2c_{0}v^{\frac{2}{3}}\\}}a^{\mu+n}v^{\sigma}g(a,v,t)\textup{d}v\textup{d}a\lesssim-\frac{1}{\epsilon_{2}}\int_{\\{a\geq
2c_{0}v^{\frac{2}{3}}\\}}a^{n}g(a,v,t)\textup{d}v\textup{d}a+C_{\epsilon_{2}}\int_{(0,\infty)^{2}}v^{-n\frac{\sigma}{\mu}}g(a,v,t)\textup{d}v\textup{d}a.$

As moments of the form $M_{0,c}$ are bounded from above, we choose
$\epsilon_{2}$ sufficiently small in order to be able to repeat the argument
in (3.46) and then we recover the region where $\\{a\geq
c_{0}v^{\frac{2}{3}}\\}:$

$\displaystyle-\int_{\\{a\geq
2c_{0}v^{\frac{2}{3}}\\}}a^{n}g(a,v,t)\textup{d}v\textup{d}a$
$\displaystyle=-\int_{\\{a\geq
c_{0}v^{\frac{2}{3}}\\}}a^{n}g(a,v,t)\textup{d}v\textup{d}a+\int_{\\{c_{0}v^{\frac{2}{3}}\leq
a<2c_{0}v^{\frac{2}{3}}\\}}a^{n}g(a,v,t)\textup{d}v\textup{d}a$
$\displaystyle\leq-\int_{\\{a\geq
c_{0}v^{\frac{2}{3}}\\}}a^{n}g(a,v,t)\textup{d}v\textup{d}a+(2c_{0})^{n}\int_{(0,\infty)^{2}}v^{\frac{2}{3}n}g(a,v,t)\textup{d}v\textup{d}a.$

We can choose $\epsilon_{1}>0$ sufficiently small so that equation (3)
becomes:

$\displaystyle\frac{\textup{d}}{\textup{d}t}M_{n,0}(g(t))\lesssim-n\frac{L(1-\gamma)}{4}R_{0}M_{n,0}(g(t))+3M_{n,0}(g(t))+C_{\epsilon_{1},\epsilon_{2},n}\leq-
M_{n,0}(g(t))+C_{\epsilon_{1},\epsilon_{2},n}$

and we conclude using a comparison argument. ∎

###### Proof of Proposition 3.25.

Let $T>0$. Let $N\in\mathbb{N}$, $N\geq 2$, fixed. Let
$S(t):U_{\epsilon,R,N}\rightarrow U_{\epsilon,R,N}$, for $t\in[0,T]$, be as in
(3.59). Proposition 3.21 guarantees that the semigroup is well-defined.
Propositions 3.26 \- 3.27 together with Propositions 3.16 \- 3.19 prove that
$S(t)\omega\subseteq\omega$ if we choose the constants
$c_{0,k},c_{1,-\alpha},c_{1,1},c_{n,0}>0$ in Proposition 3.25 to correspond to
the constants found in Propositions 3.26 \- 3.27. ∎

We now prove the stated moment bounds for the self-similar profiles found in
Theorem 2.3 and Theorem 2.5.

###### Proof of Theorem 2.3 and Theorem 2.5. Moment bounds.

The main point is to control positive powers of the area. Following the proof
of Theorem 2.3 and with the help of Proposition 3.21 and Proposition 3.25, we
can prove that there exists a self-similar profile $g_{N}$ for the two-
dimensional coagulation equation satisfying

$\displaystyle\int_{(0,\infty)^{2}}(1+a^{n})g_{N}(a,v)\textup{d}v\textup{d}a\leq
c_{n,0},$ (3.62)

for every $n\in[1,N]$. Moreover, if $\alpha>0$, we have that
$\int_{(0,\infty)^{2}}(v^{-\alpha-\tilde{\epsilon}}+v+a)g_{N}(a,v)\textup{d}v\textup{d}a$
is bounded uniformly independently of $N$, for $\tilde{\epsilon}$ as in
Proposition 3.2. If $\alpha=0$, we have that
$\int_{(0,\infty)^{2}}(v^{\gamma}+v^{\gamma+1}+a)g_{N}(a,v)\textup{d}v\textup{d}a$
is bounded uniformly independently of $N$. We can thus find a subsequence of
$\\{g_{N}\\}_{N\geq 2}$ that converges to some $\tilde{g}$ in the weak-∗
topology and $\tilde{g}$ is a self-similar profile for the two-dimensional
coagulation equation. This is since these are the moment estimates needed to
prove the existence of a self-similar profile.

As the constants $c_{n,0}$ in (3.62) are independent of $N$, for $n\leq N$, we
can conclude that there exist some constants $c_{d}>0$ such that
$\int_{(0,\infty)^{2}}a^{d}\tilde{g}(a,v)\textup{d}v\textup{d}a<c_{d}$, for
all $d\in[1,\infty).$

To control negative powers of $a$, we use Remark 3.20. For combinations of
positive powers of $a$ and powers of $v,$ we use Young’s inequality. Take
$d>0$ and $k\in\mathbb{R}$. Then:

$\displaystyle\int_{(0,\infty)^{2}}a^{d}v^{k}\tilde{g}(a,v)\textup{d}v\textup{d}a\leq\frac{d}{d+1}\int_{(0,\infty)^{2}}a^{d+1}\tilde{g}(a,v)\textup{d}v\textup{d}a+\frac{1}{d+1}\int_{(0,\infty)^{2}}v^{k(d+1)}\tilde{g}(a,v)\textup{d}v\textup{d}a.$

∎

##  4 Different asymptotic behaviors for $\mu<0$

###  4.1 Ramification theory

First, we prove existence of a weak solution for the time-dependent fusion
problem as in Definition 2.1 that satisfies the moment bounds (2.17) and
(2.18) and with initial value
$g_{\textup{in}}\in\mathscr{M}_{+}^{I}(\mathbb{R}_{>0}^{2})$. We initially
prove well-posedness for a truncated version of the time-dependent problem.
Since $\mu$ is negative, we can prove well-posedness using the truncation for
the coagulation kernel, while the cut-off in the fusion rate is not needed. So
we look at the equation:

$\displaystyle\int_{(0,\infty)^{2}}g(\eta,t)\Theta_{\epsilon}(v)\varphi(\eta)\textup{d}\eta-\frac{2}{3}\int_{(0,\infty)^{2}}g(\eta,t)\Theta_{\epsilon}(v)a\partial_{a}\varphi(\eta)\textup{d}\eta-\int_{(0,\infty)^{2}}$
$\displaystyle
g(\eta,t)\Theta_{\epsilon}(v)v\partial_{v}\varphi(\eta)\textup{d}\eta$
$\displaystyle+(1-\gamma)\langle\mathbb{K}_{R}[g],\varphi\rangle+(1-\gamma)\int_{(0,\infty)^{2}}r(\eta)(c_{0}v^{\frac{2}{3}}-a)g(\eta,t)\partial_{a}\varphi(\eta)\textup{d}\eta=\partial_{t}$
$\displaystyle\int_{(0,\infty)^{2}}g(\eta,t)\varphi(\eta)\textup{d}\eta$ (4.1)

and assume the initial value is
$g_{\textup{in},R}=g_{\textup{in}}\mathbbm{1}_{\\{[c_{0}\epsilon^{\frac{2}{3}},\infty)\times[\epsilon,2R)\\}}$.
For this form of the equation, we make use of Proposition 3.21 for the case
$N=2$ and obtain a sequence of solutions
$\\{g_{\epsilon,R}\\}_{\\{\epsilon\in(0,1),R>1\\}}$. Denote by
$\alpha_{1}=\operatorname{min}\\{-\alpha-\tilde{\epsilon},\gamma-\frac{1}{3}\\}$
and let $\tilde{m}$ as in Proposition 3.2. Using the same proof as for
Proposition 3.2, we can prove that the set

$\displaystyle\omega=\omega(\epsilon,R):=\\{\frac{1}{2}\leq
M_{0,1}(g_{\epsilon,R})\leq 1;M_{0,\alpha_{1}}(g_{\epsilon,R})\leq
c_{0,\alpha_{1}};M_{0,\tilde{m}}(g_{\epsilon,R})\leq
c_{0,\tilde{m}},M_{0,\frac{\sigma}{|\mu|}}(g_{\epsilon,R})\leq
c_{0,\frac{\sigma}{|\mu|}}\\}$ (4.2)

stays preserved in time uniformly in $\epsilon,R$.

In order to prove uniform moment bounds for negative powers of $v$, i.e. in
order to prove $M_{0,c}$, with $c\leq 0$, we need a lower bound for the
$M_{0,1}$ moment. Thus, we need to prove that the lower bound on $M_{0,1}$ is
preserved in time, independently of $\epsilon,R$ and of the initial condition
$g_{\textup{in},R}$.

To do this, we test equation (4.1) with $\varphi(a,v)=v$ and make use of the
following technical proposition.

######  Proposition 4.1.

Let $g_{\epsilon,R}$ be the solution of equation (4.1) with initial value
$g_{\textup{in},R}=g_{\textup{in}}\mathbbm{1}_{\\{[c_{0}\epsilon^{\frac{2}{3}},\infty)\times[\epsilon,2R)\\}}$
and assume in addition that $g_{\textup{in},R}\in\omega$, where $\omega$ is as
in (4.2). Then there exists $M>0$ such that:

$\displaystyle\frac{M_{0,1}(g_{\textup{in}})}{2}\leq
M_{0,1}(g_{\epsilon,R}(\cdot,\cdot,t))\leq M_{0,1}(g_{\textup{in}}),$ (4.3)

for every $t\geq 0$ and uniformly in $\epsilon<\frac{1}{M},R>M$.

###### Proof.

Fix $M>0$, sufficiently large, such that

$\displaystyle
M^{\gamma-1}\int_{(0,\infty)^{2}}v^{\gamma}g_{\textup{in}}(a,v)\textup{d}v\textup{d}a+M^{1-\tilde{m}}\int_{(0,\infty)^{2}}v^{\tilde{m}}g_{\textup{in}}(a,v)\textup{d}v\textup{d}a+M^{-\frac{2}{3}}\int_{(0,\infty)^{2}}v^{\frac{1}{3}}g_{\textup{in}}(a,v)\textup{d}v\textup{d}a\leq\frac{M_{0,1}(g_{\textup{in}})}{2},$

where $\tilde{m}$ is as in Proposition 3.2 and $\beta$ as in (1.7).

Testing in (4.1) with $\varphi(a,v)=v$, we obtain that

$\displaystyle\int_{(0,\infty)^{2}}vg_{\epsilon,R}(a,v,t)\textup{d}v\textup{d}a=\int_{(0,\infty)^{2}}vg_{\textup{in},R}(a,v)\textup{d}v\textup{d}a.$
(4.4)

Take $\epsilon<\frac{1}{M}$ and $R>M$. Since $a\geq c_{0}v^{\frac{2}{3}}$, we
have that

$\displaystyle\int_{(0,\infty)^{2}}vg_{\textup{in},R}(a,v)\textup{d}v\textup{d}a$
$\displaystyle\geq\int_{(c_{0}M^{-\frac{2}{3}},\infty)\times(\frac{1}{M},M)}vg_{\textup{in}}(a,v)\textup{d}v\textup{d}a$
$\displaystyle\geq
M_{0,1}(g_{\textup{in}})-M^{\gamma-1}\int_{(0,\infty)^{2}}v^{\gamma}g_{\textup{in}}(a,v)\textup{d}v\textup{d}a-M^{1-\tilde{m}}\int_{(0,\infty)^{2}}v^{\tilde{m}}g_{\textup{in}}(a,v)\textup{d}v\textup{d}a$
$\displaystyle-M^{-\frac{2}{3}}\int_{(0,\infty)^{2}}v^{\frac{1}{3}}g_{\textup{in}}(a,v)\textup{d}v\textup{d}a\geq\frac{M_{0,1}(g_{\textup{in}})}{2}.$
(4.5)

We combine (4.4) and (4.1) in order to conclude our proof. ∎

The estimates for the remaining moments follow as in the proof of Proposition
3.2. In particular, we obtain that there exists a function
$g\in\textup{C}([0,\infty);\mathscr{M}_{+}(\mathbb{R}_{>0}^{2}))\cap\omega$
and a subsequence of $\\{g_{\epsilon,R}\\}$ such that $g_{\epsilon,R}(t)$
converges to $g(t)$ in the weak-∗ topology, for every $t\geq 0$. We now want
to prove that $g$ satisfies equation (4.1). For this, we follow the steps used
in the proof of Theorem 2.3. The difference is that now we work with an
equation that depends on time instead of trying to prove the existence of a
self-similar profile. Therefore, unlike Theorem 2.3, we do not need to obtain
an estimate for $M_{1,0}$ independent of time, but it is enough to derive an
estimate of the form $M_{1,0}(t)\leq m(T)$, for $0\leq t\leq T$, for some
function $m:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, increasing, but finite
for any fixed time. Notice that we can expect to have $M_{1,0}$ growing as
$t\rightarrow\infty$ since this is the predicted behavior for $M_{1,0}$ in
Theorem 2.12.

