On the stability of critical chemotactic aggregation

Abstract

We consider the two dimensional parabolic-elliptic Keller–Segel model of chemotactic aggregation for radially symmetric initial data. We show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process.

Appendices

Appendix A: Estimates for the Poisson field

This appendix is devoted to the derivation of linear estimates, in particular for the Poisson field in $H^2_Q$ and ${\mathcal {E}}$. We start with weighted $H^2$ type bounds:

Lemma 7.1

($H^2$ bound) Let $p=1,2$, then for all $u\in {\mathcal {D}}(\mathbb {R}^2)$,

$$\begin{aligned} \int |x|^{2p}|\nabla ^2u|^2\lesssim \int |x|^{2p}|\Delta u|^2+\int |x|^{2p-2}|\nabla u|^2. \end{aligned}$$

(7.1)

Proof of Lemma 7.1

We integrate by parts to compute:

$$\begin{aligned}&\int x_1^{2p}(\partial _{11}u+\partial _{22}u)^2 = \int x_1^{2p}\left[ (\partial _{11}u)^2+(\partial _{22}u)^2\right] +2\int x_1^{2p}\partial _{11}u\partial _{22}u\\&\quad = \int x_1^{2p}\left[ (\partial _{11}u)^2+(\partial _{22}u)^2\right] -2\int x_1^{2p}\partial _2u\partial _{211}u\\&\quad = \int x_1^{2p}\left[ (\partial _{11}u)^2+(\partial _{22}u)^2\right] +2\int \partial _{12}u\left[ x_1^{2p}\partial _{12}u+2px_1^{2p-1}\partial _{2}u\right] \\&\quad = \int x_1^{2p}\left[ (\partial _{11}u)^2+(\partial _{22}u)^2+2(\partial _{12}u)^2\right] -2p(2p-1)\int x_1^{2p-2}(\partial _2u)^2 \end{aligned}$$

and similarly with $x_2$, and (7.1) follows.$\square$

We now turn to the linear control of the Poisson field:

Lemma 7.2

(Estimates for the Poisson field) There holds the bounds on the Poisson field:

• (i) General $L^2_Q$ bounds:

  $$\begin{aligned}&\displaystyle \Vert (1+|\mathrm{log}r|)u\Vert _{L^1}+\Vert \phi _u\Vert _{L^{\infty }(r\le 1)}+\left\| \frac{|\phi _u|}{1+|\mathrm{log}r|}\right\| _{L^{\infty }(r\ge 1)}\lesssim \Vert u\Vert _{L^2_Q},\end{aligned}$$

  (7.2)
  $$\begin{aligned}&\displaystyle \Vert \nabla \phi _u\Vert _{L^4}\lesssim \Vert u\Vert _{L^2_Q}.\qquad \end{aligned}$$

  (7.3)

• (ii) Improved $L^2_Q$ bound: if moreover $\int u=0$, then:

  $$\begin{aligned} \Vert \phi _u\Vert _{L^{\infty }}&\lesssim \Vert u\Vert _{L^2_Q},\end{aligned}$$

  (7.4)
  $$\begin{aligned} \int |\nabla \phi _u|^2&= -\int u\phi _u\lesssim \Vert u\Vert ^2_{L^2_Q}. \end{aligned}$$

  (7.5)

• (iii) Energy bound:

  $$\begin{aligned} \forall 1\le p<2,\quad \Vert \nabla \phi _u\Vert _{L^{\infty }}\le C_p\Vert u\Vert ^{1-\frac{p}{2}}_{L^{\infty }}\Vert u\Vert _{L^2}^{p-1}\Vert u\Vert ^{1-\frac{p}{2}}_{L^1}. \end{aligned}$$

  (7.6)

• (iv) Decay in the energy space: $\forall 0\le \alpha <1$,

  $$\begin{aligned} |u(x)|&\le C_{\alpha } \frac{\Vert u\Vert _{{\mathcal {E}}}}{1+|x|^\alpha },\end{aligned}$$

