}+\frac{1}{q}=\frac{1}{r} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{s_1}+\frac{1}{s_2}=\frac{1}{s}. \end{aligned}$$

Then the assumption that $f\in L^{p,s_1}(\Omega )$ and $g\in L^{q,s_2}(\Omega )$ implies that $fg \in L^{r,s}(\Omega )$, with the estimate

$$\begin{aligned} \Vert fg\Vert _{L^{r,s}(\Omega )}\leqq C(p,q,s_1,s_2)\Vert f\Vert _{L^{p,s_1}(\Omega )}\Vert g\Vert _{L^{q,s_2}(\Omega )}. \end{aligned}$$

(114)

As a result of the above Propositions, we have the following estimates with $B_{R}(0)\subset {\mathbb {R}}^3$ (which we will frequently use):

$$\begin{aligned}&\int \nolimits _{Q_{R}(0)} |\nabla f| |f| |g| \,\text {d}x\,\text {d}t \nonumber \\&\quad \leqq \ \Vert g\Vert _{L^{\infty }_{t}L^{3,\infty }_{x}(Q_{R}(0))}\Bigg ( \Vert \nabla f\Vert _{L^{2}(Q_{R}(0))}^2+ \frac{1}{R}\int \nolimits _{-R^2}^{0}\Vert \nabla f\Vert _{L^{2}(B_{R}(0))}\Vert f\Vert _{L^{2}(B_{R}(0))}\,\hbox {d}t\Bigg ), \end{aligned}$$

(115)

$$\begin{aligned}&\int \nolimits _{Q_{R}(0)} |f|^2 |g|\,\hbox {d}x\,\hbox {d}t\leqq R^{\frac{2}{3}}\Vert g\Vert _{L^{5,\infty }_{t}L^{5}_{x}(Q_{R}(0))} \Vert f\Vert _{L^{3}(Q_{R}(0))}^2. \end{aligned}$$

(116)

The first estimate is stated and proven in [11], for example. Now, we state known results for the Navier–Stokes equations with initial data in $L^{3,\infty }({\mathbb {R}}^3).$ We refer the reader to [41] and [45].

Proposition 21

There exists universal constants $\gamma ,\, K_0'\in (0,\infty )$ such that the following holds true. For all $u_{0,a}\in L^{3,\infty }_\sigma ({\mathbb {R}}^3)$, $\Vert u_{0,a}\Vert _{L^{3,\infty }}\leqq \gamma$, there exists a smooth mild solution $a\in C_{w^{*}}([0,\infty );L^{3,\infty })\cap L^\infty ((0,\infty );L^{3,\infty })$ such that $a(\cdot ,0)=u_{0,a}$ and

$$\begin{aligned}&\sup _{t\in (0,\infty )}\big (\Vert a(\cdot ,t)\Vert _{L^{3,\infty }}+t^\frac{1}{8}\Vert a(\cdot ,t)\Vert _{L^4}+t^\frac{1}{5}\Vert a(\cdot ,t)\Vert _{L^5}+t^\frac{1}{2}\Vert a(\cdot ,t)\Vert _{L^\infty }) \nonumber \\&\quad \leqq K_0'\Vert u_0\Vert _{L^{3,\infty }}. \end{aligned}$$

(117)

The mild solution is unique in the class of solutions with small enough $L^\infty (0,\infty ; L^{3,\infty })$ norm.

1.2 B.2. $L^{3,\infty }$ Initial Data: Section 2

We briefly describe the changes required for Section 2. With the above Proposition in mind concerning mild solutions, in Section 2 we can no longer assume a is in $L^{5}_{x,t}$. Instead, we assume that

$$\begin{aligned} \Vert a\Vert _{L^{\infty }(-1,0; L^{3,\infty }(B_{1}(0)))}+\Vert a\Vert _{L^{5,\infty }(-1,0; L^{5}(B_{1}(0)))}\leqq \varepsilon _{*}. \end{aligned}$$

(118)

The first adjustment regards the estimate of the pressure (Lemma 4). In particular, Hunt’s inequality can be used to show that the second and last term in (37) can be replaced by

$$\begin{aligned} C_{2}r^{\frac{1-\delta }{2}}\Bigg (\int \nolimits _{Q_{2r}(0,0)}|v|^3 \,\text {d}x\,\text {d}s\Bigg )^{\frac{1}{2}}\Vert a\Vert _{L^{5,\infty }_{t}L^{5}_{x}(Q_{1}(0,0))}^{\frac{3}{2}} \end{aligned}$$

and

$$\begin{aligned} C_{2}r^{\frac{44-5\delta }{10}}\rho ^{\frac{-39}{10}}\Bigg (\int \nolimits _{Q_{2r}(0,0)}|v|^3 \,\text {d}x\,\text {d}s\Bigg )^{\frac{1}{2}}\Vert a\Vert _{L^{5,\infty }_{t}L^{5}_{x}(Q_{1}(0,0))}^{\frac{3}{2}}. \end{aligned}$$

Now we proceed to the adjustments needed for the proof of Theorem 4. In Step 2 and Step 3 the only adjustment is to make extensive use of (115)–(116). In Step 4 we take the adjustment of Lemma 4 into account. Moreover, when estimating the pressure we have to use Hunt’s inequality to estimate $J_{4}$. In particular, this gives

$$\begin{aligned} J_{4}\leqq C\left( \sum _{k=2}^{n-1}r_{k+1}^{-4+\frac{9}{10}}\big (\varepsilon _*^\frac{2}{3}r_{k}^{3-\frac{2}{3}\delta }\big )^\frac{1}{2}\right) ^\frac{3}{2}\Vert a\Vert _{L^{5,\infty }_{t}L^{5}_{x}(Q_{1}(0,0))}^\frac{3}{2}r_{n+1}^\frac{7}{5} . \end{aligned}$$

1.3 B.3. $L^{3,\infty }$ Initial Data: Section 3

As in Section 2 we can no longer assume a is in $L^{5}_{x,t}$. Instead, we assume that

$$\begin{aligned} \sup _{0<s<\infty } s^{\frac{1}{5}}\Vert a(\cdot ,s)\Vert _{L_{5}({\mathbb {R}}^3)}\leqq \varepsilon _{**}, \end{aligned}$$

(119)

where $\varepsilon _{**}>0$ is some small universal constant. The only difference in Section 3 regards (63). In particular, we can use Young’s inequality in space, followed by O’Neil’s convolution inequality in time and finally Hunt’s inequality in time to see that the following holds: for $1<q<\infty$,

$$\begin{aligned} \Vert L(\nabla \cdot (a\otimes b))\Vert _{L^{q}_{x,t}({\mathbb {R}}^3\times (0,T))}&\leqq \ C(q)\big \Vert \Vert a(\cdot ,s)\Vert _{L^{5}_{x}}\Vert b(\cdot , s)\Vert _{L^{q}_{x}}\big \Vert _{L^{\frac{5q}{5+q},q}(0,T)}\\&\leqq \ C'(q)\sup _{0<s<\infty } s^{\frac{1}{5}}\Vert a(\cdot ,s)\Vert _{L^{5}({\mathbb {R}}^3)} \Vert b\Vert _{L^{q}_{x,t}({\mathbb {R}}^3\times (0,T))}. \end{aligned}$$

1.4 B.4. $L^{3,\infty }$ Initial Data: Section 4

First (86) must be adjusted using Hunt’s inequality as follows:

$$\begin{aligned} \begin{aligned} \Vert {{\tilde{u}}}_{0,a}\Vert _{L^{3,\infty }(B_{2}(0)\setminus B_{1}(0))}&\leqq \ C \Vert \nabla {{\tilde{u}}}_{0,a} \Vert _{L^2(B_2(0)\setminus B_1(0))}\\&\leqq \ C(\chi ) \Vert u_0\Vert _{L^2(B_{2}(0))}\leqq C(\chi )\Vert u_{0}\Vert _{L^{3,\infty }(B_{2}(0))}. \end{aligned} \end{aligned}$$

(120)

Then in (93), we must instead make use of (115)–(116).