######  Proposition 4.2.

Let $\mu<0$. There exists a solution for the weak version of the time-
dependent fusion problem as in Definition 2.1.

###### Proof.

We test equation (4.1) with $\varphi(\eta)=a$. The fusion term is non-positive
due to the isoperimetric inequality and we have that
$\langle\mathbb{K}_{R}[g_{\epsilon,R}],a\rangle=0$. Thus, we obtain:

$\displaystyle\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a\textup{d}\eta\leq\int_{(0,\infty)^{2}}g_{\textup{in}}(\eta)a\textup{d}\eta+\frac{1}{3}\int_{0}^{t}\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,s)a\textup{d}\eta\textup{d}s.$

By Grönwall’s inequality, this implies that $M_{1,0}(g_{\epsilon,R}(t))$ is
bounded for every $t\geq 0$. Thus, we can redo the argument used in Theorem
2.3 to prove that $g$ satisfies equation (2.4). ∎

We now wish to prove that $g$ satisfies the estimates in (2.18) in order to
prove Proposition 2.9.

######  Proposition 4.3.

The solutions $g_{\epsilon,R}$ of the truncated version of the time-dependent
fusion problem (4.1) satisfy (2.18) uniformly in $\epsilon,R$.

###### Proof.

In order to prove the bounds in (2.18), we prove that the estimates hold
uniformly in $\epsilon,R$ for $g_{\epsilon,R}$ and then pass to the limit. As
the ideas for different moment estimates are similar, we focus on proving
$M_{2,0}$ is uniformly bounded. Since it may not be possible to prove bounds
for $M_{2,0}$ which are independent of time, we restrict ourselves to proving
that, for some fixed $T>0$, we have that $M_{2,0}(t)$ is bounded for all
$t\leq T.$

Let $T>0$, $t\leq T$. Due to Proposition 3.21 in the case $N=2$, we are able
to test (4.1) with $\varphi(a,v)=a^{2}$. Since the fusion term is non-positive
due to the isoperimetric inequality, the equation becomes:

$\displaystyle\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a^{2}\textup{d}\eta$
$\displaystyle\leq\int_{(0,\infty)^{2}}g_{\textup{in}}(\eta)a^{2}\textup{d}\eta+(1-\frac{4}{3})\int_{0}^{t}\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,s)\Theta_{\epsilon}(v)a^{2}\textup{d}\eta\textup{d}s$
$\displaystyle+C\int_{0}^{t}\int_{(0,\infty)^{2}}av^{-\alpha}g_{\epsilon,R}(\eta,s)\textup{d}\eta\int_{(0,\infty)^{2}}a^{\prime}v^{\prime\beta}g_{\epsilon,R}(\eta^{\prime},s)\textup{d}\eta^{\prime}\textup{d}s$
$\displaystyle\leq\int_{(0,\infty)^{2}}g_{\textup{in}}(\eta)a^{2}\textup{d}\eta+C\int_{0}^{t}\int_{(0,\infty)^{2}}av^{-\alpha}g_{\epsilon,R}(\eta,s)\textup{d}\eta\int_{(0,\infty)^{2}}a^{\prime}v^{\prime\beta}g_{\epsilon,R}(\eta^{\prime},s)\textup{d}\eta^{\prime}\textup{d}s$

Thus, if $M_{1,-\alpha}$ and $M_{1,\beta}$ are bounded, we obtain that:

$\displaystyle\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a^{2}\textup{d}\eta\leq\int_{(0,\infty)^{2}}g_{\textup{in}}(\eta)a^{2}\textup{d}\eta+C(t)\leq\int_{(0,\infty)^{2}}g_{\textup{in}}(\eta)a^{2}\textup{d}\eta+C(T)$

and we can conclude using Grönwall’s inequality.

$M_{1,-\alpha}$ is bounded in the same manner as $M_{1,0}$ in Proposition 4.2.

We now estimate $M_{1,\beta}$. The contribution of the fusion term is non-
positive. We estimate the term containing the coagulation kernel using:

$\displaystyle\int_{(0,\infty)^{2}}$ $\displaystyle
K_{\epsilon,R}(a,v,a^{\prime},v^{\prime})[(a+a^{\prime})(v+v^{\prime})^{\beta}-av^{\beta}-a^{\prime}v^{\prime\beta}]g_{\epsilon,R}(\eta,t)g_{\epsilon,R}(\eta^{\prime},t^{\prime})\textup{d}\eta^{\prime}\textup{d}\eta$
$\displaystyle\leq\int_{(0,\infty)^{2}}K_{\epsilon,R}(a,v,a^{\prime},v^{\prime})[av^{\prime\beta}+a^{\prime}v^{\beta}]g_{\epsilon,R}(\eta,t)g_{\epsilon,R}(\eta^{\prime},t^{\prime})\textup{d}\eta^{\prime}\textup{d}\eta$
$\displaystyle\leq
C\int_{(0,\infty)^{2}}av^{\beta}g(\eta,t)\textup{d}\eta\int_{(0,\infty)^{2}}v^{\prime\beta-\alpha}g(\eta^{\prime},t^{\prime})\textup{d}\eta^{\prime}+C\int_{(0,\infty)^{2}}av^{-\alpha}g(\eta,t)\textup{d}\eta\int_{(0,\infty)^{2}}v^{\prime
2\beta}g(\eta^{\prime},t^{\prime})\textup{d}\eta^{\prime}.$

Since we can bound from above the moment estimates of the form $M_{0,d},$ with
$d=2\beta$ or $d=\beta-\alpha$, when testing (4.1) with $av^{\beta}$, the
equation becomes

$\displaystyle\partial_{t}M_{1,\beta}(g_{\epsilon,R}(t))$
$\displaystyle\leq(1-\frac{2}{3}-\beta)\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)\Theta_{\epsilon}(v)av^{\beta}\textup{d}\eta+C_{1}M_{1,\beta}(g_{\epsilon,R}(t))+C_{2}M_{1,-\alpha}(g_{\epsilon,R}(t))$
$\displaystyle\leq\overline{C}_{1}M_{1,\beta}(g_{\epsilon,R}(t))+\overline{C}_{2}(t),$

for some constants $C_{1},C_{2},\overline{C}_{1},\overline{C}_{2}(t)>0$, where
$\overline{C}_{2}$ is a function
$\overline{C}_{2}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, increasing, but
finite for any fixed time. Thus, for $t\in[0,T]$, we have

$\displaystyle\partial_{t}M_{1,\beta}(g_{\epsilon,R}(t))\leq\overline{C}_{1}M_{1,\beta}(g_{\epsilon,R}(t))+\overline{C}_{2}(T).$

We can conclude again using Grönwall’s inequality.

The rest of the moments in (2.18) are estimated using the same methods. The
main idea is to use an iterative argument that allows to reduce the exponents
of the powers of $a$ and $v$ to lower order terms, for which we already
obtained uniform bounds.

For example, we analyse the moment $M_{2,m},$ when $m>1$. The fusion term is
non-positive and the linear terms in (4.1) satisfy
$(1-\frac{4}{3}-m)\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)\Theta_{\epsilon}(v)a^{2}v^{m}\textup{d}\eta\leq
0$. We thus obtain:

$\displaystyle\partial_{t}M_{2,m}(g_{\epsilon,R}(t))$
$\displaystyle\lesssim\int_{(0,\infty)^{2}}\int_{(0,\infty)^{2}}K_{\epsilon,R}(\eta,\eta^{\prime})[(a+a^{\prime})^{2}(v+v^{\prime})^{m}-a^{2}v^{m}-a^{\prime
2}v^{\prime
m}]g_{\epsilon,R}(\eta,t)g_{\epsilon,R}(\eta^{\prime},t)\textup{d}\eta^{\prime}\textup{d}\eta$
$\displaystyle\leq\int_{(0,\infty)^{2}}\int_{(0,\infty)^{2}}K(\eta,\eta^{\prime})[(a+a^{\prime})^{2}(v+v^{\prime})^{m}-a^{2}v^{m}-a^{\prime
2}v^{\prime
m}]g_{\epsilon,R}(\eta,t)g_{\epsilon,R}(\eta^{\prime},t)\textup{d}\eta^{\prime}\textup{d}\eta$
(4.6)

and use the fact that

$\displaystyle(a+a^{\prime})^{2}(v+v^{\prime})^{m}-a^{2}v^{m}-a^{\prime
2}v^{\prime m}$ $\displaystyle\leq[(a+a^{\prime})^{2}-a^{2}-a^{\prime
2}](v+v^{\prime})^{m}+(a^{2}+a^{\prime
2})(v+v^{\prime})^{m}-a^{2}v^{m}-a^{\prime 2}v^{\prime m}$ $\displaystyle\leq
2aa^{\prime}(v+v^{\prime})^{m}+(a^{2}+a^{\prime
2})[(v+v^{\prime})^{m}-v^{m}-v^{\prime m}]$ $\displaystyle+(a^{2}+a^{\prime
2})(v^{m}+v^{\prime m})-a^{2}v^{m}-a^{\prime 2}v^{\prime m}$
$\displaystyle\leq
2aa^{\prime}(v+v^{\prime})^{m}+\sum_{k=1}^{k_{m}}\binom{m}{k}(a^{2}+a^{\prime
2})(v^{k}v^{\prime m-k}+v^{m-k}v^{\prime k})$ $\displaystyle+a^{2}v^{\prime
m}+a^{\prime 2}v^{m}.$

The terms containing powers of $a$ of lower order, such as $av^{k},$
$k\in[0,m]$, are bounded iteratively. The rest of the terms are of the form
$a^{2}v^{b}$, with $b-\alpha,b+\beta\in(-\alpha,m)$ or with
$b-\alpha=-\alpha$. For $b-\alpha,b+\beta\in(-\alpha,m)$, we can use Hölder’s
inequality for the integral containing the coagulation kernel:

$\displaystyle\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a^{2}v^{b+\beta}\textup{d}\eta\leq\bigg{(}\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a^{2}v^{m}\textup{d}\eta\bigg{)}^{\theta}\bigg{(}\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a^{2}v^{-\alpha}\textup{d}\eta\bigg{)}^{1-\theta},$
(4.7)

for some $\theta\in(0,1)$. We can estimate $M_{2,-\alpha}$ in a completely
analogous manner as $M_{2,0}$. The moments containing only powers of $v$ are
bounded from above.

Thus, we combine (4.1) and (4.7) to deduce that

$\displaystyle\partial_{t}M_{2,m}(g_{\epsilon,R}(t))$ $\displaystyle\lesssim
C_{1}(t)\bigg{(}\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a^{2}v^{m}\textup{d}\eta\bigg{)}^{\theta}+C_{2}(t)$
$\displaystyle\leq
C_{1}(T)\bigg{(}\int_{(0,\infty)^{2}}g_{\epsilon,R}(\eta,t)a^{2}v^{m}\textup{d}\eta\bigg{)}^{\theta}+C_{2}(T),$

for some $C_{1},C_{2}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, increasing,
but finite for any fixed time. We conclude using Grönwall’s inequality. ∎

In Proposition 4.3, we derived uniform estimates in $\epsilon,R$ for the
moments in (2.18) of $g_{\epsilon,R}$. This means that the limit $g$ will
satisfy the same moment estimates.

######  Remark 4.4.

$g$ satisfies the estimates in (2.18).

###### Proof of Proposition 2.9.

Proposition 2.9 follows by combining Propositions 4.1 \- 4.3 and Remark 4.4. ∎

We now prove a technical proposition which shows that we can test (2.4) with
$\varphi\equiv a$.

######  Proposition 4.5.

Let $T>0$. Let $\mu<0$. Then equation (2.4) holds for every
$\varphi\in\textup{C}^{1}(\mathbb{R}_{>0}^{2})$ with
$\sup_{\eta\in\mathbb{R}_{>0}^{2}}|\varphi(\eta)+a\partial_{a}\varphi(\eta)|\leq
Ca$ and $\sup_{\eta\in\mathbb{R}_{>0}^{2}}|\partial_{v}\varphi(\eta,t)|\leq C$
if (2.18) holds.