  (7.7)
  $$\begin{aligned} |\nabla \phi _u(x)|&\le C_{\alpha } \frac{\Vert u\Vert _{{\mathcal {E}}}}{1+|x|^{\frac{\alpha }{2}}}. \end{aligned}$$

  (7.8)

Proof of Lemma 7.2

Proof of (i)–(ii): By Cauchy–Schwarz:

$$\begin{aligned} \int (1+|\mathrm{log}r|)|u(y)|\lesssim \Vert u\Vert _{L^2_Q}\left( \int (1+|\mathrm{log}r|)^2Q\right) ^{\frac{1}{2}}\lesssim \Vert u\Vert _{L^2_Q}. \end{aligned}$$

For $|x|\le 1$, we estimate using Cauchy–Schwarz:

$$\begin{aligned} |\phi _{u}(x)|&\lesssim \int _{|x-y|\le 1}|\mathrm{log}(|x-y|)|u(y)|dy+ \int _{|x-y|\ge 1}|\mathrm{log}|x-y|||u(y)|dy\\&\lesssim \Vert u\Vert _{L^2_Q}+\int (1+|\mathrm{log}|y||)|u(y)|\lesssim \Vert u\Vert _{L^2_Q}. \end{aligned}$$

For $|x|\ge 1$, we rewrite

$$\begin{aligned}&\left| \phi _u(x)-\frac{\mathrm{log}|x|}{2\pi }\int u\right| \\&\quad \lesssim \int |\mathrm{log}\left( \frac{|x-y|}{|x|}\right) ||u(y)|dy\\&\quad \lesssim \int _{|x-y|\ge \frac{|x|}{2}} |\mathrm{log}\left( \frac{|x-y|}{|x|}\right) ||u(y)|dy+\int _{|x-y|\le \frac{|x|}{2}} |\mathrm{log}\left( \frac{|x-y|}{|x|}\right) ||u(y)|dy. \end{aligned}$$

For the outer term, we estimate:

$$\begin{aligned} \int _{|x-y|\ge \frac{|x|}{2}} |\mathrm{log}\left( \frac{|x-y|}{|x|}\right) ||u(y)|dy&\lesssim \int _{|x-y|\ge \frac{|x|}{2}} \left( \frac{|x-y|}{|x|}\right) ^{\frac{3}{4}}|u(y)|dy\\&\lesssim \int \left( 1+\frac{|y|^{\frac{3}{4}}}{|x|^{\frac{3}{4}}}\right) |u(y)|dy\\&\lesssim \Vert u\Vert _{L^1}+ \frac{\Vert u\Vert _{L^2_Q}}{|x|^{\frac{3}{4}}}\lesssim \Vert u\Vert _{L^2_Q}. \end{aligned}$$

On the singularity, we estimate using a simple change of variables:

$$\begin{aligned}&\int _{|x-y|\le \frac{|x|}{2}} |\mathrm{log}\left( \frac{|x-y|}{|x|}\right) ||u(y)|dy\\&\quad \lesssim \left( \int _{|x-y|\le \frac{|x|}{2}}|\mathrm{log}\left( \frac{|x-y|}{|x|}\right) |^2dy\right) ^{\frac{1}{2}}\left( \int _{|x-y|\le \frac{|x|}{2}} |u(y)|^2dy\right) ^{\frac{1}{2}}\\&\quad \lesssim \left( \Vert x\Vert ^2\int _{|z|\le \frac{1}{2}}(\mathrm{log}|z|)^2dz\int _{|x-y|\le \frac{|x|}{2}}|u(y)|^2dy\right) ^{\frac{1}{2}}\lesssim \left( \int |y|^2|u(y)|^2dy\right) ^{\frac{1}{2}}\\&\quad \lesssim \Vert u\Vert _{L^2_Q} \end{aligned}$$

and this concludes the proof of (7.2) and (7.4).