Appendix C: The Case of Besov Initial Data

1.1 C.1. Preliminaries

Let $d, m \in {\mathbb {N}}\setminus \{0\}$. We begin by recalling the definition of the homogeneous Besov spaces$\dot{B}^s_{p,q}({\mathbb {R}}^d;{\mathbb {R}}^m)$. There exists a non-negative radial function $\varphi \in C^\infty ({\mathbb {R}}^d)$ supported on the annulus $\{ \xi \in {\mathbb {R}}^d : 3/4 \leqq |\xi | \leqq 8/3 \}$ such that

$$\begin{aligned} \sum _{j \in {\mathbb {Z}}} \varphi (2^{-j} \xi ) = 1, \quad \xi \in {\mathbb {R}}^3 \setminus \{0\}. \end{aligned}$$

(121)

The homogeneous Littlewood-Paley projectors ${{\dot{\Delta }}}_j$ are defined by

$$\begin{aligned} {{\dot{\Delta }}}_j f = \varphi (2^{-j} D) f, \quad j \in {\mathbb {Z}}, \end{aligned}$$

(122)

for all tempered distributions f on ${\mathbb {R}}^d$ with values in ${\mathbb {R}}^m$. The notation $\varphi (2^{-j}D) f$ denotes convolution with the inverse Fourier transform of $\varphi (2^{-j}\cdot )$ with f.

Let $p,q \in [1,\infty ]$ and $s \in (-\infty ,d/p)$.⁴ The homogeneous Besov space $\dot{B}^s_{p,q}({\mathbb {R}}^d;{\mathbb {R}}^m)$ consists of all tempered distributions f on ${\mathbb {R}}^d$ with values in ${\mathbb {R}}^m$ satisfying

$$\begin{aligned} \Vert {f}\Vert _{\dot{B}^s_{p,q}({\mathbb {R}}^d;{\mathbb {R}}^m)} :=\Big (\sum _{j\in {\mathbb {Z}}} \big ( 2^{js}\Vert {\dot{\Delta }}_{j} f\Vert _{L^{p}}\big )^{q}\Big )^{\frac{1}{q}}. \end{aligned}$$

(123)

and such that $\sum _{j \in {\mathbb {Z}}} {{\dot{\Delta }}}_j f$ converges to f in the sense of tempered distributions on ${\mathbb {R}}^d$ with values in ${\mathbb {R}}^m$. In this range of indices, $\dot{B}^{s}_{p,q}({\mathbb {R}}^d;{\mathbb {R}}^m)$ is a Banach space. When $s \geqq 3/p$ and $q > 1$, the spaces must be considered modulo polynomials. Note that other reasonable choices of the function $\varphi$ defining ${{\dot{\Delta }}}_j$ lead to equivalent norms.

We now recall a particularly useful property of Besov spaces, that is, their characterization in terms of the heat kernel. For all $s \in (-\infty ,0)$, there exists a constant $c := c(s) > 0$ such that for all tempered distributions f on ${\mathbb {R}}^3$,

$$\begin{aligned} c^{-1} \sup _{t> 0} t^{-\frac{s}{2}} \Vert e^{t\Delta } f\Vert _{L^p({\mathbb {R}}^3)} \leqq \Vert {f}\Vert _{\dot{B}^s_{p,\infty }({\mathbb {R}}^3)} \leqq c \sup _{t > 0} t^{-\frac{s}{2}}\Vert {e^{t\Delta } f}\Vert _{L^p({\mathbb {R}}^3)}. \end{aligned}$$

(124)

Let $\Omega \subset {\mathbb {R}}^3$ be a domain with sufficiently smooth boundary. We say $u\in {\dot{B}}^{s}_{p,q}(\Omega )$ if

• (*) (extension property) there exists $E(u)\in {\dot{B}}^{s}_{p,q}({\mathbb {R}}^3)$ such that $E(u)=u$ on $\Omega$ as distributions.

Then

$$\begin{aligned} \Vert u\Vert _{{\dot{B}}^{s}_{p,q}(\Omega )}:=\inf {\{\Vert E(u)\Vert _{{\dot{B}}^{s}_{p,q}({\mathbb {R}}^3)}:\,\,\,\,\,E(u)\,\,\,\,\,\text {satisfies}\,\,\,\,\,(*)\}}. \end{aligned}$$

In what follows, we will mostly use just one feature of the definition of Besov spaces on bounded domains:

$$\begin{aligned}&u_{0}\in {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(B_{2}(0)),\,\,\,\,\,\,\varphi \in C_{0}^{\infty }(B_{2}(0)) \Rightarrow \Vert \varphi u_{0}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)} \nonumber \\&\quad \leqq C(\varphi ) \Vert u_{0}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(B_{2}(0))}. \end{aligned}$$

(125)

The proof of this uses the definition of Besov spaces on bounded domains and the fact that for $\varphi$ in the Schwartz class

$$\begin{aligned} \Vert f\varphi \Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)}\leqq C(\varphi )\Vert f\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)}. \end{aligned}$$

The proof of this is along the lines of Proposition 2.3 of [36].

We will also make use of a decomposition result for Homogeneous Besov spaces. The statement without (130) can be found in [2]; see also [4].

Lemma 22

Let $p \in (3,\infty )$. There exist $\gamma _1, \gamma _2 > 0$, and $C > 0$, each depending only on p, such that for each divergence-free vector field $g \in {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }\cap L^{2}({\mathbb {R}}^3)$ and $N>0$, there exist divergence-free vector fields ${\bar{g}}^{N}\in {\dot{B}}^{-1+\delta _{2}}_{\infty ,\infty }({\mathbb {R}}^3)\cap {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)\cap L^{2}({\mathbb {R}}^3)$ and ${\widetilde{g}}^{N}\in L^2({\mathbb {R}}^3)\cap {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)$ with the following properties:

$$\begin{aligned} g= & {} {\widetilde{g}}^{N} + {\bar{g}}^{N}, \end{aligned}$$

(126)

$$\begin{aligned} \Vert {\widetilde{g}}^{N}\Vert _{L^2({\mathbb {R}}^3)}\leqq & {} C N^{-\gamma _{2}} \Vert {g}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }}, \end{aligned}$$

(127)

$$\begin{aligned} \Vert {\bar{g}}^{N}\Vert _{{\dot{B}}^{-1+\delta _{2}}_{\infty ,\infty }({\mathbb {R}}^3)}\leqq & {} C N^{\gamma _{1}} \Vert {g}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }}. \end{aligned}$$

(128)

Furthermore,

$$\begin{aligned} \Vert {\widetilde{g}}^{N}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)}, \Vert {\bar{g}}^{N}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)}\leqq C \Vert {g}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }} \end{aligned}$$

(129)

and

$$\begin{aligned} \Vert {\widetilde{g}}^{N}\Vert _{L^{2}({\mathbb {R}}^3)}, \Vert {\bar{g}}^{N}\Vert _{L^{2}({\mathbb {R}}^3)}\leqq C \Vert {g}\Vert _{L^{2}}. \end{aligned}$$

(130)

Finally, we state known results for the Navier–Stokes equations with data in ${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3).$ We refer the reader to [45], for example.