###### Proof.

Let $T>0$, $t\in[0,T]$. We construct a sequence of functions
$\\{\zeta_{n}\\}_{n\in\mathbb{N}}\subset\textup{C}_{\textup{c}}^{1}(\mathbb{R}_{>0}^{2})$
such that $\zeta_{n}(\eta)=1$ when $\eta\in[\frac{1}{n},n]^{2}$ and
$\zeta_{n}(\eta)=0$ when $\eta\not\in[\frac{1}{2n},2n]^{2}$. The idea is to
use Lebesgue’s dominated convergence theorem in (2.4) for the functions
$\varphi_{n}=\zeta_{n}\varphi$. We thus show below only the needed estimates
for the proof. The term with the coagulation kernel in (2.4) can be bounded
directly by

$\displaystyle\big{|}\langle\mathbb{K}[g](t),\varphi_{n}\rangle\big{|}\leq
C\sup_{s\in[0,T]}M_{1,-\alpha}(g(s))\sup_{s\in[0,T]}M_{0,\beta}(g(s))+C\sup_{s\in[0,T]}M_{0,-\alpha}(g(s))\sup_{s\in[0,T]}M_{1,\beta}(g(s))\leq
C(T).$

We now wish to control the fusion term in (2.4), namely

$\displaystyle\int_{(0,\infty)^{2}}|r(\eta)(c_{0}v^{\frac{2}{3}}-a)\partial_{a}\varphi_{n}(\eta)|g(\eta,s)\textup{d}\eta.$

Notice that we can construct $\zeta_{n}$ such that
$|a\partial_{a}\zeta_{n}(\eta)|\leq C,$ for some constant independent of
$n\in\mathbb{N}.$ Moreover, we know that the fusion kernel satisfies (1.10)
and that $a\geq c_{0}v^{\frac{2}{3}}$. Thus

$\displaystyle|r(a,v)(c_{0}v^{\frac{2}{3}}-a)|\lesssim
a^{\mu+1}v^{\sigma}+a^{\mu}v^{\sigma+\frac{2}{3}}\leq 2a^{\mu+1}v^{\sigma}\leq
2c_{0}^{\mu}av^{\gamma-1}.$ (4.8)

Using the above inequality, we find that the following upper bound holds

$\displaystyle|r(\eta)(c_{0}v^{\frac{2}{3}}-a)\partial_{a}\varphi_{n}(\eta)|\lesssim
av^{\gamma-1}|\partial_{a}\varphi(\eta)|+av^{\gamma-1}|\varphi(\eta)\partial_{a}\zeta_{n}(\eta)|$

up to a multiplicative constant. We then use Young’s inequality to deduce that

$\displaystyle av^{\gamma-1}|\partial_{a}\varphi(\eta)|\lesssim
av^{\beta}+av^{-\alpha-1}$

and

$\displaystyle av^{\gamma-1}|\varphi(\eta)\partial_{a}\zeta_{n}(\eta)|\leq
Cav^{\gamma-1}\lesssim av^{\beta}+av^{-\alpha-1}.$

Since we have that

$\displaystyle\int_{(0,\infty)^{2}}(av^{\beta}+av^{-\alpha-1})g(\eta,s)\textup{d}\eta\leq\sup_{s\in[0,T]}M_{1,\beta}(g(s))+\sup_{s\in[0,T]}M_{1,-\alpha-1}(g(s))<\infty,$

the moment estimates in (2.18) suffice in order to conclude our proof. ∎

We now have all the necessary parts in order to prove Theorem 2.12. We first
prove that the total area goes to infinity as $t\rightarrow\infty$. More
precisely, we will obtain a lower bound for the total surface area that
increases exponentially in time.

###### Proof of Theorem 2.12.

From Proposition 4.5, we notice that the estimates (2.18) on $g$ allow us to
test with $\varphi(a,v,t)=a$.

Since $\langle\mathbb{K}[g],a\rangle=0$ and using (1.10), equation (2.4)
becomes

$\displaystyle\partial_{t}\int_{(0,\infty)^{2}}g(\eta,t)a\textup{d}\eta$
$\displaystyle=\int_{(0,\infty)^{2}}g(\eta,t)a\textup{d}\eta-\frac{2}{3}\int_{(0,\infty)^{2}}g(\eta,t)a\textup{d}\eta+(1-\gamma)\int_{(0,\infty)^{2}}r(\eta)(c_{0}v^{\frac{2}{3}}-a)g(\eta,t)\textup{d}\eta$
$\displaystyle\geq\frac{1}{3}\int_{(0,\infty)^{2}}g(\eta,t)a\textup{d}\eta-(1-\gamma)R_{1}\int_{(0,\infty)^{2}}a^{-|\mu|+1}v^{\sigma}g(\eta,t)\textup{d}\eta.$

We distinguish now two cases:

  * •

Case 1:

If $\mu\in(-1,0),$ we can apply Young’s inequality to obtain:

$\displaystyle
a^{1+\mu}v^{\sigma}\leq\frac{\epsilon}{p}a+\frac{\epsilon^{-\frac{q}{p}}}{q}v^{\frac{\sigma}{|\mu|}},$
(4.9)

where $p=\frac{1}{1-|\mu|}$ and $q=\frac{1}{|\mu|}$. Thus, there exists a
constant $C_{\epsilon}>0$ for which:

$\displaystyle-\int_{(0,\infty)^{2}}a^{\mu+1}v^{\sigma}g(a,v,t)\textup{d}v\textup{d}a\gtrsim-\frac{\epsilon}{p}\int_{(0,\infty)^{2}}ag(a,v,t)\textup{d}v\textup{d}a-C_{\epsilon}\int_{(0,\infty)^{2}}v^{-\frac{\sigma}{\mu}}g(a,v,t)\textup{d}v\textup{d}a.$

Choose $\epsilon(R_{1})$ such that
$(1-\gamma)R_{1}\frac{\epsilon(R_{1})}{p}<\tilde{\delta},$ for some
$\tilde{\delta}\in(0,\frac{1}{3})$. Then, condition $(\ref{cond2_non-
existence})$ implies:

$\displaystyle\partial_{t}\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta\geq$
$\displaystyle(\frac{1}{3}-\tilde{\delta})\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta-\tilde{C}_{\epsilon},$
(4.10)

where
$\tilde{C}_{\epsilon}=(1-\gamma)R_{1}\frac{\epsilon(R_{1})^{-\frac{q}{p}}}{q}\sup_{t\geq
0}M_{0,\frac{\sigma}{|\mu|}}(g(t))$.

  * •

Case 2:

If $\mu\leq-1$, using (1.12), we have that $a^{1+\mu}v^{\sigma}\leq
c_{0}^{1+\mu}v^{\gamma-\frac{1}{3}}$ and the equation for $g$ becomes:

$\displaystyle\partial_{t}\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta\geq\frac{1}{3}\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta-C\int_{(0,\infty)^{2}}v^{\gamma-\frac{1}{3}}g(\eta,t)\textup{d}\eta.$

The moment $M_{0,\gamma-\frac{1}{3}}$ is bounded from above, cf. (4.2), so we
are in the same situation as in Case 1.

We define
$C(R_{1}):=\frac{(1-\gamma)R_{1}}{\frac{1}{3}-\tilde{\delta}}\frac{\epsilon(R_{1})^{-\frac{q}{p}}}{q}\sup_{t\geq
0}M_{0,\frac{\sigma}{|\mu|}}(g(t))$, where $\tilde{\delta}$ is as in (4.10),
when $\mu\in(-1,0)$ and $C(R_{1}):=3(1-\gamma)R_{1}c_{0}^{1+\mu}\sup_{t\geq
0}M_{0,\gamma-\frac{1}{3}}(g(t)),$ when $\mu\leq-1$.

We have that, if $\mu\in(-1,0)$,

$\displaystyle\partial_{t}\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta\geq$
$\displaystyle(\frac{1}{3}-\tilde{\delta})\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta-(\frac{1}{3}-\tilde{\delta})C(R_{1}).$

If $\mu\leq-1$, we have

$\displaystyle\partial_{t}\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta\geq$
$\displaystyle\frac{1}{3}\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta-\frac{1}{3}C(R_{1}).$

Thus, if we choose an initial condition such that

$\displaystyle\int_{(0,\infty)^{2}}ag_{\text{in}}(a,v)\textup{d}v\textup{d}a\geq
2C(R_{1})$

then

$\displaystyle\int_{(0,\infty)^{2}}ag(a,v,t)\textup{d}v\textup{d}a\geq
e^{(\frac{1}{3}-\tilde{\delta})t}C(R_{1}),$ (4.11)

if $\mu\in(-1,0)$ and

$\displaystyle\int_{(0,\infty)^{2}}ag(a,v,t)\textup{d}v\textup{d}a\geq
e^{\frac{1}{3}t}C(R_{1}),$

when $\mu\leq-1$.

We now prove that we can improve the lower bound found in (4.11) by removing
the $\tilde{\delta}$ in the exponential. This will conclude the proof of
Theorem 2.12.

In the following, we do not write explicitly the constants that are not
necessarily relevant for the proof. We want to improve the lower bound in
(4.9) in the case when $\mu\in(-1,0)$. For $\mu\in(-1,0)$, we let $\epsilon$
to decrease to $0$ as $t\rightarrow\infty$ in (4.9).

More precisely, we take a function
$\epsilon(t):=\epsilon_{1}\textup{e}^{-\epsilon_{1}\frac{|\mu|}{1-|\mu|}t}\in(0,\epsilon_{1}],$
for some constant

$\displaystyle\epsilon_{1}\in(0,\frac{1}{12}).$ (4.12)

This means that
$\epsilon(t)^{-\frac{q}{p}}=\epsilon_{1}^{-\frac{q}{p}}\textup{e}^{\epsilon_{1}t}$
in (4.9) and thus:

$\displaystyle
a^{1+\mu}v^{\sigma}\leq\frac{\epsilon(t)}{p}a+C_{\epsilon_{1},\mu}\frac{\textup{e}^{\epsilon_{1}t}}{q}v^{\frac{\sigma}{|\mu|}},$

where $C_{\epsilon_{1},\mu}$ is a constant depending on $\epsilon_{1}$ and
$\mu$. We denote $A(t):=\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta$
and equation (2.4) then becomes:

$\displaystyle\partial_{t}A(t)\geq$
$\displaystyle\bigg{(}\frac{1}{3}-\epsilon(t)\bigg{)}A(t)-\tilde{C}\textup{e}^{\epsilon_{1}t},$
(4.13)

where $\tilde{C}$ is a constant depending on $\epsilon_{1}$, $\mu$ and
$M_{0,\frac{\sigma}{|\mu|}}(g(t))$. We note that
$\int_{0}^{t}\epsilon(s)\textup{d}s=\frac{1-|\mu|}{|\mu|}\bigg{(}1-\textup{e}^{-\epsilon_{1}\frac{|\mu|}{1-|\mu|}t}\bigg{)}$.

Since $\epsilon(t)\leq\epsilon_{1}$, for every $t\geq 0$, we have that
$\int_{0}^{t}\epsilon(s)\textup{d}s\leq\epsilon_{1}t$. Multiplying (4.13) with
$\textup{e}^{-\frac{1}{3}t+\int_{0}^{t}\epsilon(s)\textup{d}s}$, we obtain:

$\displaystyle\textup{e}^{-\frac{1}{3}t+\int_{0}^{t}\epsilon(s)\textup{d}s}\bigg{[}\partial_{t}A(t)-(\frac{1}{3}-\epsilon(t))A(t)\bigg{]}\geq-\tilde{C}\textup{e}^{\epsilon_{1}t}\textup{e}^{-\frac{1}{3}t+\int_{0}^{t}\epsilon(s)\textup{d}s}\geq-\tilde{C}\textup{e}^{2\epsilon_{1}t}\textup{e}^{-\frac{1}{3}t}\geq-\tilde{C}\textup{e}^{-\frac{1}{6}t},$
(4.14)

since $\epsilon_{1}$ satisfies (4.12). Then we integrate in time from $0$ to
$t$:

$\displaystyle\textup{e}^{-\frac{1}{3}t+\int_{0}^{t}\epsilon(s)\textup{d}s}A(t)-A(0)\geq
6\tilde{C}\textup{e}^{-\frac{1}{6}t}-6\tilde{C}$

and thus

$\displaystyle A(t)\geq
A(0)\textup{e}^{\frac{1}{3}t-\int_{0}^{t}\epsilon(s)\textup{d}s}-6\tilde{C}\textup{e}^{\frac{1}{3}t-\int_{0}^{t}\epsilon(s)\textup{d}s}+6\tilde{C}\textup{e}^{\frac{1}{6}t-\int_{0}^{t}\epsilon(s)\textup{d}s}\geq
A(0)\textup{e}^{\frac{1}{3}t-\int_{0}^{t}\epsilon(s)\textup{d}s}-6\tilde{C}\textup{e}^{\frac{1}{3}t-\int_{0}^{t}\epsilon(s)\textup{d}s}.$

Using the definition of $A(t)$, this can be written as

$\displaystyle\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta\geq\bigg{(}\int_{\mathbb{R}^{2}_{>0}}g(\eta,0)a\textup{d}\eta-6\tilde{C}\bigg{)}\textup{e}^{\frac{1}{3}t-\int_{0}^{t}\epsilon(s)\textup{d}s}.$

This implies that, if we start with sufficiently large total surface area,
$\int_{\mathbb{R}^{2}_{>0}}g(\eta,0)a\textup{d}\eta\geq 12\tilde{C}$, we have:

$\displaystyle\int_{\mathbb{R}^{2}_{>0}}g(\eta,t)a\textup{d}\eta\geq
6\tilde{C}\textup{e}^{\frac{1}{3}t-\int_{0}^{t}\epsilon(s)\textup{d}s}\gtrsim
C\textup{e}^{\frac{1}{3}t},$

where we used that
$1-\textup{e}^{-\epsilon_{1}\frac{|\mu|}{1-|\mu|}t}\in[0,1]$ and thus
$\textup{e}^{-\int_{0}^{t}\epsilon(s)\textup{d}s}\geq\textup{e}^{-\frac{1-|\mu|}{|\mu|}}$.