We then estimate from the 2 dimensional Hardy–Littlewood–Sobolev and Hölder:

$$\begin{aligned} \Vert \nabla \phi _u\Vert _{L^4}\lesssim \Vert \frac{1}{|x|}\star u\Vert _{L^4}\lesssim \Vert u\Vert _{L^{\frac{4}{3}}}\lesssim \Vert u\Vert _{L^2_Q}\Vert Q\Vert _{L^2}^{\frac{1}{2}} \end{aligned}$$

and (7.3) is proved. Let a smooth cut off function $\chi (x)=1$ for $|x|\le 1$, $\chi (x)=0$ for $|x|\ge 2$, and $\chi _R(x)=\chi (\frac{x}{R})$, then from (7.4):

$$\begin{aligned} \int \chi _R|\nabla \phi _{u}|^2&= \frac{1}{2}\int \Delta \chi _R\phi _{u}^2-\int \chi _Ru\phi _{u}\\ \nonumber&\lesssim \Vert \phi _u\Vert ^2_{L^{\infty }}+\Vert \phi _u\Vert _{L^{\infty }}\Vert u\Vert _{L^1}\lesssim \Vert u\Vert _{L^2_Q}^2 \end{aligned}$$

(7.9)

with constants independent of $R>0$, and hence

$$\begin{aligned} \int |\nabla \phi _u|^2<+\infty . \end{aligned}$$

(7.10)

Moreover, integrating by parts:

$$\begin{aligned} \int _{|x|\le R}|\nabla \phi _u|^2=R\int _0^{2\pi }\phi _u\partial _r\phi _ud\theta -\int _{|x|\le R}u\phi _u. \end{aligned}$$

From (7.10), we can find a sequence $R_n\rightarrow \infty$ such that

$$\begin{aligned} R^2_n\int _0^{2\pi }|\partial _r\phi _{u}|^2\rightarrow 0 \end{aligned}$$

and then from (7.4):

$$\begin{aligned} R_n\left| \int _0^{2\pi }\phi _u\partial _r\phi _u\right| \lesssim \Vert \phi _u\Vert _{L^{\infty } }\left( R^2_n\int _0^{2\pi }|\partial _r\phi _{u}|^2\right) ^{\frac{1}{2}}\rightarrow 0 \end{aligned}$$

and thus

$$\begin{aligned} \int |\nabla \phi _u|^2=\mathop {{\mathop {\mathrm{lim}}}}\limits _{R_n\rightarrow +\infty }\int _{|x|\le R_n}|\nabla \phi _u|^2=-\int u\phi _u. \end{aligned}$$

The estimate (7.5) now follows from (7.4).

Proof of (iii): Let $1\le p<2$, we estimate in brute force:

$$\begin{aligned} |\nabla \phi _u|&\lesssim \frac{1}{|x|}\star |u|\lesssim \int _{|x-y|\le R}\frac{|u(y)|}{|x-y|}dy+\int _{|x-y|\ge R}\frac{|u(y)|}{|x-y|}dy\\&\lesssim \Vert u\Vert _{L^{\infty }}\int _{|z|\le R}\frac{1}{|z|}+\Vert u\Vert _{L^{p}}\left( \,\,\int _{|z|\ge R}\frac{1}{|z|^{p'}}\right) ^{\frac{1}{p'}} \lesssim R\Vert u\Vert _{L^{\infty }}+\frac{\Vert u\Vert _{L^p}}{R^{1-\frac{2}{p'}}}. \end{aligned}$$

We optimize in $R$ and interpolate:

$$\begin{aligned} \Vert \nabla \phi _u\Vert _{L^{\infty }}\le C_p \Vert u\Vert _{L^p}^{\frac{p}{2}}\Vert u\Vert _{L^{\infty }}^{1-\frac{p}{2}}\lesssim C_p\Vert u\Vert ^{1-\frac{p}{2}}_{L^{\infty }}\Vert u\Vert _{L^2}^{p-1}\Vert u\Vert ^{1-\frac{p}{2}}_{L^1}, \end{aligned}$$

this is (7.6).