Proposition 23

Let $S_{mild}\in (0,\infty )$ and $p\in (3,\infty )$. There exists two constants $\gamma (p)\in (0,\infty )$ and $K_0''(p)\in (0,\infty )$ such that the following holds true. For all divergence-free $u_{0,a}\in {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)$,

$$\begin{aligned} \sup _{0<t<S_{mild}} t^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert e^{t\Delta }u_{0,a}\Vert _{L^{p}}\leqq \gamma (p), \end{aligned}$$

there exists a smooth mild solution $a\in C_{w^{*}}([0,S_{mild});{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty })\cap L^\infty ((0,S_{mild});{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty })$ such that $a(\cdot ,0)=u_{0,a}$ and

$$\begin{aligned}&\sup _{t\in (0,S_{mild})}\big (\Vert a(\cdot ,t)\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }}+ t^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert a(\cdot ,t)\Vert _{L^{p}}+t^\frac{1}{2}\Vert a(\cdot ,t)\Vert _{L^\infty }\big ) \nonumber \\&\quad \leqq K_0''(p)\sup _{t\in (0,S_{mild})} t^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert e^{t\Delta }u_{0,a}\Vert _{L^{p}} . \end{aligned}$$

(131)

The mild solution is unique in the class of solutions with sufficiently small

$$\begin{aligned} \sup _{t\in (0,S_{mild})}t^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert a(\cdot ,t)\Vert _{L^{p}} \end{aligned}$$

norm.

1.2 C.2. Besov Initial Data: Section 3

In this section we should now assume

$$\begin{aligned} \sup _{0<s<T} s^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert a(\cdot ,s)\Vert _{L^{p}({\mathbb {R}}^3)}\leqq \varepsilon _{**}, \end{aligned}$$

where $\varepsilon _{**}>0$ is some small universal constant. With this adjustment, the arguments in Section 3 are the same as in the case of $L^{3,\infty }$ initial data.

1.3 C.3. Besov Initial Data: Section 4

1.3.1 C.3.1. The Extension Operator

Throughout this part we assume

$$\begin{aligned} \Vert u_{0}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(B_{2}(0))}\leqq \gamma _{univ}\,\,\,\,\,\,\,\text {and}\,\,\,\,\,\,\,\, \sup _{{\bar{x}}\in \mathbb {R}^3} \Vert u_{0}\Vert _{L^{2}(B_{2}({\bar{x}}))}\leqq M. \end{aligned}$$

For convenience, we assume without loss of generality that $p\in (6,\infty )$. Let $\chi \in C^\infty _c({\mathbb {R}}^3)$ be a cut-off function such that

$$\begin{aligned} 0\leqq \chi \leqq 1,\quad {{\,\mathrm{supp}\,}}\chi \subset B_2(0),\quad \chi =1\ \text{ on }\ B_\frac{3}{2}(0),\quad |\nabla \chi |\leqq K_3, \end{aligned}$$

where $K_3\in (0,\infty )$ is a universal constant. From the preliminaries we have

$$\begin{aligned} \Vert \chi u_{0}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)}\leqq C(\chi )\gamma _{univ}\leqq \gamma {'}_{univ}. \end{aligned}$$

(132)

Obviously,

$$\begin{aligned} \Vert \chi u_{0}\Vert _{L^{2}({\mathbb {R}}^3)}\leqq C\Vert u_{0}\Vert _{L^{2}(B_{2}(0))}. \end{aligned}$$

Then, we introduce ${{\tilde{u}}}_{0,a}$ given by Bogovskii’s lemma, such that

$$\begin{aligned} \begin{aligned}&\nabla \cdot {{\tilde{u}}}_{0,a}=u_0\cdot \nabla \chi ,\quad {{\tilde{u}}}_{0,a}=0\ \text{ on }\ \partial (B_2(0)\setminus B_\frac{3}{2}(0)),\\&\Vert {{\tilde{u}}}_{0,a}\Vert _{L^6(B_{2}(0)\setminus B_{\frac{3}{2}}(0)))}\leqq K_4\Vert u_0\cdot \nabla \chi \Vert _{L^2(B_2(0)\setminus B_{\frac{3}{2}}(0))}\leqq K_3K_4\Vert u_0\Vert _{L^{2}(B_{2}(0))},\\&\Vert {{\tilde{u}}}_{0,a}\Vert _{L^2(B_{2}(0)\setminus B_{\frac{3}{2}}(0))}\leqq K_4\Vert u_0\cdot \nabla \chi \Vert _{L^2(B_2(0)\setminus B_\frac{3}{2}(0))}\leqq K_3K_4\Vert u_0\Vert _{L^2(B_{2}(0))}, \end{aligned} \end{aligned}$$

(133)

where $K_4\in (0,\infty )$ is a universal constant. We extend ${{\tilde{u}}}_{0,a}$ by 0 and let

$$\begin{aligned} u_{0,a}=u_0\chi -{{\tilde{u}}}_{0,a}. \end{aligned}$$

Clearly,

$$\begin{aligned} \Vert u_{0,a}\Vert _{L_{2}({\mathbb {R}}^3)}\leqq (C+K_{3}K_{4})\Vert u_{0}\Vert _{L^{2}(B_2(0))}. \end{aligned}$$

(134)

Since ${\tilde{u}}_{0,a}$ has compact support and $L_{3}\hookrightarrow {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }$ we have

$$\begin{aligned} \Vert u_{0,a}\Vert _{{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }({\mathbb {R}}^3)}\leqq \gamma '_{univ}+ K_{3}K_{4}\Vert u_{0}\Vert _{L^{2}(B_{2}(0))}. \end{aligned}$$

(135)

Using (132)–(133), together with the heat flow characterisation of Besov spaces gives

$$\begin{aligned} \sup \limits _{0<t<T} t^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert e^{t\Delta } u_{0,a}\Vert _{L^{p}({\mathbb {R}}^3)}\leqq C(\chi )\gamma _{univ}+ K_{3}K_{4} MT^{\frac{1}{4}}. \end{aligned}$$

(136)