This concludes the proof of Theorem 2.12. ∎

###  4.2 Self-similarity in the case of fast fusion

In this subsection we prove Theorem 2.7.

We look at the following truncated version of equation (2.4):

$\displaystyle\int_{(0,\infty)^{2}}g(\eta,t)\Theta_{\epsilon}(v)\varphi(\eta)\textup{d}\eta-\frac{2}{3}\int_{(0,\infty)^{2}}g(\eta,t)\tilde{\Theta}_{\epsilon}(a,v)a\partial_{a}\varphi(\eta)\textup{d}\eta-\int_{(0,\infty)^{2}}$
$\displaystyle
g(\eta,t)\Theta_{\epsilon}(v)v\partial_{v}\varphi(\eta)\textup{d}\eta$
$\displaystyle+(1-\gamma)\langle\mathbb{K}_{\epsilon,R}[g],\varphi\rangle+(1-\gamma)\int_{(0,\infty)^{2}}r(\eta)(c_{0}v^{\frac{2}{3}}-a)g(\eta,t)\partial_{a}\varphi(\eta)\textup{d}\eta=\partial_{t}$
$\displaystyle\int_{(0,\infty)^{2}}g(\eta,t)\varphi(\eta)\textup{d}\eta,$
(4.15)

where $\mathbb{K}_{\epsilon,R}$ is the term in (3.6) with coagulation kernel
$K_{\epsilon,R}$ defined in (3.1), $\Theta_{\epsilon}$ was defined in (3.4)
and $\varphi\in\textup{C}^{1}_{\textup{c}}(\mathbb{R}_{>0}^{2})$. We define a
continuous function
$\tilde{\Theta}_{\epsilon}:\mathbb{R}^{2}_{>0}\rightarrow[0,1]$ such that:

$\displaystyle\tilde{\Theta}_{\epsilon}(a,v)=\begin{cases}0&a\leq
1-\epsilon\textup{ and }v\leq\epsilon;\\\
\Theta_{\epsilon}(v)&a=c_{0}v^{\frac{2}{3}};\\\ 1&a>1\textup{ or }v\geq
2\epsilon,\end{cases}$ (4.16)

which satisfies $\tilde{\Theta}_{\epsilon}(a,v)\geq\Theta_{\epsilon}(v).$ The
choice in (4.16) was made in order for (4.2) to preserve the isoperimetric
inequality in time. This means that if $g$ solves (4.2) and
$g(0)\in\mathscr{M}_{+}^{I}(\mathbb{R}_{>0}^{2}),$ then
$g(t)\in\mathscr{M}_{+}^{I}(\mathbb{R}_{>0}^{2})$, for any $t\geq 0$.
Additionally, $\tilde{\Theta}_{\epsilon}$ does not vanish in the region
$\\{a>1\\}$. This form will simplify the proof of the moment estimates.

Notice that, differently from Section 3, we do not truncate the fusion rate
$r(\eta)$ since $r(\eta)(c_{0}v^{\frac{2}{3}}-a)$ grows at most linearly.

The statement of Proposition 3.21 holds if we replace equation (3) with (4.2).
This is obtained using the same proof and the growth rate of
$r(\eta)(c_{0}v^{\frac{2}{3}}-a)$ in (4.2). Thus, we prove the existence of
solutions of (4.2) as in Proposition 3.21 with $N=2$, which we will denote by
$g_{\epsilon,R}$. The choice $N=2$ was made since we want to test equation
(4.2) with $a^{2}$ in order to obtain uniform moment bounds later in the
proof.

To prove Theorem 2.7 we will follow the strategy of Theorem 2.3. The main
point is to find an invariant set that allows us to pass to the limit as
$\epsilon\rightarrow 0$ and $R\rightarrow\infty$ in the two-dimensional
coagulation equation. In this case, it will be harder to estimate the moment
$M_{1,0}$ than in the case when $\mu>0$. This is because of the fact that, in
the case of (3), the moment estimates are a consequence of the fast growth of
the fusion ratio $r(\eta)$. This fast growth does not take place in (4.2). We
will be able to replace the fast growth of $r(\eta)$ with the choice of a
sufficiently large constant $\lambda$.

We first prove some elementary inequalities that will be useful for the proof
of Theorem 2.7.

######  Proposition 4.6.

Let $\mu\leq 0$. For $\delta_{1}\in(0,1)$, there exist $\lambda$ sufficiently
large (depending on $\delta_{1}$) and some constants
$m_{1}=\alpha(\mu-2)-\sigma,$ $m_{2}=\beta(2-\mu)-\sigma$ such that:

$\displaystyle av^{-\alpha}$ $\displaystyle\leq\frac{1-\gamma}{6}\lambda
a^{\mu+1}v^{\sigma-\alpha}+\delta_{1}a^{2}+v^{m_{1}};$ (4.17) $\displaystyle
av^{\beta}$
$\displaystyle\leq\frac{1-\gamma}{2(K_{0}c_{0,\beta-\alpha}+2)}\lambda
a^{\mu+1}v^{\sigma+\beta}+\delta_{1}a^{2}+v^{m_{2}},$ (4.18)

where $c_{0,\beta-\alpha}$ is a constant that will be fixed in Lemma 4.7 and
$K_{0}$ is as in (1.7).

###### Proof.

We prove (4.17) in detail. The strategy to prove (4.17) and (4.18) is the
same. The proof is as follows.

For any $\delta_{2},\delta_{3}\in(0,1)$, there exist $C_{\delta_{2}}>0$ and
$C_{\delta_{3}}>0$ such that

$\displaystyle av^{-\alpha}\leq
C_{\delta_{2}}a^{\frac{\mu+1}{2}}v^{-\alpha\frac{3-\mu}{2}}+\delta_{2}a^{2}$

and

$\displaystyle a^{\frac{\mu+1}{2}}v^{-\alpha\frac{3-\mu}{2}}\leq
C_{\delta_{3}}a^{\mu+1}v^{\sigma-\alpha}+\delta_{3}v^{m_{1}},$

with $m_{1}=\alpha(\mu-2)-\sigma$. Combining the two inequalities, we obtain

$\displaystyle av^{-\alpha}\leq
C_{\delta_{2}}C_{\delta_{3}}a^{\mu+1}v^{\sigma-\alpha}+C_{\delta_{2}}\delta_{3}v^{m_{1}}+\delta_{2}a^{2}.$

To conclude the argument, we choose $\lambda$ sufficiently large such that
$\frac{1-\gamma}{6}\lambda\geq C_{\delta_{2}}C_{\delta_{3}}$ and choose
$\delta_{3}$ such that $C_{\delta_{2}}\delta_{3}\leq 1$. ∎

We now make use of Proposition 4.6 to prove that we have uniform bounds in
$\epsilon$ and $R$ for the solutions of equation (4.2).

######  Lemma 4.7.

Let $K_{\epsilon,R}$ be as in (3.1) and assume $r$ satisfies the assumptions
of Theorem 2.7. Let $T>0$. We define a semigroup
$S(t):U_{\epsilon,R,2}\rightarrow U_{\epsilon,R,2}$ in the following way:
$S(t)g_{\textup{in},R}=g_{\epsilon,R}(\cdot,\cdot,t)$, for all $t\in[0,T]$ and
with $U_{\epsilon,R,2}$ defined as in (3.56) with $N=2$. Take $\mu\leq 0$, fix
$\tilde{\epsilon}\in(0,1)$, let $m_{i}\in\mathbb{R}$, $i=\overline{1,2}$, to
be the coefficients found in Proposition 4.6 and denote
$\tilde{\alpha}:=\alpha+\tilde{\epsilon}$. Then there exist constants
$c_{0,-\tilde{\alpha}},$ $c_{0,m_{i}},$ $c_{2,0},$ $c_{1,-\alpha},$
$c_{1,\beta}>0$, $i\in\\{1,2\\}$, for which the set

$\displaystyle\tilde{\omega}=$
$\displaystyle\\{M_{0,1}(g_{\epsilon,R})=1;M_{0,-\tilde{\alpha}}(g_{\epsilon,R})\leq
c_{0,-\tilde{\alpha}};M_{0,2}(g_{\epsilon,R})\leq
c_{0,2};M_{0,m_{i}}(g_{\epsilon,R})\leq c_{0,m_{i}};$ $\displaystyle
M_{2,0}(g_{\epsilon,R})\leq c_{2,0};M_{1,-\alpha}(g_{\epsilon,R})\leq
c_{1,-\alpha};M_{1,\beta}(g_{\epsilon,R})\leq c_{1,\beta}\\},$

$i\in\\{1,2\\}$, is preserved in time uniformly in $\epsilon,R$ under equation
(4.2), or equivalently $S(t)\tilde{\omega}\subseteq\tilde{\omega},$ for all
$t\in[0,T]$.

###### Proof.

Uniform bounds for moments of the form $M_{0,d},$ with $d\in\mathbb{R}$, are
derived as in Propositions 3.16 \- 3.18. We now find uniform bounds for
$M_{2,0}(g),$ $M_{1,-\alpha}(g)$ and $M_{1,\beta}(g)$. For
$\varphi\in\textup{C}^{1}(\mathbb{R}^{2}_{>0})$, whose growth in $a$ is of
order $1+a^{2}$ and with $\partial_{a}\varphi(a,v)\geq 0$, we have that:

$\displaystyle\partial_{t}$
$\displaystyle\int_{(0,\infty)^{2}}g(\eta,t)\varphi(\eta)\textup{d}\eta\leq\int_{(0,\infty)^{2}}g(\eta,t)\Theta_{\epsilon}(v)\varphi(\eta)\textup{d}\eta-\frac{2}{3}\int_{(0,\infty)^{2}}g(\eta,t)\tilde{\Theta}_{\epsilon}(a,v)a\partial_{a}\varphi(\eta)\textup{d}\eta$
$\displaystyle-\int_{(0,\infty)^{2}}g(\eta,t)\Theta_{\epsilon}(v)v\partial_{v}\varphi(\eta)\textup{d}\eta+(1-\gamma)\langle\mathbb{K}_{\epsilon,R}[g],\varphi\rangle+\lambda(1-\gamma)\int_{(0,\infty)^{2}}a^{\mu}v^{\sigma}(c_{0}v^{\frac{2}{3}}-a)g(\eta,t)\partial_{a}\varphi(\eta)\textup{d}\eta.$

Thus we obtain:

$\displaystyle\partial_{t}M_{2,0}(g(t))$
$\displaystyle\leq-\frac{1}{3}\int_{(0,\infty)^{2}}g(\eta,t)a^{2}\textup{d}\eta+\frac{4}{3}\int_{\\{a\leq
1\\}}g(\eta,t)a^{2}\textup{d}\eta$
$\displaystyle+2K_{0}(1-\gamma)\int_{(0,\infty)^{2}}g(\eta,t)av^{-\alpha}\textup{d}\eta\int_{(0,\infty)^{2}}g(\eta^{\prime},t)a^{\prime}v^{\prime\beta}\textup{d}\eta^{\prime}.$
$\displaystyle\partial_{t}M_{1,-\alpha}(g(t))$
$\displaystyle\leq(\frac{1}{3}+\alpha)\int_{(0,\infty)^{2}}g(\eta,t)\Theta_{\epsilon}(v)av^{-\alpha}\textup{d}\eta+(1-\gamma)\lambda\int_{(0,\infty)^{2}}a^{\mu}v^{\sigma-\alpha}(c_{0}v^{\frac{2}{3}}-a)g(\eta,t)\textup{d}\eta.$
$\displaystyle\partial_{t}M_{1,\beta}(g(t))$
$\displaystyle\leq(\frac{1}{3}-\beta)\int_{(0,\infty)^{2}}g(\eta,t)\Theta_{\epsilon}(v)av^{\beta}\textup{d}\eta+(1-\gamma)\lambda\int_{(0,\infty)^{2}}a^{\mu}v^{\sigma+\beta}(c_{0}v^{\frac{2}{3}}-a)g(\eta,t)\textup{d}\eta$
$\displaystyle+K_{0}(1-\gamma)\int_{(0,\infty)^{2}}g(\eta,t)av^{\beta}\textup{d}\eta\int_{(0,\infty)^{2}}g(\eta^{\prime},t)v^{\prime\beta-\alpha}\textup{d}\eta^{\prime}$
$\displaystyle+K_{0}(1-\gamma)\int_{(0,\infty)^{2}}g(\eta,t)av^{-\alpha}\textup{d}\eta\int_{(0,\infty)^{2}}g(\eta^{\prime},t)v^{\prime
2\beta}\textup{d}\eta^{\prime}.$ (4.19)

Notice that in (4.19) the truncation on the kernel does not create problems:
for $M_{1,-\alpha}$ the term with the coagulation kernel is non-positive and
for $M_{2,0},M_{1,\beta}$ we used $K_{\epsilon,R}\leq K$.