Proof of (iv): By density, it suffices to prove (7.7) for $u\in {\mathcal {D}}(\mathbb {R}^2)$. Let $(v_{i,j}=x_i\partial _ju)_{1\le i,j\le 2}$, we estimate from (7.1) with $p=2$:

$$\begin{aligned}&\int |v_{i,j}|^2+\int |\nabla v_{i,j}|^2\lesssim \int (1+|x|^2)|\nabla u|^2+\int (1+|x|^4)\\&\quad \times \left[ (\partial _{11}u)^2+(\partial _{22}u)^2+(\partial _{12}u)^2\right] \lesssim \Vert u\Vert _{H^2_Q}^2 \end{aligned}$$

and thus from Sobolev:

$$\begin{aligned} \forall p>2,\quad \Vert v_{i,j}\Vert _{L^p}\lesssim \Vert v_{i,j}\Vert _{H^1}\lesssim \Vert u\Vert _{H^2_Q}. \end{aligned}$$

(7.11)

We now recall the standard Sobolev bound, see for example [8]:

$$\begin{aligned} \forall p>2,\quad \forall f\in {\mathcal {D}}(\mathbb {R}^2),\quad |f(x)-f(y)|\lesssim |x-y|^{1-\frac{2}{p}}\Vert \nabla f\Vert _{L^p}. \end{aligned}$$

We may find $|a|\le 1$ such that

$$\begin{aligned} |f(a)|\lesssim \left( \,\,\int _{|y|\le 1}|f(y)|^2dy\right) ^{\frac{1}{2}} \end{aligned}$$

and hence the growth estimate:

$$\begin{aligned} |f(x)|\lesssim \left( \,\,\int _{|y|\le 1}|f(y)|^2dy\right) ^{\frac{1}{2}}+|x|^{1-\frac{2}{p}}\Vert \nabla f\Vert _{L^p}. \end{aligned}$$

We apply this to $f_{i}=x_iu$ and conclude from (7.11), (9.4): $\forall p>2$ and $i=1,2$:

$$\begin{aligned} |x_iu|&\lesssim \left( \,\,\int _{|y|\le 1}|x_iu|^2dy\right) ^{\frac{1}{2}}+|x|^{1-\frac{2}{p}}\Vert \nabla (x_iu)\Vert _{L^p}\\&\lesssim \Vert u\Vert _{L^{\infty }}+|x|^{1-\frac{2}{p}}\left[ \Vert v_{i,j}\Vert _{L^p}+\Vert u\Vert _{L^p}\right] \lesssim (1+|x|^{1-\frac{2}{p}})\Vert u\Vert _{{\mathcal {E}}} \end{aligned}$$

and hence the decay:

$$\begin{aligned} |u(x)|\lesssim \frac{ \Vert u\Vert _{{\mathcal {E}}}}{1+|x|^{\frac{2}{p}}} \end{aligned}$$

which yields (7.7).

Let $0\le \alpha <1$ and $|x|\gg 1$, we estimate the Poisson field in brute force using (7.7):

$$\begin{aligned} |\nabla \phi _u(x)|&\lesssim \int \frac{|u(y)|}{|x-y|}=\int _{|x-y|>|x|^{\frac{\alpha }{2}}} \frac{|u(y)|}{|x-y|}+\int _{|x-y|<|x|^{\frac{\alpha }{2}}} \frac{|u(y)|}{|x-y|}\\&\lesssim \frac{\Vert u\Vert _{L^1}}{|x|^{\frac{\alpha }{2}}}+\int _{|y|\ge \frac{|x|}{2},\quad |x-y|<|x|^{\frac{\alpha }{2}}}\frac{|u(y)|}{|x-y|}\lesssim \frac{\Vert u\Vert _{L^1}}{|x|^{\frac{\alpha }{2}}}+\frac{\Vert u\Vert _{{\mathcal {E}}}}{1+|x|^{\alpha }}\int _{|z|\le |x|^{\frac{\alpha }{2}}}\frac{dz}{|z|}\\&\lesssim \frac{\Vert u\Vert _{{\mathcal {E}}}}{1+|x|^{\frac{\alpha }{2}}} \end{aligned}$$

and (7.8) is proved.$\square$

Appendix B: Hardy bounds

We recall some standard weighted Hardy inequalities:

Lemma 8.1

(Weighted Hardy inequality) There holds the Hardy bounds:

$$\begin{aligned} \forall \alpha&> -2,\quad \int r^{\alpha +2}|\partial _ru|^2\ge \frac{(2+\alpha )^2}{4} \int r^{\alpha }u^2,\end{aligned}$$

(8.1)

$$\begin{aligned} \int |\Delta \phi |^2&\gtrsim \int \frac{|\nabla \phi |^2}{r^2(1+|\mathrm{log}r|)^2}-\int \frac{|\nabla \phi |^2}{1+r^4},\end{aligned}$$

(8.2)

$$\begin{aligned} \int \frac{|\nabla \phi |^2}{r^2(1+|\mathrm{log}r|)^2}&\gtrsim \int \frac{\phi ^2}{(1+r^4)(1+|\mathrm{log}r|)^2}-\int \frac{\phi ^2}{1+r^6}. \end{aligned}$$

(8.3)

Proof of Lemma 8.1

Let $u\in {\mathcal {C}}^{\infty }_c(\mathbb {R}^2)$. We integrate by parts to estimate:

$$\begin{aligned} \frac{\alpha +2}{2}\int r^{\alpha } u^2=-\int r^{\alpha +1}u\partial _ru\le \left( \int r^{\alpha }u^2\right) ^{\frac{1}{2}}\left( \int r^{\alpha +2}(\partial _ru)^2\right) ^{\frac{1}{2}} \end{aligned}$$

and (8.1) follows.

Let now $\phi \in {\mathcal {D}}(\mathbb {R}^2)$ and consider the radial continuous and piecewise ${\mathcal {C}}^1$ function

$$\begin{aligned} f(r)=\left\{ \begin{array}{ll}\frac{1}{r(1-\mathrm{log}r)}\ \ \hbox {for}\ \ 0<r\le 1,\\ \frac{1}{r(1+\mathrm{log}r)}\ \ \hbox {for}\ \ r\ge 1\end{array}\right. ,\ \ F(x)=f(r)\frac{x}{r} \end{aligned}$$

then

$$\begin{aligned} \nabla \cdot F(x)=\left\{ \begin{array}{ll}\frac{1}{r^2(1-\mathrm{log}r)^2}\ \ \hbox {for}\ \ 0<r< 1,\\ -\frac{1}{r^2(1+\mathrm{log}r)^2}\ \ \hbox {for}\ \ r> 1\end{array}\right. , \end{aligned}$$

and thus:

$$\begin{aligned} \int \frac{1}{r^2(1+|\mathrm{log}r|)^2}|\nabla \phi |^2&\lesssim \int _{|x|\le 1}\nabla \cdot F |\nabla \phi |^2-\int _{|x|\ge 1}\nabla \cdot F |\nabla \phi |^2\\&\lesssim \int _{r=1}f|\nabla \phi |^2d\sigma +\int |f||\nabla (|\nabla \phi |^2)|\\&\lesssim \int _{r=1}f|\nabla \phi |^2d\sigma +\left( \int |f|^2|\nabla \phi |^2\right) ^{\frac{1}{2}}\left( \int |\nabla ^2\phi |^2\right) ^{\frac{1}{2}} \end{aligned}$$

Now by Sobolev:

$$\begin{aligned} \int _{r=1}f|\nabla \phi |^2d\sigma \lesssim \int _{\frac{1}{2}\le r\le 2}(|\nabla \phi |^2+|\nabla ^2\phi |^2) \end{aligned}$$

and thus

$$\begin{aligned}&\int \frac{1}{r^2(1+|\mathrm{log}r|)^2}|\nabla \phi |^2\\&\quad \lesssim \int _{\frac{1}{2}\le r\le 2}|\nabla \phi |^2+\int |\Delta \phi |^2+\left( \int \frac{1}{r^2(1+|\mathrm{log}r|)^2}|\nabla \phi |^2\right) ^{\frac{1}{2}}\left( \int |\Delta \phi |^2\right) ^{\frac{1}{2}} \end{aligned}$$

which implies (8.2).