Using this and Proposition 23, there exists ${\hat{T}}(M,\gamma )$ and a mild solution $a(\cdot , u_{0,a})$ associated to $u_{0,a}$ on ${\mathbb {R}}^3 \times (0, {\hat{T}}(M,\gamma ))$. Furthermore,

$$\begin{aligned}&\sup \limits _{s\in (0,{\hat{T}})}\big ( s^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert a(\cdot ,s)\Vert _{L_{p}}+ s^{\frac{1}{2}}\Vert a(\cdot ,s)\Vert _{L_{\infty }}\big ) \\&\quad \leqq K_{0}''(p)\sup \limits _{s\in (0,{\hat{T}})} s^{\frac{1}{2}\left( 1-\frac{3}{p}\right) }\Vert e^{s\Delta } u_{0,a}\Vert _{L^{p}({\mathbb {R}}^3)}. \end{aligned}$$

Moreover, (134)–(136) and Theorem 3.1 of [5] imply that for $t\in (0, {\hat{T}}(M,\gamma ))$:

$$\begin{aligned} \Vert a(\cdot ,t)-e^{t\Delta } u_{0,a}\Vert _{L^{2}({\mathbb {R}}^3)}^2+ \int \nolimits _{0}^{t}\int \nolimits _{{\mathbb {R}}^3} |\nabla (a-e^{t\Delta } u_{0,a})|^2 \,\text {d}x\,\text {d}t'\leqq C(M, {\hat{T}},p) t^{\frac{1}{p-2}}.\nonumber \\ \end{aligned}$$

(137)

1.3.2 C.3.2. Local Decay Estimates Near the Initial Time

Now clearly $v=u-a$ has zero initial data locally on the ball $B_{\frac{3}{2}}(0)$. We next wish to show that for $t\in (0, \min (1,{\hat{T}},S_{lews}))$ we have

$$\begin{aligned} \Vert v(\cdot ,t)\Vert _{L_{2}(B_{1}(0))}^2+ \int \nolimits _{0}^{t}\int \nolimits _{B_{1}(0)} |\nabla v|^2 \,\text {d}x\,\text {d}t' \leqq C(M, {\hat{T}}, \gamma _{univ}) t^{\nu (p)}. \end{aligned}$$

for some $\nu (p)>0$. With (137) in mind, it is sufficient to show that for $t\in (0, \min ( 1, S_{lews}) )$:

$$\begin{aligned}&\Vert u(\cdot ,t)-e^{t\Delta }u_{0,a}\Vert _{L_{2}(B_{1}(0))}^2+ \int \nolimits _{0}^{t}\int \nolimits _{B_{1}(0)} |\nabla (u-e^{t'\Delta }u_{0,a})|^2 \,\text {d}x\,\text {d}t' \\&\quad \leqq C(M, {\hat{T}}, \gamma _{univ}) t^{\nu (p)}. \end{aligned}$$

In order to show this, we use splitting arguments inspired by the work of Cálderón [10]. The arguments we present here closely follow those presented in [5, 23] and [2]. According to Lemma 22 and (134)–(135), we can split $u_{0,a}$ into two divergence-free pieces:

$$\begin{aligned} u_{0,a}= & {} \widetilde{u_{0,a}}^{N}+\overline{u_{0,a}}^{N} \end{aligned}$$

(138)

$$\begin{aligned} \Vert \widetilde{u_{0,a}}^{N}\Vert _{L^{2}}\leqq & {} C(M,\gamma )N^{-\gamma _{2}} \end{aligned}$$

(139)

$$\begin{aligned} \Vert \overline{u_{0,a}}^{N}\Vert _{{\dot{B}}^{-1+\delta _{2}}_{\infty ,\infty }}\leqq & {} C(M,\gamma ) N^{\gamma _{1}} \end{aligned}$$

(140)

$$\begin{aligned} \Vert \overline{u_{0,a}}^{N}\Vert _{L^{2}}+\Vert \widetilde{u_{0,a}}^{N}\Vert _{L^{2}}\leqq & {} C(M). \end{aligned}$$

(141)

Define $u^{N}:= u- e^{t\Delta }\overline{u_{0,a}}^{N}$. Then

$$\begin{aligned}&\partial _{t}u^{N}-\Delta u^{N}+ u^{N}\cdot \nabla u^{N}+ e^{t\Delta }\overline{u_{0,a}}^{N}\cdot \nabla u^{N}+u^{N}\cdot \nabla e^{t\Delta }\overline{u_{0,a}}^{N}\\&\quad + \nabla p= -e^{t\Delta }\overline{u_{0,a}}^{N}\cdot \nabla e^{t\Delta }\overline{u_{0,a}}^{N},\\&\nabla \cdot u^{N}=0 \\&u^{N}(x,0)= \widetilde{u_{0,a}}^{N}\,\,\,\,\,\text {in}\,\,\,\,\,\,\, B_{\frac{3}{2}}(0). \end{aligned}$$

We remark that p is the pressure associated to the original local energy solution u. From Proposition 17 and (141), we have

$$\begin{aligned}&\sup _{s\in (0,S_{lews})}\sup _{{{\bar{x}}}\in {\mathbb {R}}^3}\int \nolimits _{B_1({{\bar{x}}})}\frac{|u^{N}(x,s)|^2}{2}\, \,\hbox {d}x\nonumber \\&\qquad +\sup _{{{\bar{x}}}\in {\mathbb {R}}^3}\int \nolimits _0^{S_{lews}}\int \nolimits _{B_1({{\bar{x}}})}|\nabla u^{N}(x,s)|^2\, \,\hbox {d}x\,\hbox {d}s\leqq C(M). \end{aligned}$$

(142)

Let $\phi \in C^\infty _c({\mathbb {R}}^3)$ such that $0\leqq \phi \leqq 1$, ${{\,\mathrm{supp}\,}}\phi \subset B_\frac{3}{2}(0)$, $\phi =1$ on $B_1(0)$, $|\nabla (\phi ^2)|\leqq K_5$ and $|\Delta (\phi ^2)|\leqq K_5'$ where $K_5,\ K_5'\in (0,\infty )$ are a universal constants.

For $t\in (0,\min (1, S_{lews}))$ we have:

$$\begin{aligned}&\int \nolimits _{{\mathbb {R}}^3}|u^N(\cdot ,t)|^2\phi ^2 \,\hbox {d}x+2\int \nolimits _{0}^t\int \nolimits _{{\mathbb {R}}^3}|\nabla u^{N}|^2\phi ^2 \,\text {d}x\,\text {d}s \leqq \Vert \widetilde{u_{0,a}^{N}}\Vert _{L_{2}}^{2}\\&\qquad + \int \nolimits _{0}^t\int \nolimits _{{\mathbb {R}}^3}|u^{N}|^2\Delta (\phi ^2) \,\text {d}x\,\text {d}s+ \int \nolimits _0^t\int \nolimits _{{\mathbb {R}}^3}|u^{N}|^2u^{N}\cdot \nabla (\phi ^2) \,\text {d}x\,\text {d}s\\&\qquad +\int \nolimits _0^t\int \nolimits _{{\mathbb {R}}^3}2qu^{N}\cdot \nabla (\phi ^2) \,\text {d}x\,\text {d}s -\int \nolimits _0^t\int \nolimits _{{\mathbb {R}}^3}(e^{s\Delta }\overline{u_{0,a}}^{N}\cdot \nabla u^{N})\cdot u^{N}\phi ^2 \,\text {d}x\,\text {d}s\\&\qquad +\int \nolimits _0^t\int \nolimits _{{\mathbb {R}}^3}(e^{s\Delta }\overline{u_{0,a}}^{N}\otimes u^{N}):\nabla u^{N} \phi ^2 \,\text {d}x\,\text {d}s \\&\qquad +\int \nolimits _0^t\int \nolimits _{{\mathbb {R}}^3}(e^{s\Delta }\overline{u_{0,a}}^{N}\otimes u^{N}):u^{N}\otimes \nabla (\phi ^2) \,\text {d}x\,\text {d}s\\&\qquad +\int \nolimits _0^t\int \nolimits _{{\mathbb {R}}^3}(e^{s\Delta }\overline{u_{0,a}}^{N}\otimes e^{s\Delta }\overline{u_{0,a}}^{N}):\nabla u^{N} \phi ^2 \,\text {d}x\,\text {d}s\\&\qquad +\int \nolimits _0^t\int \nolimits _{{\mathbb {R}}^3}(e^{s\Delta }\overline{u_{0,a}}^{N}\otimes e^{s\Delta }\overline{u_{0,a}}^{N}):u^{N}\otimes \nabla (\phi ^2) \,\text {d}x\,\text {d}s\\&\quad = I_{0}+I_1+\ldots \ I_8, \end{aligned}$$

Using (139), we have $I_{0}\leqq C(M, \gamma ) N^{-2\gamma _{2}}.$ Using (142) and the same arguments as in Section 4 gives

$$\begin{aligned} |I_{1}|+|I_{2}|+|I_{3}|\leqq C(M)t^{\frac{1}{10}}. \end{aligned}$$

Furthermore, using (142) and (140) we obtain

$$\begin{aligned} |I_{4}|+|I_{5}|+|I_{6}|\leqq C(M,\gamma , \delta _{2}) N^{\gamma _{1}}t^{\frac{\delta _{2}}{2}}. \end{aligned}$$

Next, we may use (140)–(141) to see that

$$\begin{aligned} \int \nolimits _{0}^{t}\int \nolimits _{{\mathbb {R}}^3} |e^{t\Delta }\overline{u_{0,a}}^{N}|^4 \,\text {d}x\,\text {d}t'\leqq C(M,\gamma ,\delta _{2}) N^{2\gamma _{1}} t^{\delta _{2}}. \end{aligned}$$

This may be used with (142) to show that

$$\begin{aligned} |I_{7}|+|I_{8}|\leqq C(M,\gamma , \delta _{2})N^{\gamma _{1}}t^{\frac{\delta _{2}}{2}}. \end{aligned}$$

Thus for $t\in (0,\min (1, S_{lews}))$ we have

$$\begin{aligned}&\Vert u^{N}(\cdot ,t)\Vert ^2_{L^{2}(B_{1}(0))}+\int \nolimits _{0}^{t}\int \nolimits _{B_{1}(0)} |\nabla u^{N}(x,t')|^2 \,\text {d}x\,\text {d}t'\\&\quad \leqq C(M,\gamma , \delta _{2})\big (N^{-2\gamma _{2}}+(1+N^{\gamma _{1}})t^{\min \big (\frac{1}{10}, \frac{\delta _{2}}{2}\big )}\big ). \end{aligned}$$

Noting that $u-e^{t\Delta } u_{0,a}= u^{N}- \widetilde{u_{0,a}}^{N}$, we thus obtain for $t\in (0,\min (1, S_{lews}))$ and $N\in (0,\infty )$ that

$$\begin{aligned}&\Vert u(\cdot ,t)-e^{t\Delta } u_{0,a}\Vert ^2_{L^{2}(B_{1}(0))}+\int \nolimits _{0}^{t}\int \nolimits _{B_{1}(0)} |\nabla (u-e^{t'\Delta }u_{0,a})|^2 \,\text {d}x\,\text {d}t'\\&\quad \leqq C(M,\gamma , \delta _{2})\left( N^{-2\gamma _{2}}+(1+N^{\gamma _{1}})t^{\min \left( \frac{1}{10}, \frac{\delta _{2}}{2}\right) }\right) . \end{aligned}$$

Choosing $N= t^{-\beta }$, where $\beta >0$ is sufficiently small, then yields the desired estimate (C.3.2).

1.4 C.4. Besov Initial Data: Section 2

In this section we give the adjustments needed to prove Theorem 4 in the case of a drift a, which rather than satisfying the global $L^5(Q_1(0,0))$ bound (27), just satisfies

$$\begin{aligned} \sup _{s\in (-1,0)}|s-t_0|^\frac{1}{5}\Vert a(\cdot ,s)\Vert _{L^5(B_1(0))}<\infty \end{aligned}$$

and small, for a fixed $t_0\in [-1,0]$. This extension is needed to deal with the case of locally Besov initial data $\dot{B}^{-1+\frac{3}{p}}_{p,\infty }$, with $p=5$, for which the mild solution just satisfies (131). We actually prove the following theorem which allows to handle any $p\in (3,\infty )$:

Theorem 6

Let $t_0\in [-1,0]$ and $\eta \in (0,1)$ be fixed. For all $\delta \in (0,3)$, there exists $C_*(\delta )\in (0,\infty )$, for all $E\in (0,\infty )$, there exists $\varepsilon _*(\delta ,\eta ,E)\in (0,\infty )$, for all a such that

$$\begin{aligned} \sup _{s\in (-1,0)}|s-t_0|^{\frac{1}{2}}\Vert a(\cdot ,s)\Vert _{L^\infty (B_1(0))}<\infty \end{aligned}$$

and all local suitable solutions v to (24) in $Q_1(0,0)$ such that⁵

$$\begin{aligned}&\int \nolimits _{B_1(0)}|v(x,s)|^2\,\hbox {d}x+\int \nolimits _{-1}^{s}\int \nolimits _{B_1(0)}|\nabla v|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\leqq \ E(s-t_0)_+^\eta ,\quad \forall s\in (-1,0), \end{aligned}$$

(143)

$$\begin{aligned}&\int \nolimits _{-1}^s\int \nolimits _{B_1(0)}|q|^\frac{3}{2}\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\leqq \ E(s-t_0)_+^{\frac{3}{4}\eta },\quad \forall s\in (-1,0), \end{aligned}$$

(144)

the conditions

$$\begin{aligned} \sup _{s\in (-1,0)}|s-t_0|^{\frac{1}{2}}\Vert a(\cdot ,s)\Vert _{L^\infty (B_1(0))} \leqq \varepsilon _* \end{aligned}$$

(145)

and

$$\begin{aligned} \int \nolimits _{Q_1(0,0)}|v|^3+|q|^\frac{3}{2}\,\text {d}x\,\text {d}s\leqq \varepsilon _* \end{aligned}$$

(146)

imply that for all $({{\bar{x}}},t)\in {{\bar{Q}}}_{1/2}(0,0)$, for all $r\in (0,\frac{1}{4}]$,

(147)

We note that (143) implies in particular that for all $s\in (-1,0)$, $v(\cdot ,s)=0$ almost everywhere on $B_1(0)$. As was emphasized just below Theorem 4, the constant $C_*$ only depends on $\delta$, because it arises when going from scale $r_n$ to $r\in (r_{n+1},r_n)$.