Making use of (4.17) and (4.18), (4.19) becomes:

$\displaystyle\partial_{t}M_{2,0}(g(t))$
$\displaystyle\leq-\frac{1}{3}M_{2,0}(g(t))+\frac{4}{3}M_{0,0}(g(t))+2K_{0}(1-\gamma)M_{1,-\alpha}(g(t))M_{1,\beta}(g(t)).$
$\displaystyle\partial_{t}M_{1,-\alpha}(g(t))$
$\displaystyle\leq(\frac{1}{3}+\alpha)M_{1,-\alpha}(g(t))-\frac{(1-\gamma)\lambda}{2}\int_{\\{a\geq
2c_{0}v^{\frac{2}{3}}\\}}a^{\mu+1}v^{\sigma-\alpha}g(\eta,t)\textup{d}\eta$
$\displaystyle\leq-M_{1,-\alpha}(g(t))+6c_{0}M_{0,\frac{2}{3}-\alpha}(g(t))$
$\displaystyle+3\int_{\\{a\geq
2c_{0}v^{\frac{2}{3}}\\}}av^{-\alpha}g(\eta,t)\textup{d}\eta-\frac{(1-\gamma)\lambda}{2}\int_{\\{a\geq
2c_{0}v^{\frac{2}{3}}\\}}a^{\mu+1}v^{\sigma-\alpha}g(\eta,t)\textup{d}\eta$
$\displaystyle\leq-
M_{1,-\alpha}(g(t))+3\delta_{1}M_{2,0}(g(t))+3M_{0,m_{1}}(g(t))+6c_{0}M_{0,\frac{2}{3}-\alpha}(g(t)),$
(4.20)

where we have used that $\alpha<1$ (see (1.8)) and thus
$2c_{0}(\alpha+\frac{4}{3})\leq 6c_{0}$ and $\alpha+\frac{4}{3}\leq 3$, and

$\displaystyle\partial_{t}M_{1,\beta}(g(t))$ $\displaystyle\leq-
M_{1,\beta}(g(t))+2M_{1,\beta}(g(t))-\frac{(1-\gamma)\lambda}{2}\int_{\\{a\geq
2c_{0}v^{\frac{2}{3}}\\}}a^{\mu+1}v^{\sigma+\beta}g(\eta,t)\textup{d}\eta$
$\displaystyle+K_{0}(1-\gamma)c_{0,\beta-\alpha}\int_{(0,\infty)^{2}}g(\eta,t)av^{\beta}\textup{d}\eta+K_{0}(1-\gamma)c_{0,2\beta}\int_{(0,\infty)^{2}}g(\eta,t)av^{-\alpha}\textup{d}\eta$
$\displaystyle\leq-M_{1,\beta}(g(t))-\frac{(1-\gamma)\lambda}{2}\int_{\\{a\geq
2c_{0}v^{\frac{2}{3}}\\}}a^{\mu+1}v^{\sigma+\beta}g(\eta,t)\textup{d}\eta+K_{0}c_{0,2\beta}M_{1,-\alpha}(g(t))$
$\displaystyle+(K_{0}c_{0,\beta-\alpha}+2)M_{1,\beta}(g(t))$
$\displaystyle\leq-
M_{1,\beta}(g(t))+K_{0}c_{0,2\beta}M_{1,-\alpha}(g(t))+2C(c_{0,\beta-\alpha})c_{0}M_{0,\frac{2}{3}+\beta}(g(t))+C(c_{0,\beta-\alpha})\delta_{1}M_{2,0}(g(t))$
$\displaystyle+C(c_{0,\beta-\alpha})M_{0,m_{2}}(g(t)),$ (4.21)

where $C(c_{0,\beta-\alpha})=K_{0}c_{0,\beta-\alpha}+2.$ As we have that
$M_{0,0},$ $M_{0,\frac{2}{3}-\alpha},$ $M_{0,m_{1}},$ $M_{0,m_{2}},$
$M_{0,\frac{2}{3}+\beta}$ are uniformly bounded from above, (4.20) and (4.21)
become:

$\displaystyle\partial_{t}M_{2,0}(g(t))$
$\displaystyle\leq-\frac{1}{3}M_{2,0}(g(t))+2K_{0}(1-\gamma)M_{1,-\alpha}(g(t))M_{1,\beta}(g(t))+C;$
$\displaystyle\partial_{t}M_{1,-\alpha}(g(t))$ $\displaystyle\leq-
M_{1,-\alpha}(g(t))+3\delta_{1}M_{2,0}(g(t))+C;$
$\displaystyle\partial_{t}M_{1,\beta}(g(t))$ $\displaystyle\leq-
M_{1,\beta}(g(t))+K_{0}c_{0,2\beta}M_{1,-\alpha}(g(t))+C(c_{0,\beta-\alpha})\delta_{1}M_{2,0}(g(t))+C,$
(4.22)

where $C>0$ is a constant depending only on $\gamma,c_{0}$ and the upper bound
of some moments of the form $M_{0,d},$ with $d\in\mathbb{R}$. Using (4.22) we
are able to find a region that stays invariant in time.

In order to give some insight about the proof, we consider first the case
$\delta_{1}=0$, since the result for $\delta_{1}>0$, sufficiently small,
follows by a perturbative argument. (4.22) becomes

$\displaystyle\partial_{t}M_{2,0}(g(t))$
$\displaystyle\leq-\frac{1}{3}M_{2,0}(g(t))+2K_{0}(1-\gamma)M_{1,-\alpha}(g(t))M_{1,\beta}(g(t))+C;$
$\displaystyle\partial_{t}M_{1,-\alpha}(g(t))$ $\displaystyle\leq-
M_{1,-\alpha}(g(t))+C;$ $\displaystyle\partial_{t}M_{1,\beta}(g(t))$
$\displaystyle\leq-M_{1,\beta}(g(t))+K_{0}c_{0,2\beta}M_{1,-\alpha}(g(t))+C.$
(4.23)

Notice that from (4.23) we deduce that the set $\\{M_{1,-\alpha}(g)\leq C,$
$M_{1,\beta}(g)\leq C(K_{0}c_{0,2\beta}+1),$ $M_{2,0}(g)\leq
3[C+2(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]\\}$ stays invariant in time.

Let $\delta_{1}>0$. We prove that the region defined by
$\\{M_{1,-\alpha}(g)\leq 2C,$ $M_{1,\beta}(g)\leq 2C(K_{0}c_{0,2\beta}+1),$
$M_{2,0}(g)\leq 3[C+32(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]\\}$ is
invariant in time.

Assume that $M_{1,-\alpha}(g(0))\leq 2C,$ $M_{1,\beta}(g(0))\leq
2C(K_{0}c_{0,2\beta}+1)$ and $M_{2,0}(g(0))\leq
3[C+32(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]$. By continuity in time, for
all $s$ sufficiently small, we obtain that $M_{1,-\alpha}(g(s))\leq 4C,$
$M_{1,\beta}(g(s))\leq 4C(K_{0}c_{0,2\beta}+1)$. Plugging this in (4.22), we
obtain that

$\displaystyle\partial_{s}M_{2,0}(g(s))$
$\displaystyle\leq-\frac{1}{3}M_{2,0}(g(s))+32K_{0}(1-\gamma)C^{2}(K_{0}c_{0,2\beta}+1)+C,$
(4.24)

for all times $s$ small enough. Thus, the region $M_{2,0}(g(s))\leq
3[C+32(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]$ is invariant for small
enough times. We now make use again of (4.22) and the newly obtained bound for
$M_{2,0}$, to deduce that

$\displaystyle\partial_{s}M_{1,-\alpha}(g(s))$ $\displaystyle\leq-
M_{1,-\alpha}(g(s))+9\delta_{1}[C+32(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]+C;$
$\displaystyle\partial_{s}M_{1,\beta}(g(s))$ $\displaystyle\leq-
M_{1,\beta}(g(s))+K_{0}c_{0,2\beta}M_{1,-\alpha}(g(s))$
$\displaystyle+3C(c_{0,\beta-\alpha})\delta_{1}[C+32(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]+C,$
(4.25)

for sufficiently small times. Choosing $\delta_{1}$ sufficiently small so that
$9\delta_{1}[C+32(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]\leq C$ and
$3C(c_{0,\beta-\alpha})\delta_{1}[C+32(1-\gamma)K_{0}C^{2}(K_{0}c_{0,2\beta}+1)]\leq
C$, we obtain that the regions $M_{1,-\alpha}(g(s))\leq 2C$ and
$M_{1,\beta}(g(s))\leq 2C(K_{0}c_{0,2\beta}+1)$ are invariant in time for all
$s$ sufficiently small. We are now able to iterate the argument, extending it
to all times. These bounds are independent of $\epsilon,R$. ∎

We now have all the necessary parts to conclude the proof of Theorem 2.7.

###### Proof of Theorem 2.7.

We can uniformly bound $M_{1,0}(g_{\epsilon,R}(t))\leq
M_{1,-\alpha}(g_{\epsilon,R}(t))+M_{1,\beta}(g_{\epsilon,R}(t))$, since
$M_{1,-\alpha}(g_{\epsilon,R}(t))$ and $M_{1,\beta}(g_{\epsilon,R}(t))$ are
bounded. This is enough to enable us to follow the steps of the proof of
Theorem 2.3 to conclude that there exists a self-similar profile for the two-
dimensional coagulation equation in this case. This finishes the proof of
Theorem 2.7. ∎

##  Appendix A Formal rescaling properties

It is worthwhile to mention that the equation (1.1) has some useful rescaling
properties. More explicitly, if $f$ satisfies (1.1) and
$\int_{(0,\infty)^{2}}vf(a,v,t)\textup{d}v\textup{d}a=v_{0}$, then we can
define a set of functions $\tilde{f}_{k}$

$\displaystyle
f(a,v,t)=\frac{v_{0}}{k^{\frac{8}{3}}}\tilde{f}_{k}(\frac{a}{k^{\frac{2}{3}}},\frac{v}{k},k^{\gamma-1}v_{0}t).$
(A.1)

The rescaling (A.1) can be used to remove the dependence on $v_{0}$, the
problem being reduced to one where the total volume of particles is equal to
$1$. We have that $\tilde{f}_{k}$ solves (1.1) with the fusion kernel $r$
replaced by $\frac{r}{v_{0}}$ and

$\displaystyle\int_{(0,\infty)^{2}}v\tilde{f}_{k}(a,v,t)\textup{d}v\textup{d}a=1.$

Notice that, if $f$ satisfies (1.1) and $g$ is defined as

$\displaystyle
f(a,v,t)=\frac{1}{t^{\frac{8}{3}\xi}}g(\frac{a}{t^{\frac{2}{3}\xi}},\frac{v}{t^{\xi}},\xi\log(t)),$
(A.2)

up to a translation in time, then $g$ satisfies (2.1). Combining (A.1) and
(A.2), we obtain that $\tilde{g}_{k}$, which can be expressed in terms of $g$
in the following manner

$\displaystyle\tilde{g}_{k}(\tilde{A},\tilde{V},\tilde{\tau})=v_{0}^{\frac{8}{3}\xi-1}g(v_{0}^{\frac{2}{3}\xi}\tilde{A},v_{0}^{\xi}\tilde{V},\tau),$
(A.3)

where $\tau=\xi\log(t)$ and $\tilde{\tau}=\xi\log(k^{\gamma-1}v_{0}t)$,
satisfies (2.1) with total volume equal to $1$. In particular, if we choose
$k=v_{0}^{\xi}$, we have $\tilde{\tau}=\tau$.

We obtain the following rescaling property for $\tilde{g}_{k}$ in terms of
$g$:

$\displaystyle
M_{y_{1},y_{2}}(\tilde{g}_{k}(\tilde{\tau}))=v_{0}^{\frac{\gamma-\frac{2}{3}y_{1}-y_{2}}{1-\gamma}}M_{y_{1},y_{2}}(g(\tau)),$
(A.4)

for $y_{1},y_{2}\in\mathbb{R}$, which we make use of in Proposition 2.9 and
Theorem 2.12. Notice that in the relation between the moments of $g$ and the
moments of $\tilde{g}_{k}$ the dependence on the variable $k$ vanishes and the
reason we choose to fix $k=v_{0}^{\xi}$ is in order to avoid the time
translation.