Similarly, for $r\ge r_0$ large enough:

$$\begin{aligned} \int _{r\ge r_0}\frac{\phi ^2}{(1+r^4)(1+|\mathrm{log}r|)^2}&\lesssim -\int _{r\ge r_0}\nabla \cdot \left[ \frac{1}{(1+r^3)(1+|\mathrm{log}r|)^2}\frac{y}{|y|}\right] \phi ^2\\&\lesssim \int _{r=r_0}|\phi |^2d\sigma +\int _{r\ge r_0}\frac{|\phi ||\nabla \phi |}{(1+r^3)(1+|\mathrm{log}r|)^2} \end{aligned}$$

and (8.3) follows again from Cauchy Schwarz and Sobolev.

This concludes the proof of Lemma 8.1.

Appendix C: Interpolation bounds

We collect in this appendix the bootstrap bounds on $\varepsilon$ which are a consequence of the spectral estimates of Proposition 2.8 and further interpolation estimates.

Proposition 9.1

(Interpolation bounds) There holds:

• (i) $H^2_Q$ bound:

  $$\begin{aligned}&\int (1+r^4)|\nabla ^2\varepsilon |^2+\int (1+r^2)|\nabla \varepsilon |^2+\int \varepsilon ^2\nonumber \\&\quad +\int |\Delta \phi _\varepsilon |^2+\int \frac{|\nabla \phi _\varepsilon |^2}{r^2(1+|\mathrm{log}r|)^2}\lesssim C(M)\Vert \varepsilon _2\Vert _{L^2_Q}^2,\end{aligned}$$

  (9.1)
  $$\begin{aligned}&\int (1+r^2)|\varepsilon _1|^2\lesssim C(M)\Vert \varepsilon _2\Vert _{L^2_Q}^2. \end{aligned}$$

  (9.2)

• (ii) $L^{\infty }$ bounds:

  $$\begin{aligned} \forall 0<\eta \le \frac{1}{2},\quad \Vert \nabla \phi _\varepsilon \Vert _{L^{\infty }}\le C_\eta \Vert \varepsilon _2\Vert ^{1-\eta }_{L^2_Q}, \end{aligned}$$

  (9.3)

  and $\forall 0\le \alpha <\frac{1}{2}$,

  $$\begin{aligned} \Vert (1+|x|^{\alpha })\varepsilon \Vert _{L^{\infty }}+\left\| (1+|x|^{\frac{\alpha }{2}})\nabla \phi _\varepsilon (x)\right\| _{L^{\infty }}\lesssim \delta (\alpha ^*). \end{aligned}$$

  (9.4)

• (iii) $H^2_Q$ bound with logarithmic loss:

  $$\begin{aligned} \int (1+\mathrm{log}(1+r))^C\frac{|\nabla \phi _\varepsilon |^2}{r^2(1+|\mathrm{log}r|)^2}\lesssim |\mathrm{log}b|^C\left( \Vert \varepsilon _2\Vert _{L^2_Q}^2+b^{10}\right) , \end{aligned}$$

  (9.5)

• (iv) $L^{\infty }$ bound with loss:

  $$\begin{aligned} \left\| \frac{\nabla \phi _\varepsilon }{1+|x|}\right\| ^2_{L^{\infty }}\lesssim |\mathrm{log}b|^C\left( \Vert \varepsilon _2\Vert _{L^2_Q}^2+b^{10}\right) . \end{aligned}$$

  (9.6)

• (v) Weighted bound with loss:

  $$\begin{aligned} \int \frac{|\varepsilon |}{1+r}\lesssim C(M)\sqrt{|\mathrm{log}b|}\Vert \varepsilon _2\Vert _{L^2_Q}+b^{10}. \end{aligned}$$

  (9.7)