The proof goes through using the same general scheme as in Section 2. The main difficulty is that the bound (145) does not imply $a\in L^\frac{2}{1-\frac{3}{p}}(-1,0;L^p(B_1(0)))$. Hence estimates on the term

$$\begin{aligned} I_4=2\int \nolimits _{Q_\frac{1}{2}({{\bar{x}}},t)}|a||v||\nabla v||\phi _n|\,\text {d}x\,\text {d}s \end{aligned}$$

carried out in Section 2 do not work as such any longer. One possible way out is to use estimate (C.3.2), which allows us to remove the singularity due to (145). Consequently, there are two main modifications to the argument in Section 2. The first modification is on the bounds ($A_k$) and ($B_k$) which are iterated. The second modification is on Lemma 4 for the pressure.

Let $({{\bar{x}}},t)\in {{\bar{Q}}}_{1/2}(0,0)$ be fixed for the rest of this section. For all $n\in {\mathbb {N}}$, we let $r_n:=2^{-n}$. Our aim is to propagate for $k\geqq 2$ the following three bounds:

and

where

for $\eta '=\frac{\eta }{6}\in (0,\frac{1}{6})$ and constants $\varepsilon _*(\delta ,\eta ,E),\, C_B(\delta ,\eta )\in (0,\infty )$ to be chosen. Notice that the power $\frac{3}{4}\eta '$ in ($A_k''$) is worse than the corresponding power in ($A_k'$). This fact appears in Step 4 of the proof of Theorem 6. It is due to the fourth term in the right hand side of (149) below.

We also need the following modification of Lemma 4:

Lemma 24

(Pressure estimate). There exists a constant $C_2'\in (0,\infty )$ such that for all $\rho \in (0,\infty )$, and for all a such that

$$\begin{aligned} M_a:=\sup _{s\in (-1,0)}|s-t_0|^{\frac{1}{2}}\Vert a(\cdot ,s)\Vert _{L^\infty (B_1(0))}<\infty , \end{aligned}$$

(148)

and for all weak solutions $q\in L^\frac{3}{2}(Q_\rho (0,0))$ to

$$\begin{aligned} -\Delta q=\nabla \cdot \nabla \cdot (v\otimes v)+\nabla \cdot \nabla \cdot (a\otimes v)+\nabla \cdot \nabla \cdot (v\otimes a)\quad \text{ in }\quad Q_\rho (0,0), \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned}&r^{-\frac{1+\delta }{2}}\int \nolimits _{-r^2}^s\int \nolimits _{B_{r}(0)}|q-(q)_r({{\hat{s}}})|^\frac{3}{2}\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\quad \leqq \ C_2'r^{-\frac{1+\delta }{2}}\int \nolimits _{-r^2}^s\int \nolimits _{B_{2r}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\qquad +C_2'r^{\frac{3}{4}-\frac{\delta }{2}}M_a^\frac{3}{2}\Bigg (\int \nolimits _{-r^2}^s\frac{1}{|{{\hat{s}}}-t_0|}\int \nolimits _{B_{2r}(0)}|v(x,{{\hat{s}}})|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{3}{4}\\&\qquad +C_2'r^{6-\frac{\delta }{2}}\Bigg (\sup _{-r^2<{{\hat{s}}}<s}\int \nolimits _{2r<|x|<\rho }\frac{|v(x,{{\hat{s}}})|^2}{|x|^4}\,\text {d}x\Bigg )^\frac{3}{2}\\&\qquad +C_2'r^{4-\frac{\delta }{2}}M_a^\frac{3}{2}\int \nolimits _{-r^2}^s\frac{1}{|{{\hat{s}}}-t_0|^\frac{3}{4}}\Bigg (\int \nolimits _{2r<|x|<\rho }\frac{|v(x,{{\hat{s}}})|}{|x|^4}\,\text {d}x\Bigg )^\frac{3}{2}\,\hbox {d}{{\hat{s}}}\\&\qquad +C_2'r^{4-\frac{\delta }{2}}\rho ^{-\frac{9}{2}}\int \nolimits _{-r^2}^s\int \nolimits _{B_\rho (0)}|v|^3+|q|^\frac{3}{2}\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\qquad +C_2'r^{4-\frac{\delta }{2}}\rho ^{-\frac{15}{4}}M_a^\frac{3}{2}\int \nolimits _{-r^2}^s\frac{1}{|{{\hat{s}}}-t_0|^\frac{3}{4}}\Bigg (\int \nolimits _{B_\rho (0)\setminus B_{\frac{\rho }{2}}(0)}|v(x,{{\hat{s}}})|^2\,\text {d}x\Bigg )^\frac{3}{4}\text {d}{{\hat{s}}} \end{aligned} \end{aligned}$$

(149)

for all $s\in (-r^2,0)$, for all $0<r\leqq \rho /2$.

Sketch of the proof of Theorem 6

In the whole proof, we define$M_a$as in (148). Notice that by assumption (145), $M_a\leqq \varepsilon _*$. Let us sketch the main differences with respect to the proof of Theorem 4 in Section 2. We focus on the case when $({{\bar{x}}},t)=(0,0)$, but the argument for general $({{\bar{x}}},t)\in Q_{\frac{1}{2}}(0,0)$ follows along the same lines.

This step is slightly different from the analogous step in Section 2. Indeed, assumption (146) does not imply any rate of decay near the time $t_0$. Therefore, we have to combine (146), to get the smallness with respect to $\varepsilon _*$, with (143) or (144), to get the decay rate in time. To do so, one has to give up a bit of the power $\eta$. Indeed, instead of $(s-t_0)_+^\eta$, the decay rate in ($A_k'$) is $(s-t_0)_+^\frac{\eta }{6}$. We have

$$\begin{aligned} \frac{1}{r_2^2}\int \nolimits _{-r_2^2}^s\int \nolimits _{B_{r_2}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}}&\leqq \ \frac{1}{r_2^2}\Bigg (\int \nolimits _{-r_2^2}^s\int \nolimits _{B_{r_2}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{1}{6}\Bigg (\int \nolimits _{-r_2^2}^s\int \nolimits _{B_{r_2}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{5}{6}\\&\leqq \ C\big (E^\frac{3}{2}(s-t_0)_+^{\frac{3}{2}\eta }\big )^\frac{1}{6}\varepsilon _*^\frac{5}{6}\\&\leqq \ \frac{1}{2}(s-t_0)_+^{\frac{3}{2}\eta '}\varepsilon _*^\frac{2}{3}r_2^{3-\delta }. \end{aligned}$$

The estimate for the pressure using (144) is similar.

We do not give the details for this step. Similar calculations are done below in Step 3. Notice that the terms $I_4$ and $I_5$ have to be estimated using (143) and ($A_k'$) for $k=2$. The smallness of a given by (145) enables us to absorb some constants by choosing $\varepsilon _*$ small enough.