The existence of a function $g_{\textup{in}}$ which satisfies (2.15), (2.20)
and has volume $v_{0}$ is equivalent by means of the scaling (A.3) to finding
a function $\tilde{g}_{\textup{in}}(\tilde{A},\tilde{V})$ such that all these
inequalities hold with $v_{0}=1$. The existence of such a function can be
easily seen if we choose its support in an appropriate region contained in
$\tilde{V}\in[\frac{1}{2},1]$ and with $\tilde{A}$ sufficiently large.

Thus, we can assume without loss of generality that the total volume of the
particles is equal to $1$, keeping in mind that the constants $R_{0}$ and
$R_{1}$ in (1.10) change by a factor $v_{0}$.

##  Appendix B Proof of some results used to obtain the existence of solutions
for the truncated time-dependent problem

###### Proof of Proposition 3.6.

The following remark is used in order to prove the stated properties:

$\displaystyle\Phi(V,V^{\prime},t)\textup{e}^{h_{\epsilon}(l_{2}(V,V^{\prime},t),t)}=\textup{e}^{h_{\epsilon}(V,t)+h_{\epsilon}(V^{\prime},t)}.$
(B.1)

We make use of (3.20), (3.21), the kernel bound and the inequality
$|e^{-x_{1}}-e^{-x_{2}}|\leq|x_{1}-x_{2}|,$ for $x_{1},x_{2}\geq 0$.

We first prove that the map stays in the ball of radius $2M$. In order to
bound $J_{1}[G]$ in (3.23), we use that $x_{t,V}(A)\leq A,$ for every
$t\in[0,\tau]$ and the assumption $\tau\leq\ln{2}$. For the term $J_{2}[G]$,
we notice that
$1+x_{t,V}(A)+x_{t,V^{\prime}}(A^{\prime})\leq(1+x_{t,V}(A))(1+x_{t,V^{\prime}}(A^{\prime}))$.
Thus, for every $\varphi\in\textup{C}_{\textup{c}}(\mathbb{R}^{2}_{>0})$ with
$||\varphi||_{\infty}\leq 1$, we have:

$\displaystyle\int_{(0,\infty)^{2}}\textup{e}^{h_{\epsilon}(V,t)}(1+x_{t,V}(A))J[G](A,V,t)\varphi(A,V)\textup{d}V\textup{d}A$
$\displaystyle\leq
2||\varphi||_{\infty}\int_{(0,\infty)^{2}}(1+a)g_{\textup{in},R}(a,v)\textup{d}v\textup{d}a$
$\displaystyle+\frac{1-\gamma}{2}||K_{R}||_{\infty}||G||^{2}||\varphi||_{\infty}\tau$
$\displaystyle\leq M+(1-\gamma)2M^{2}||K_{R}||_{\infty}\tau\leq M+1\leq 2M.$

We now prove that the map $J$, which was defined in (3.23), is contractive:

$\displaystyle\bigg{|}\int_{(0,\infty)^{2}}\textup{e}^{h_{\epsilon}(V,t)}(1$
$\displaystyle+x_{t,V}(A))[J[F](A,V,t)-J[G](A,V,t)]\varphi(A,V)\textup{d}V\textup{d}A\bigg{|}$
$\displaystyle\leq
2\int_{(0,\infty)^{2}}(1+a)g_{\textup{in},R}(a,v)\textup{d}v\textup{d}a||K_{R}||_{\infty}||\varphi||_{\infty}||F-G||(\textup{e}^{\tau}-1)$
$\displaystyle+\frac{1-\gamma}{2}||K_{R}||_{\infty}(||F||+||G||)||\varphi||_{\infty}||F-G||\tau+\frac{1-\gamma}{2}||K_{R}||_{\infty}^{2}||G||^{2}||\varphi||_{\infty}||F-G||\tau(\textup{e}^{\tau}-1)$
$\displaystyle\leq(M||K_{R}||_{\infty}+2M(1-\gamma)||K_{R}||_{\infty}+||K_{R}||_{\infty})(\textup{e}^{\tau}-1)||F-G||<\frac{1}{2}||F-G||.$

∎

###### Proof of Proposition 3.7.

By testing with $\textup{e}^{h_{\epsilon}(V,t)}x_{t,v},$ for $t\in[0,\tau],$
and using (B.1), we obtain that:

$\displaystyle\int_{\mathbb{R}_{>0}^{2}}G_{\epsilon,R,\delta}(\eta,t)\textup{e}^{h_{\epsilon}(V,t)}x_{t,V}(A)\textup{d}\eta$
$\displaystyle\leq
2\int_{\mathbb{R}_{>0}^{2}}g_{\textup{in},R}(\eta)A\textup{d}\eta+\frac{1-\gamma}{2}\int_{0}^{t}\int_{\mathbb{R}_{>0}^{2}}\int_{\mathbb{R}_{>0}^{2}}K_{R}(\phi_{s}(\eta),\phi_{s}(\eta^{\prime}))G_{\epsilon,R,\delta}(\textup{d}\eta,s)$
$\displaystyle
G_{\epsilon,R,\delta}(\textup{d}\eta^{\prime},s)\textup{e}^{h_{\epsilon}(V,s)+h_{\epsilon}(V^{\prime},s)}[(x_{s,V}(A)+x_{s,V^{\prime}}(A^{\prime}))-x_{s,V}(A)-x_{s,V^{\prime}}(A^{\prime})]\textup{d}s.$

Thus:

$\displaystyle\int_{\mathbb{R}_{>0}^{2}}G_{\epsilon,R,\delta}(A,V,t)\textup{e}^{h_{\epsilon}(V,t)}x_{t,V}(A)\textup{d}V\textup{d}A$
$\displaystyle\leq
2\int_{\mathbb{R}_{>0}^{2}}g_{\textup{in},R}(A,V)A\textup{d}V\textup{d}A,$

which is bounded by assumption. Similarly, testing with
$\textup{e}^{h_{\epsilon}(V,t)}$, we get:

$\displaystyle\int_{\mathbb{R}_{>0}^{2}}G_{\epsilon,R,\delta}(A,V,t)\textup{e}^{h_{\epsilon}(V,t)}\textup{d}V\textup{d}A$
$\displaystyle\leq
2\int_{\mathbb{R}_{>0}^{2}}g_{\textup{in},R}(A,V)\textup{d}V\textup{d}A.$

The condition imposed on $g_{\textup{in},R}$ in order to prove Proposition 3.6
was
$2\int_{(0,\infty)^{2}}(1+a)g_{\textup{in},R}(a,v)\textup{d}v\textup{d}a\leq
M$. Thus, we can replace $g_{\textup{in},R}$ by
$G_{\epsilon,R,\delta}(\cdot,\cdot,\tau)$ and then iterate the argument to
extend the solution to all times. ∎

###### Proof of Proposition 3.14.

Let $s,t\in[0,T]$. We denote by $g(\cdot,t):=S(t)g(\cdot)$. Let
$n\in\mathbb{N}$ be sufficiently large. Let
$\varphi\in\textup{C}_{\textup{c}}(\mathbb{R}_{>0}^{2})$ and
$\varphi_{n}\in\textup{C}^{1}_{\textup{c}}(\mathbb{R}_{>0}^{2})$ be such that
$\sup_{(a,v)\in\mathbb{R}_{>0}^{2}}|\varphi_{n}(a,v)-\varphi(a,v)|\leq\frac{1}{n}$.
Then

$\displaystyle\bigg{|}\int_{(0,\infty)^{2}}$
$\displaystyle[S(t)g(\eta)-S(s)g(\eta)]\varphi(\eta)\textup{d}\eta\bigg{|}\leq$
$\displaystyle\leq\bigg{|}\int_{(0,\infty)^{2}}[g(\eta,t)-g(\eta,s)]\varphi_{n}(\eta)\textup{d}\eta\bigg{|}+\bigg{|}\int_{(0,\infty)^{2}}[g(\eta,t)-g(\eta,s)][\varphi_{n}(\eta)-\varphi(\eta)]\textup{d}\eta\bigg{|}$
$\displaystyle\leq\bigg{|}\int_{(0,\infty)^{2}}[g(\eta,t)-g(\eta,s)]\varphi_{n}(\eta)\textup{d}\eta\bigg{|}+\frac{2}{n}\sup_{r\in[0,T]}\int_{(0,\infty)^{2}}g(\eta,r)\textup{d}\eta$
$\displaystyle\leq\bigg{|}\int_{(0,\infty)^{2}}[g(\eta,t)-g(\eta,s)]\varphi_{n}(\eta)\textup{d}\eta\bigg{|}+\frac{C}{n},$
(B.2)

where we used that we can bound
$\sup_{r\in[0,T]}\int_{(0,\infty)^{2}}g(\eta,r)\textup{d}\eta$ from above. On
the other hand,

$\displaystyle\bigg{|}\int_{(0,\infty)^{2}}[g(\eta,t)-g(\eta,s)]\varphi_{n}(\eta)\textup{d}\eta\bigg{|}$
$\displaystyle\leq\int_{s}^{t}\int_{(0,\infty)^{2}}|g(\eta,r)\varphi_{n}(\eta)|\textup{d}\eta\textup{d}r+\frac{2}{3}\int_{s}^{t}\int_{(0,\infty)^{2}}|g(\eta,r)a\partial_{a}\varphi_{n}(\eta)|\textup{d}\eta\textup{d}r$
$\displaystyle+\int_{s}^{t}\int_{(0,\infty)^{2}}|g(\eta,r)v\partial_{v}\varphi_{n}(\eta)|\textup{d}\eta\textup{d}r+(1-\gamma)\int_{s}^{t}|\langle\mathbb{K}_{\epsilon,R}[g](r),\varphi_{n}\rangle|\textup{d}r$
$\displaystyle+(1-\gamma)\int_{s}^{t}\int_{(0,\infty)^{2}}r_{\delta}(\eta)|c_{0}v^{\frac{2}{3}}-a|g(\eta,r)|\partial_{a}\varphi_{n}(\eta)|\textup{d}\eta\textup{d}r.$

As $K_{\epsilon,R}$ is bounded and $g_{\epsilon,R,\delta}$ has compact support
in the $v$ variable, we can find a constant $C(\epsilon,R,\delta)$, which
depends on $\epsilon,R,\delta$ such that

$\displaystyle\bigg{|}\int_{(0,\infty)^{2}}[g(\eta,t)-g(\eta,s)]\varphi_{n}(\eta)\textup{d}\eta\bigg{|}$
$\displaystyle\leq\frac{2}{3}\int_{s}^{t}\int_{(0,\infty)^{2}}|g(\eta,r)a\partial_{a}\varphi_{n}(\eta)|\textup{d}\eta\textup{d}r+C(\epsilon,R,\delta)|t-s|$
$\displaystyle+\frac{1}{\delta}C(\epsilon,R,\delta)\int_{s}^{t}\int_{(0,\infty)^{2}}|1+a|g(\eta,r)|\partial_{a}\varphi_{n}(\eta)|\textup{d}\eta\textup{d}r.$

As
$\sup_{r\in[0,T]}\int_{(0,\infty)^{2}}(1+a)g_{\epsilon,R,\delta}(\eta,r)\textup{d}\eta$
is bounded from above, we find that there exists $C(\epsilon,R,\delta)$
depending on the written parameters, such that

$\displaystyle\bigg{|}\int_{(0,\infty)^{2}}$
$\displaystyle[g(\eta,t)-g(\eta,s)]\varphi_{n}(\eta)\textup{d}\eta\bigg{|}\leq
C(\epsilon,R,\delta)|t-s|.$ (B.3)

Combining (B.3) with (B), we obtain the continuity in time of the map
$t\rightarrow\int_{(0,\infty)^{2}}g(\eta,t)\varphi(\eta)\textup{d}\eta$, if
$\varphi\in\textup{C}_{\textup{c}}(\mathbb{R}_{>0}^{2})$. We now extend the
argument to all functions $\varphi\in\textup{C}_{0}(\mathbb{R}_{>0}^{2})$. Let
$\varphi\in\textup{C}_{0}(\mathbb{R}_{>0}^{2}),$ with
$||\varphi||_{\infty}\leq 1.$ Due to the support of $g\in U_{\epsilon,R},$ it
is enough to cut the function $\varphi$ for large values of $a$ in order to
make it compactly supported. Let $\chi_{M}(a)$ be as in (3.31). Then