Proof of Proposition 9.1

Proof of (i): The estimate (9.1) follows directly from (2.58), (2.60), our choice of orthogonality conditions (4.6) and (7.1) with $p=2$. We then estimate from the definitions (5.36), (5.37) and (9.1):

$$\begin{aligned} \int (1+r^2)|\varepsilon _1|^2&\lesssim \int (1+r^2)\left[ |\nabla \varepsilon |^2+|\nabla \phi _Q|^2|\varepsilon |^2+Q^2|\nabla \phi _\varepsilon |^2\right] \\&\lesssim \Vert \varepsilon \Vert ^2_{H^2_Q}+\int \frac{|\nabla \phi _\varepsilon |^2}{r^2(1+|\mathrm{log}r|)^2}\lesssim C(M)\Vert \varepsilon _2\Vert _{L^2_Q}^2. \end{aligned}$$

Proof of (ii): Let $p=2(1-\eta )\in [1,2)$, then from (7.6), Sobolev, (9.1) and the bootstrap bound (4.8):

$$\begin{aligned} \Vert \nabla \phi _\varepsilon \Vert _{L^{\infty }}\lesssim C_p(M)\Vert \varepsilon _2\Vert _{L^2_Q}^{\frac{p}{2}}\Vert \varepsilon \Vert ^{1-\frac{p}{2}}_{L^1}\lesssim C_\eta \Vert \varepsilon _2\Vert ^{1-\eta }_{L^2_Q}. \end{aligned}$$

The decay bound (9.4) follows from the interpolation bounds (7.7), (7.8), the $H^2_Q$ bound (9.1) and the bootstrap bounds (4.8), (4.9).

Proof of (iii): The lossy bound (9.5) follows from (9.1), (9.4) with $\alpha =\frac{1}{2}$. Indeed, let $B=b^{-100}$, then:

$$\begin{aligned}&\int (1+\mathrm{log}(1+r))^C\frac{|\nabla \phi _\varepsilon |^2}{r^2(1+|\mathrm{log}r|)^2}\\&\quad \lesssim \int _{r\le B}(1+\mathrm{log}(1+r))^C\frac{|\nabla \phi _\varepsilon |^2}{r^2(1+|\mathrm{log}r|)^2}+\int _{r\ge B}(1+\mathrm{log}(1+r))^C\frac{|\nabla \phi _\varepsilon |^2}{r^2(1+|\mathrm{log}r|)^2}\\&\quad \lesssim |\mathrm{log}b|^{C_1(C)}\int \frac{|\nabla \phi _\varepsilon |^2}{r^2(1+|\mathrm{log}r|)^2}+\int _{r\ge B}\frac{(1+|\mathrm{log}r|)^C}{r^{2+\frac{1}{2}}}\\&\quad \lesssim |\mathrm{log}b|^{C_1(C)}\left[ \Vert \varepsilon _2\Vert _{L^2_Q}^2+\frac{1}{\sqrt{B}}\right] \lesssim |\mathrm{log}b|^{C_1(C)}\left( \Vert \varepsilon _2\Vert _{L^2_Q}^2+b^{10}\right) . \end{aligned}$$

Proof of (iv): From Sobolev:

$$\begin{aligned} \left\| \frac{\nabla \phi _\varepsilon }{1+|x|}\right\| ^2_{L^{\infty }}\lesssim \left\| \frac{\nabla \phi _\varepsilon }{1+|x|}\right\| ^2_{H^2}\lesssim \int \frac{|\nabla \phi _\varepsilon |^2}{1+r^2}+\Vert \varepsilon \Vert _{H^2}^2 \end{aligned}$$

and (9.6) follows from (9.5).

Proof of (9.7): We use the global $L^1$ bound (4.8), (9.1) and Cauchy Schwarz to estimate:

$$\begin{aligned} \int \frac{|\varepsilon |}{1+r}&\lesssim \Vert \varepsilon \Vert _{L^2}\left( \int _{r\le b^{-20}}\frac{1}{r^2}\right) ^{\frac{1}{2}}+\int _{r\ge b^{-20}}\frac{|\varepsilon |}{1+r}\\&\lesssim C(M)\sqrt{|\mathrm{log}b|}\Vert \varepsilon _2\Vert _{L^2_Q}+b^{20}\Vert \varepsilon \Vert _{L^1} \end{aligned}$$

and (9.7) is proved.$\square$