Thanks to the local energy inequality (25), we have for all $s\in (-r_n^2,0)$,

$$\begin{aligned}&C_1^{-1}r_n^{-1}\int \nolimits _{B_{r_n}(0)}|v(x,s)|^2\,\hbox {d}x+C_1^{-1}r_n^{-1}\int \nolimits _{-r_n^2}^s\int \nolimits _{B_{r_n}(0)}|\nabla v|^2\,\text {d}x\,\text {d}s\\&\quad \leqq \ C_1r_n^2\int \nolimits _{-(\frac{1}{2})^2}^s\int \nolimits _{B_{1/2}(0)}|v|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}+\int \nolimits _{-(\frac{1}{2})^2}^s\int \nolimits _{B_{1/2}(0)}|v|^3|\nabla \phi _n|\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\qquad +2\left| \int \nolimits _{-(\frac{1}{2})^2}^s\int \nolimits _{B_{1/2}(0)}v\cdot \nabla \phi _n q\,\text {d}x\,\text {d}{{\hat{s}}}\right| +2\int \nolimits _{-(\frac{1}{2})^2}^s\int \nolimits _{B_{1/2}(0)}|a||v||\nabla v||\phi _n|\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\qquad +\int \nolimits _{-(\frac{1}{2})^2}^s\int \nolimits _{B_{1/2}(0)}|a||v|^2|\nabla \phi _n|\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\quad =\ I_1'+\ldots \, I_5'. \end{aligned}$$

For $I_1'$, we have

$$\begin{aligned} |I_1'|\leqq \ C_1(s-t_0)^\frac{1}{3}_+\varepsilon _*^\frac{2}{3}r_n^{2-\frac{2}{3}\delta }. \end{aligned}$$

The term $I_2'$ is immediate following the estimates of Section 2. Let us write some details for $I_3'$. We have

$$\begin{aligned} |I_3'|&\leqq \ Cr_n^2\sum _{k=2}^nr_{k-1}^{-4}\Bigg (\int \nolimits _{-r_{k-1}^2}^s\int \nolimits _{B_{r_{k-1}}(0)}|q-(q)_{r_{k-1}}({{\hat{s}}})|^\frac{3}{2}\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{2}{3} \\&\quad \Bigg (\int \nolimits _{-r_{k-1}^2}^s\int \nolimits _{B_{r_{k-1}}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{1}{3}\\&\quad +Cr_n^{-2}\Bigg (\int \nolimits _{-r_{n}^2}^s\int \nolimits _{B_{r_{n}}(0)}|q-(q)_{r_{n}}({{\hat{s}}})|^\frac{3}{2}\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{2}{3}\Bigg (\int \nolimits _{-r_{n}^2}^s\int \nolimits _{B_{r_{n}}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{1}{3}\\&\leqq \ C(\delta )(\varepsilon _*^\frac{2}{3})^\frac{2}{3}r_n^{-2}\big (r_n^{\frac{7}{2}-\frac{\delta }{2}}(s-t_0)_+^{\frac{3}{4}\eta '}\big )^\frac{2}{3}(\varepsilon _*^\frac{2}{3})^\frac{1}{3}\big (r_n^{5-\delta }(s-t_0)_+^{\frac{3}{2}\eta '}\big )^\frac{1}{3}\\&=C(\delta )(s-t_0)_+^{\eta '}\varepsilon _*^\frac{2}{3}r_n^{2-\frac{2}{3}\delta }. \end{aligned}$$

Some changes are necessary in order to deal with $I_4'$ and $I_5'$. For $I_4'$,

$$\begin{aligned} |I_4'|&\leqq \ Cr_n^2\sum _{k=1}^nr_{k+1}^{-3}\int \nolimits _{-r_k^2}^s\int \nolimits _{B_{r_k}(0)}|a||v||\nabla v|\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\leqq \ Cr_n^2M_a\sum _{k=1}^nr_{k+1}^{-3}\int \nolimits _{-r_k^2}^s\frac{1}{|{{\hat{s}}}-t_0|^\frac{1}{2}}\Bigg (\int \nolimits _{B_{r_k}(0)}|v(\cdot ,{{\hat{s}}})|^2\,\hbox {d}x\Bigg )^\frac{1}{2}\\&\quad \Bigg (\int \nolimits _{B_{r_k}(0)}|\nabla v(\cdot ,{{\hat{s}}})|^2\,\hbox {d}x\Bigg )^\frac{1}{2}\,\text {d}{{\hat{s}}}\\&\leqq \ Cr_n^2M_a\varepsilon _*^\frac{1}{3}\sum _{k=1}^nr_{k+1}^{-3}r_k^{\frac{3}{2}-\frac{\delta }{3}}\Bigg (\int \nolimits _{t_0}^s\frac{1}{({{\hat{s}}}-t_0)^{1-\eta '}}\,\text {d}{{\hat{s}}}\Bigg )^\frac{1}{2}\\&\quad \Bigg (\int \nolimits _{-r_k^2}^s\int \nolimits _{B_{r_k}(0)}|\nabla v(\cdot ,{{\hat{s}}})|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{1}{2}\\&\leqq \ C(\eta ',\delta )(s-t_0)_{+}^{\eta '}\varepsilon _*^{1+\frac{2}{3}}r_n^{2-\frac{2}{3}\delta }, \end{aligned}$$

using the fact that $M_a\leqq \varepsilon _*$ by assumption. Finally,

$$\begin{aligned} |I_5'|&\leqq \ Cr_n^2\sum _{k=1}^nr_{k+1}^{-4}\int \nolimits _{-r_k^2}^s\int \nolimits _{B_{r_k}(0)}|a||v|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\leqq \ Cr_n^2M_a\sum _{k=1}^nr_{k+1}^{-4}\int \nolimits _{-r_k^2}^s\frac{1}{|{{\hat{s}}}-t_0|^\frac{1}{2}}\int \nolimits _{B_{r_k}(0)}|v(\cdot ,{{\hat{s}}})|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\leqq \ C(\eta ',\delta )(s-t_0)_+^{\frac{1}{2}+\eta '}\varepsilon _*^{1+\frac{2}{3}}r_n^{2-\frac{2}{3}\delta }. \end{aligned}$$

This concludes Step 3.

We first prove the estimate ($A_k'$). We have

$$\begin{aligned}&\int \nolimits _{-r_{n+1}^2}^s\int \nolimits _{B_{r_{n+1}}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}} \\&\quad \leqq \ r_{n+1}^{-\frac{3}{2}}\int \nolimits _{-r_{n+1}^2}^s\Bigg (\int \nolimits _{B_{r_{n+1}}(0)}|v|^2\,\hbox {d}x\Bigg )^\frac{3}{2}\,\text {d}{{\hat{s}}}\\&\qquad +\int \nolimits _{-r_{n+1}^2}^s\Bigg (\int \nolimits _{B_{r_{n+1}}(0)}|v|^2\,\hbox {d}x\Bigg )^\frac{3}{4}\Bigg (\int \nolimits _{B_{r_{n+1}}(0)}|\nabla v|^2\,\hbox {d}x\Bigg )^\frac{3}{4}\,\text {d}{{\hat{s}}}\\&\quad \leqq \ r_{n+1}^{-\frac{3}{2}}\int \nolimits _{-r_{n+1}^2}^s\big (C_B\varepsilon _*^\frac{2}{3}({{\hat{s}}}-t_0)_+^{\eta '}r_{n+1}^{3-\frac{2}{3}\delta }\big )^\frac{3}{2}\,\text {d}{{\hat{s}}}\\&\qquad +\Bigg (\int \nolimits _{-r_{n+1}^2}^s\big (C_B\varepsilon _*^\frac{2}{3}({{\hat{s}}}-t_0)_+^{\eta '} r_{n+1}^{3-\frac{2}{3}\delta }\big )^3\,\text {d}{{\hat{s}}}\Bigg )^\frac{1}{4}\Bigg (\int \nolimits _{-r_{n+1}^2}^s\int \nolimits _{B_{r_{n+1}}(0)}|\nabla v|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{3}{4}\\&\quad \leqq \ 2C_B^\frac{3}{2}\varepsilon _*(s-t_0)_{+}^{\frac{3}{2}\eta '}r_{n+1}^{5-\delta }, \end{aligned}$$