$\displaystyle\bigg{|}\int_{(0,\infty)^{2}}$
$\displaystyle[g(\eta,t)-g(\eta,s)]\varphi(\eta)\textup{d}\eta\bigg{|}\leq$
$\displaystyle\leq\bigg{|}\int_{(0,\infty)^{2}}[g(\eta,t)-g(\eta,s)]\varphi(\eta)\chi_{M}(a)\textup{d}\eta\bigg{|}+2\int_{(M,\infty)\times[\epsilon,2R]}\big{|}[g(\eta,t)-g(\eta,s)]\varphi(\eta)\big{|}\textup{d}\eta$
$\displaystyle\leq
C|t-s|+2M^{-1}\int_{(M,\infty)\times[\epsilon,2R]}a[g(\eta,t)+g(\eta,s)]\textup{d}\eta$
$\displaystyle\leq C|t-s|+c(t)M^{-1}.$

We conclude the argument by taking $M$ sufficiently large. ∎

###### Proof of Proposition 3.24.

Let $n\in\mathbb{N}$. By Lemma 3.23, we have that:

$\displaystyle\int_{\mathbb{R}_{>0}^{2}}G_{\epsilon,R,\delta}$
$\displaystyle(\eta,t)\textup{e}^{h_{\epsilon}(V,t)}x_{t,V}(A)^{n}\textup{d}\eta\leq$
$\displaystyle\leq$ $\displaystyle
2\int_{\mathbb{R}_{>0}^{2}}g_{\textup{in},R}(\eta)A^{n}\textup{d}\eta+\frac{1-\gamma}{2}\int_{0}^{t}\int_{\mathbb{R}_{>0}^{2}}\int_{\mathbb{R}_{>0}^{2}}K_{R}(\phi_{s}(A,V),\phi_{s}(A^{\prime},V^{\prime}))G_{\epsilon,R,\delta}(\eta^{\prime},s)G_{\epsilon,R,\delta}(\eta,s)$
$\displaystyle\textup{e}^{h_{\epsilon}(V,s)+h_{\epsilon}(V^{\prime},s)}[(x_{s,V}(A)+x_{s,V^{\prime}}(A^{\prime}))^{n}-x_{s,V}(A)^{n}-x_{s,V^{\prime}}(A^{\prime})^{n}]\textup{d}\eta^{\prime}\textup{d}\eta\textup{d}s$
$\displaystyle\leq$ $\displaystyle
2\int_{\mathbb{R}_{>0}^{2}}g_{\textup{in},R}(\eta)A^{n}\textup{d}\eta+(1-\gamma)||K_{R}||_{\infty}\sum_{k=1}^{k_{n}}\binom{n}{k}\int_{0}^{t}\int_{\mathbb{R}_{>0}^{2}}\textup{e}^{h_{\epsilon}(V,s)}x_{s,V}^{k}(A)G_{\epsilon,R,\delta}(\eta,s)\textup{d}\eta$
$\displaystyle\int_{\mathbb{R}_{>0}^{2}}\textup{e}^{h_{\epsilon}(V^{\prime},s)}x_{s,V^{\prime}}^{n-k}(A^{\prime})G_{\epsilon,R,\delta}(\eta^{\prime},s)\textup{d}\eta^{\prime}\textup{d}s,$

where $k_{n}$ is taken as in Lemma 3.23. The above computations show that, in
order to find an upper bound for
$\int_{\mathbb{R}_{>0}^{2}}G_{\epsilon,R,\delta}(\eta,t)\textup{e}^{h_{\epsilon}(V,t)}x_{t,V}(A)^{n}\textup{d}\eta$,
it is enough to estimate
$\int_{\mathbb{R}_{>0}^{2}}G_{\epsilon,R,\delta}(\eta,t)\textup{e}^{h_{\epsilon}(V,t)}x_{t,V}(A)^{k}\textup{d}\eta$,
where $k\in[1,n-1]$. As such, we can use an iteration argument for the
exponents of $x_{t,V}(A)$. We then use that

$\int_{(0,\infty)^{2}}G_{\epsilon,R,\delta}(A,V,t,)e^{h_{\epsilon}(V,t)}x_{t,V}(A)\textup{d}V\textup{d}A\leq
2\int_{(0,\infty)^{2}}ag_{\textup{in},R}(a,v)\textup{d}v\textup{d}a$

as in the proof of Proposition 3.7, which can be found in this appendix. In
this manner, we derive suitable moment estimates which allow to extend the
obtained solution to all times by iterating the argument. ∎

##  Appendix C Some technical results used to prove the existence of self-
similar profiles

######  Remark C.1.

In order to simplify the notation we will write $g$ and $g_{\textup{in}}$
instead of $g_{\epsilon,R,\delta}$ and $g_{\textup{in},R}$, respectively, in
the following computations. It is relevant to take into account that
$g_{\epsilon,R,\delta}:=g$ is supported in the region
$(a,v)\in[c_{0}\epsilon^{\frac{2}{3}},\infty)\times[\epsilon,2R)$.

###### Proof of Proposition 3.16.

We obtain (3.35) by testing (3) with $\varphi(a,v)=v$. In order to derive
(3.36), we test (3) with $\varphi(a,v)=v^{\gamma}$.

By Remark 3.15 and since $g$ is supported in the region
$(a,v)\in[c_{0}\epsilon^{\frac{2}{3}},\infty)\times[\epsilon,2R)$, we can
ignore the dependence of $K_{\epsilon,R}$ on $R$:

$\displaystyle\frac{(1-\gamma)}{2}\int_{(0,\infty)^{2}}\int_{(0,\infty)^{2}}K_{\epsilon,R}(a,v,a^{\prime},v^{\prime})\xi_{R}(v+v^{\prime})g(a,v,t)g(a^{\prime},v^{\prime},t)[(v+v^{\prime})^{\gamma}-v^{\gamma}-v^{\prime\gamma}]\textup{d}v^{\prime}\textup{d}a^{\prime}\textup{d}v\textup{d}a$
$\displaystyle\leq$
$\displaystyle\frac{K_{1}(1-\gamma)}{2}\int_{(0,\infty)^{2}}\int_{(0,\infty)^{2}}(v^{-\alpha}v^{\prime\beta}+v^{\beta}v^{\prime-\alpha})\xi_{R}(v+v^{\prime})g(a,v,t)g(a^{\prime},v^{\prime},t)[(v+v^{\prime})^{\gamma}-v^{\gamma}-v^{\prime\gamma}]\textup{d}v^{\prime}\textup{d}a^{\prime}\textup{d}v\textup{d}a$
$\displaystyle\leq$
$\displaystyle\frac{K_{1}(1-\gamma)}{2}\int_{\\{v<\frac{R}{2}\\}}\int_{\\{v^{\prime}<\frac{R}{2}\\}}(v^{-\alpha}v^{\prime\beta}+v^{\beta}v^{\prime-\alpha})g(a,v,t)g(a^{\prime},v^{\prime},t)[(v+v^{\prime})^{\gamma}-v^{\gamma}-v^{\prime\gamma}]\textup{d}v^{\prime}\textup{d}a^{\prime}\textup{d}v\textup{d}a.$

Suppose $v\geq v^{\prime}$ and take $z:=\frac{v^{\prime}}{v}\in(0,1]$. We
have:

$\displaystyle(v^{-\alpha}v^{\prime\beta}+v^{\beta}v^{\prime-\alpha})((v+v^{\prime})^{\gamma}-v^{\gamma}-v^{\prime\gamma})$
$\displaystyle\leq
v^{\gamma}(z^{\beta}+z^{-\alpha})v^{\gamma}((1+z)^{\gamma}-1-z^{\gamma})$
$\displaystyle\leq-(1-\gamma)v^{2\gamma}z^{\gamma}=-(1-\gamma)v^{\gamma}v^{\prime\gamma},$

since $(1+z)^{\gamma}-1-z^{\gamma}\leq 0$ and $z^{-\alpha}\geq 1$. We also
used that:

$(1+z)^{\gamma}-1=\gamma\int_{1}^{1+z}s^{\gamma-1}\textup{d}s\leq\gamma
z\leq\gamma z^{\gamma}.$

Equation (3) now becomes:

$\displaystyle\frac{\textup{d}}{\textup{d}t}M_{0,\gamma}(g(t))$
$\displaystyle\leq(1-\gamma)\int_{(0,\infty)^{2}}v^{\gamma}\Theta_{\epsilon}(v)g(a,v,t)\textup{d}v\textup{d}a-\frac{(1-\gamma)^{2}K_{1}}{2}\bigg{(}\int_{\\{v<\frac{R}{2}\\}}v^{\gamma}g(a,v,t)\textup{d}v\textup{d}a\bigg{)}^{2}$
$\displaystyle\leq(1-\gamma)M_{0,\gamma}(g(t))-\frac{(1-\gamma)^{2}K_{1}}{4}M_{0,\gamma}(g(t))^{2}+\frac{(1-\gamma)^{2}K_{1}}{2}\bigg{(}\int_{\\{v>\frac{R}{2}\\}}v^{\gamma}g(a,v,t)\textup{d}v\textup{d}a\bigg{)}^{2}.$

We can control the region where $v>\frac{R}{2}$ using (3.35), namely:

$\displaystyle\int_{\\{v>\frac{R}{2}\\}}v^{\gamma}g(a,v,t)\textup{d}v\textup{d}a\leq\bigg{(}\frac{R}{2}\bigg{)}^{\gamma-1}\int_{(0,\infty)^{2}}vg(a,v,t)\textup{d}v\textup{d}a\leq\bigg{(}\frac{R}{2}\bigg{)}^{\gamma-1}M_{0,1}(g_{\textup{in}}),$

thus finding a constant $C>0$, independent of $\epsilon\in(0,1),R>1$ and
$\delta\in(0,1)$, for which

$\displaystyle\frac{\textup{d}}{\textup{d}t}M_{0,\gamma}(g(t))\leq(1-\gamma)M_{0,\gamma}(g(t))-\frac{(1-\gamma)^{2}K_{1}}{4}M_{0,\gamma}(g(t))^{2}+C.$

We then conclude that there exists a constant $C_{0,\gamma}$, independent of
$\epsilon,R$ and $\delta,$ for which the region $M_{0,\gamma}(g(t))\leq
C_{0,\gamma}$ is invariant in time. ∎

###### Proof of Proposition 3.17.

We want to bound

$K(a,v,a^{\prime},v^{\prime})[(v+v^{\prime})^{m}-v^{m}-v^{\prime m}].$

Assume $v\geq v^{\prime}$. Denote $z:=\frac{v^{\prime}}{v}\in(0,1]$ and
observe that:

$\displaystyle K(a,v,a^{\prime},v^{\prime})[(v+v^{\prime})^{m}-v^{m}-v^{\prime
m}]\leq$ $\displaystyle
K(a,v,a^{\prime},v^{\prime})v^{m}[(1+z)^{m}-1-z^{m}]\leq
2C_{m}K(a,v,a^{\prime},v^{\prime})v^{m}z$ $\displaystyle\leq$ $\displaystyle
2C_{m}K_{0}v^{\gamma}(z^{\beta}+z^{-\alpha})v^{m}z\leq
4C_{m}K_{0}v^{\gamma+m}z^{-\alpha+1}$ $\displaystyle\leq$ $\displaystyle
4C_{m}K_{0}(v^{\beta+m-1}v^{\prime-\alpha+1}+v^{\prime\beta+m-1}v^{-\alpha+1}),$
(C.1)

where we used that

$(1+z)^{m}-1=m\int_{1}^{1+z}s^{m-1}\textup{d}s\leq m(1+z)^{m-1}z\leq
C_{m}(1+z^{m-1})z,$

since $m>1$ and $z\in(0,1]$. By symmetry, $(\ref{onegreater})$ holds for all
$(v,v^{\prime})\in(0,\infty)^{2}$.

Equation (3) becomes:

$\displaystyle\frac{\textup{d}}{\textup{d}t}M_{0,m}(g(t))$
$\displaystyle\leq-(m-1)M_{0,m}(g(t))+(m-1)\int_{\\{v\leq
2\epsilon\\}}v^{m}g(\eta,t)\textup{d}\eta+2(1-\gamma)C_{m}K_{0}M_{0,\beta+m-1}(g)M_{0,-\alpha+1}(g)$
$\displaystyle\leq-(m-1)M_{0,m}(g(t))+(m-1)2^{m-1}M_{0,1}(g_{\textup{in}})+CM_{0,\beta+m-1}(g(t))M_{0,-\alpha+1}(g(t)).$
(C.2)

By (3.35) and (3.36), we obtain that $M_{0,-\alpha+1}(g(t))$ is uniformly
bounded as $-\alpha+1\in(\gamma,1]$.

Additionally, we have that $m+\beta-1\in(\gamma,m)$. Let $C_{0,\gamma}$ be the
constant found in Proposition 3.16, then there exists $\theta\in(0,1)$ such
that:

$\displaystyle M_{0,m+\beta-1}(g(t))\leq
M_{0,\gamma}(g(t))^{1-\theta}M_{0,m}(g(t))^{\theta}\leq
C_{0,\gamma}^{1-\theta}M_{0,m}(g(t))^{\theta}.$ (C.3)

Hence, combining (C) and (C.3), we find an invariant region in time for the
moment $M_{0,m}$. ∎

###### Proof of Proposition 3.18.

Let $C_{0,\tilde{m}}$ be the constant found in (3.37), with $\tilde{m}>1$ as
in Proposition 3.2, and let $\beta<1$ as in (1.7). For all $t\geq 0:$

$\displaystyle 1$ $\displaystyle=M_{0,1}(g(t))\leq
N^{1-\beta}\int_{(0,\infty)\times(0,N)}v^{\beta}g(a,v,t)\textup{d}v\textup{d}a+N^{1-\tilde{m}}\int_{(0,\infty)\times(N,\infty)}v^{\tilde{m}}g(a,v,t)\textup{d}v\textup{d}a$
$\displaystyle\leq
N^{1-\beta}M_{0,\beta}(g(t))+N^{1-\tilde{m}}\operatorname{max}\\{M_{0,\tilde{m}}(g_{\textup{in}}),C_{0,\tilde{m}}\\}.$
(C.4)

Thus, for $N>0$ sufficiently large, we obtain

$\displaystyle M_{0,\beta}(g(t))\geq\frac{1}{2N^{1-\beta}}.$ (C.5)

We analyse the term

$\displaystyle\int_{(0,\infty)^{2}}\int_{(0,\infty)^{2}}K_{\epsilon,R}(a,v,a^{\prime},v^{\prime})\xi_{R}(v+v^{\prime})g(a,v,t)g(a^{\prime},v^{\prime},t)[(v+v^{\prime})^{-l}-v^{-l}-v^{\prime-l}]\textup{d}v^{\prime}\textup{d}a^{\prime}\textup{d}v\textup{d}a.$

We make use of Remark 3.15: From the definition of $K_{\epsilon,R}$ in (3.1)
and the support of $g$, we have that $K_{\epsilon,R}=K$ when
$v,v^{\prime}\in[\epsilon,2R)^{2}$. Since
$(v+v^{\prime})^{-l}-v^{-l}-v^{\prime-l}\leq 0$, we can use the lower bound
for $K$ in (1.7). We obtain:

$\displaystyle\int_{(0,\infty)^{2}}\int_{(0,\infty)^{2}}K_{\epsilon,R}(a,v,a^{\prime},v^{\prime})\xi_{R}(v+v^{\prime})g(a,v,t)g(a^{\prime},v^{\prime},t)[(v+v^{\prime})^{-l}-v^{-l}-v^{\prime-l}]\textup{d}v^{\prime}\textup{d}a^{\prime}\textup{d}v\textup{d}a$
$\displaystyle\leq$ $\displaystyle
K_{1}\int_{\\{v<\frac{R}{2}\\}}\int_{\\{v^{\prime}<\frac{R}{2}\\}}(v^{-\alpha}v^{\prime\beta}+v^{\beta}v^{\prime-\alpha})g(a,v,t)g(a^{\prime},v^{\prime},t)[(v+v^{\prime})^{-l}-v^{-l}-v^{\prime-l}]\textup{d}v^{\prime}\textup{d}a^{\prime}\textup{d}v\textup{d}a$
$\displaystyle\leq$ $\displaystyle
K_{1}\int_{\\{v<\frac{R}{2}\\}}\int_{\\{v^{\prime}<\frac{R}{2}\\}}(-v^{-\alpha-l}v^{\prime\beta}-v^{\prime-\alpha-l}v^{\beta})g(a,v,t)g(a^{\prime},v^{\prime},t)\textup{d}v^{\prime}\textup{d}a^{\prime}\textup{d}v\textup{d}a$

and (3) thus becomes:

$\displaystyle\frac{\textup{d}}{\textup{d}t}M_{0,-l}(g(t))\leq(1+l)M_{0,-l}(g(t))-(1-\gamma)K_{1}\int_{\\{v<\frac{R}{2}\\}}v^{-\alpha-l}g(a,v,t)\textup{d}v\textup{d}a\int_{\\{v<\frac{R}{2}\\}}v^{\beta}g(a,v,t)\textup{d}v\textup{d}a.$
(C.6)

By (C.5)

$\displaystyle\int_{\\{v<\frac{R}{2}\\}}v^{\beta}g(a,v,t)\textup{d}v\textup{d}a\geq\frac{1}{2N^{1-\beta}}-\bigg{(}\frac{R}{2}\bigg{)}^{\beta-1}\geq\frac{1}{4N^{1-\beta}},$
(C.7)

for all $R>1$ that are sufficiently large.

As $-l\in(-\alpha-l,1)$, there exists $\theta\in(0,1)$ for which
$M_{0,-l}(g(t))\leq$ $M_{0,1}(g(t))^{1-\theta}M_{0,-\alpha-l}(g(t))^{\theta}$
$\leq$ $M_{0,1}(g_{\textup{in}})^{1-\theta}M_{0,-\alpha-l}(g(t))^{\theta}$.
Combining this with (C.6) and (C.7), there exists a constant $C>0$ such that:

$\frac{\textup{d}}{\textup{d}t}M_{0,-l}(g(t))\leq(1+l)M_{0,-l}(g(t))-C\bigg{(}\int_{\\{v<\frac{R}{2}\\}}v^{-l}g(a,v,t)\textup{d}v\textup{d}a\bigg{)}^{\frac{1}{\theta}}.$

We use $x^{\frac{1}{\theta}}\geq
2^{1-\frac{1}{\theta}}(x+y)^{\frac{1}{\theta}}-y^{\frac{1}{\theta}}$ with
$x=\int_{\\{v<\frac{R}{2}\\}}v^{-l}g(a,v,t)\textup{d}v\textup{d}a$ and
$y=\int_{\\{v>\frac{R}{2}\\}}v^{-l}g(a,v,t)\textup{d}v\textup{d}a$. We can
bound $\int_{\\{v>\frac{R}{2}\\}}v^{-l}g(a,v,t)\textup{d}v\textup{d}a$ from
above because $R>1$ and $M_{0,1}(g(t))$ is uniformly bounded. Thus

$\displaystyle\frac{\textup{d}}{\textup{d}t}M_{0,-l}(g(t))\leq(1+l)M_{0,-l}(g(t))-C_{1}M_{0,-l}(g(t))^{\frac{1}{\theta}}+2^{l+1}CM_{0,1}(g_{\textup{in}})$

and we conclude using the uniform bound on $M_{0,1}$ and then a comparison
argument. ∎

### Acknowledgements

The authors would like to thank B. Niethammer for useful suggestions regarding
the contents of the paper. The authors gratefully acknowledge the financial
support of the collaborative research centre The mathematics of emerging
effects (CRC 1060, Project-ID 211504053) and Bonn International Graduate
School of Mathematics at the Hausdorff Center for Mathematics (EXC 2047/1,
Project-ID 390685813) funded through the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation).

#### Statements and Declarations

Conflict of interest The authors declare that they have no conflict of
interest.

Data availability Data sharing not applicable to this article as no datasets
were generated or analysed during the current study.

## References

  * [1] J. Banasiak, W. Lamb and P. Laurençot  “Analytic Methods for Coagulation-Fragmentation Models, Volume II”  Boca Raton: CRC Press, 2019
  * [2] A. Bobylev, I. Gamba and V. Panferov  “Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions”  In _Journal of Statistical Physics_ 116, 2004, pp. 1651–1682
  * [3] I. Cristian  “Mathematical theory for two-dimensional coagulation equations”  Master’s thesis, University of Bonn, 2021
  * [4] I. Cristian and J. J. L. Velázquez  “Fast fusion in a two-dimensional coagulation model”  In _Preprint, arxiv: 2303.09475_ , 2023
  * [5] M. Escobedo, P. Laurençot, S. Mischler and B. Perthame  “Gelation and mass conservation in coagulation-fragmentation models”  In _Journal of Differential Equations_ 195.1, 2003, pp. 143–174
  * [6] M. Escobedo and S. Mischler  “Dust and self-similarity for the Smoluchowski coagulation equation”  In _Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire_ 23.3, 2006, pp. 331–362
  * [7] M. Escobedo, S. Mischler and M. Rodriguez Ricard  “On self-similarity and stationary problem for fragmentation and coagulation models”  In _Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire_ 22.1, 2005, pp. 99 –125
  * [8] M. A. Ferreira, J. Lukkarinen, A. Nota and J. J. L. Velázquez  “Localization in stationary non-equilibrium solutions for multicomponent coagulation systems”  In _Communications in Mathematical Physics_ 388.1, 2021, pp. 479–506
  * [9] M. A. Ferreira, J. Lukkarinen, A. Nota and J. J. L. Velázquez  “Asymptotic localization in multicomponent mass conserving coagulation equations”  In _Preprint, arxiv: 2203.08076_ , 2022
  * [10] M. A. Ferreira, J. Lukkarinen, A. Nota and J. J. L. Velázquez  “Non-equilibrium stationary solutions for multicomponent coagulation systems with injection”  In _Journal of Statistical Physics_ 190.98, 2023, pp. 1–35
  * [11] M. A. Ferreira, J. Lukkarinen, A. Nota and J. J. L. Velázquez  “Stationary non-equilibrium solutions for coagulation systems”  In _Archive for Rational Mechanics and Analysis_ 240, 2021, pp. 809–875
  * [12] N. Fournier and P. Laurençot  “Existence of self-similar solutions to Smoluchowski’s coagulation equation”  In _Communications in Mathematical Physics_ 256, 2005, pp. 589–609
  * [13] S. K. Friedlander  “Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics”  New York: Oxford University Press, 2000
  * [14] S.K. Friedlander and W. Koch  “The effect of particle coalescence on the surface area of a coagulating aerosol”  In _Journal of Colloid and Interface Science_ 140.2, 1990, pp. 419 –427
  * [15] H. Gajewski  “On a first order partial differential equation with nonlocal nonlinearity”  In _Mathematische Nachrichten_ 111.1, 1983, pp. 289–300
  * [16] M. Herrmann, P. Laurençot and B. Niethammer  “Self-similar solutions with fat tails for a coagulation equation with nonlocal drift”  In _Comptes Rendus Mathematique_ 347.15, 2009, pp. 909–914
  * [17] M. Herrmann, B. Niethammer and J. J. L. Velázquez  “Self-similar solutions for the LSW model with encounters”  In _Journal of Differential Equations_ 247.8, 2009, pp. 2282–2309
  * [18] P. Laurençot  “The Lifshitz-Slyozov equation with encounters”  In _Mathematical Models and Methods in Applied Sciences_ 11.4, 2001, pp. 731–748
  * [19] I. M. Lifshitz and V. V. Slyozov  “The kinetics of precipitation from supersaturated solid solutions”  In _Journal of Physics and Chemistry of Solids_ 19.1, 1961, pp. 35–50
  * [20] A. A. Lushnikov and M. Kulmala  “Nucleation burst in a coagulating system”  In _Physical Review E_ 62 American Physical Society, 2000, pp. 4932–4939
  * [21] G. Menon and R. L. Pego  “Approach to self-similarity in Smoluchowski's coagulation equations”  In _Communications on Pure and Applied Mathematics_ 57.9, 2004, pp. 1197–1232
  * [22] G. Menon and R. L. Pego  “Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence”  In _SIAM Review_ 48.4, 2006, pp. 745–768
  * [23] G. Menon and R. L. Pego  “The scaling attractor and ultimate dynamics for Smoluchowski’s coagulation equations”  In _Journal of Nonlinear Science_ 18.2, 2008, pp. 143–190
  * [24] B. Niethammer and J. J. L. Velázquez  “Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels”  In _Communications in Mathematical Physics_ 318.2 Springer ScienceBusiness Media LLC, 2012, pp. 505–532
  * [25] J. R. Norris  “Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent”  In _Annals of Applied Probability_ 9.1 The Institute of Mathematical Statistics, 1999, pp. 78–109
  * [26] M. V. Smoluchowski  “Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen”  In _Zeitschrift fur Physik_ 17, 1916, pp. 557–585
  * [27] I. W. Stewart  “A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels”  In _Mathematical Methods in the Applied Sciences_ 11.5, 1989, pp. 627–648
  * [28] S. Throm  “Uniqueness of measure solutions for multi-component coagulation equations”  In _Preprint: 2303.00775_ arXiv, 2023
  * [29] J. Wattis  “An introduction to mathematical models of coagulation–fragmentation processes: A discrete deterministic mean-field approach”  In _Physica D: Nonlinear Phenomena_ 222, 2006, pp. 1–20
  * [30] J. Wattis  “Exact solutions for cluster-growth kinetics with evolving size and shape profiles”  In _Journal of Physics A: Mathematical and General_ 39, 2006, pp. 7283–7298