which proves ($A_k'$) for $k=n+1$ by choosing $\varepsilon _*$ sufficiently small. Let us prove the estimate for the pressure using the bound of Lemma 24. We take $r=r_{n+1}$ and $\rho =\frac{1}{4}$. We have

$$\begin{aligned} \begin{aligned}&r_{n+1}^{-\frac{1+\delta }{2}}\int \nolimits _{-r_{n+1}^2}^s\int \nolimits _{B_{r_{n+1}}(0)}|q-(q)_{r_{n+1}}({{\hat{s}}})|^\frac{3}{2}\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\quad \leqq \ C_2'r_{n+1}^{-\frac{1+\delta }{2}}\int \nolimits _{-r_{n+1}^2}^s\int \nolimits _{B_{r_{n}}(0)}|v|^3\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\qquad +C_2' r_{n+1}^{\frac{3}{4}-\frac{\delta }{2}}M_a^\frac{3}{2}\Bigg (\int \nolimits _{-r_{n+1}^2}^s\frac{1}{|{{\hat{s}}}-t_0|}\int \nolimits _{B_{{r_{n}}}(0)}|v(x,{{\hat{s}}})|^2\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\Bigg )^\frac{3}{4}\\&\qquad +C_2'r_{n+1}^{6-\frac{\delta }{2}}\Bigg (\sup _{-r_{n+1}^2<{{\hat{s}}}<s}\int \nolimits _{{r_{n}}<|x|<\frac{1}{4}}\frac{|v(x,{{\hat{s}}})|^2}{|x|^4}\,\hbox {d}x\Bigg )^\frac{3}{2}\\&\qquad +C_2'r_{n+1}^{4-\frac{\delta }{2}}M_a^\frac{3}{2}\int \nolimits _{-r_{n+1}^2}^s\frac{1}{|{{\hat{s}}}-t_0|^\frac{3}{4}}\Bigg (\int \nolimits _{r_{n}<|x|<\frac{1}{4}}\frac{|v(x,{{\hat{s}}})|}{|x|^4}\,\hbox {d}x\Bigg )^\frac{3}{2}\,\text {d}{{\hat{s}}}\\&\qquad +2^9C_2'r_{n+1}^{4-\frac{\delta }{2}}\int \nolimits _{-r_{n+1}^2}^s\int \nolimits _{B_{\frac{1}{4}}(0)}|v|^3+|q|^\frac{3}{2}\,\hbox {d}x\,\hbox {d}{{\hat{s}}}\\&\qquad +2^{\frac{15}{2}}C_2'r_{n+1}^{4-\frac{\delta }{2}}M_a^\frac{3}{2}\int \nolimits _{-r_{n+1}^2}^s\frac{1}{|{{\hat{s}}}-t_0|^\frac{3}{4}}\Bigg (\int \nolimits _{B_{\frac{1}{4}}(0)\setminus B_{\frac{1}{8}}(0)}|v(x,{{\hat{s}}})|^2\,\hbox {d}x\Bigg )^\frac{3}{4}\,\text {d}{{\hat{s}}}\\&\quad =\ J_1'+\ldots \ J_6', \end{aligned} \end{aligned}$$

for all $s\in (-r_{n+1}^2,0)$. We concentrate on the estimates for $J_2'$, $J_3'$ and $J_4'$. The estimate of $J_1'$ is similar to the one just done above. The estimates of $J_5'$ and $J_6'$ do not pose any additional difficulty. For $J_2'$, we have

$$\begin{aligned} |J_2'|&\leqq \ C_2'r_{n+1}^{\frac{3}{4}-\frac{\delta }{2}}\varepsilon _*^\frac{3}{2}C_B^\frac{3}{4}\varepsilon _*^\frac{1}{2}r_n^{\frac{9}{4}-\frac{\delta }{2}}\Bigg (\int \nolimits _{t_0}^s\frac{1}{({{\hat{s}}}-t_0)^{1-\eta '}}\,\text {d}{{\hat{s}}}\Bigg )^\frac{3}{4}\leqq C(\eta ')C_B^\frac{3}{4}\varepsilon _*^2( s-t_0)_+^{\frac{3}{4}\eta '}r_{n+1}^{3-\delta }. \end{aligned}$$

For $J_3'$, the estimate is very close to the bound for $J_3$ in Section 2. We also split the integral into rings. This yields

$$\begin{aligned} |J_3'|&\leqq \ C_2'r_{n+1}^{6-\frac{\delta }{2}}\varepsilon _*C_B^\frac{2}{3}C(\delta )(s-t_0)_+^{\frac{3}{2}\eta '}r_{n+1}^{-\frac{3}{2}-\delta }\leqq C_2'C_B^\frac{2}{3}C(\delta )\varepsilon _*(s-t_0)_+^{\frac{3}{2}\eta '}r_{n+1}^{3-\delta }. \end{aligned}$$

Finally for $J_4'$ splitting again into rings leads to

$$\begin{aligned} |J_4'|&\leqq \ C_2'r_{n+1}^{4-\frac{\delta }{2}}\varepsilon _*^\frac{3}{2}C_B^\frac{3}{4}C(\delta )\varepsilon _*^\frac{1}{2}r_{n+1}^{-\frac{3}{2}-\frac{\delta }{2}}\int \nolimits _{-r_{n+1}^2}^s\frac{({{\hat{s}}}-t_0)_+^{\frac{3}{4}\eta '}}{|s-t_0|^\frac{3}{4}}\,\text {d}{{\hat{s}}}\\&\leqq \ C_2'r_{n+1}^{4-\frac{\delta }{2}}\varepsilon _*^\frac{3}{2}C_B^\frac{3}{4}C(\delta )\varepsilon _*^\frac{1}{2}r_{n+1}^{-\frac{3}{2}-\frac{\delta }{2}}(s-t_0)_+^{\frac{3}{4}\eta '}r_{n+1}^\frac{1}{2}\\&\leqq \ C_2'C_B^\frac{3}{4}C(\delta )\varepsilon _*(s-t_0)_+^{\frac{3}{4}\eta '}r_n^{3-\delta }. \end{aligned}$$

Hence the estimate ($A_k''$) follows for $\varepsilon _*$ sufficiently small. $\quad \square$