# Fast Diffusion leads to partial mass concentration in Keller-Segel type
stationary solutions

J. A. Carrillo, M. G. Delgadino, R. L. Frank, M. Lewin

###### Abstract

We show that partial mass concentration can happen for stationary solutions of
aggregation-diffusion equations with homogeneous attractive kernels in the
fast diffusion range. More precisely, we prove that the free energy admits a
radial global minimizer in the set of probability measures which may have part
of its mass concentrated in a Dirac delta at a given point. In the case of the
quartic interaction potential, we find the exact range of the diffusion
exponent where concentration occurs in space dimensions $N\geq 6$. We then
provide numerical computations which suggest the occurrence of mass
concentration in all dimensions $N\geq 3$, for homogeneous interaction
potentials with higher power.

Nonlinear aggregation-diffusion equations of the form

$\partial_{t}\rho=\Delta\rho^{q}+\,\nabla\cdot\left(\rho\,\nabla
W\ast\rho\right)\,,$ (1)

are ubiquitous in continuous descriptions of populations, with applications in
mathematical biology, gravitational collapse and statistical mechanics [24,
31, 36, 23, 7, 39, 37, 22, 5, 15, 13]. Here $\rho(t)$ is a time-dependent
probability measure over ${\mathbb{R}}^{N}$, $q>0$ is the diffusion exponent
regulating if the diffusion is slow ($q>1$), linear ($q=1$) or fast ($0<q<1$)
for small values of the density, and $W$ is the aggregation kernel describing
some attraction between the agents of the population. In this work, $W$ will
be taken equal to the homogeneous potential
$W_{\lambda}(x):=|x|^{\lambda}/\lambda$ with $\lambda>-N$. A positive
$\lambda$ corresponds to a bounded interaction potential at the origin,
whereas $-N<\lambda<0$ provide locally integrable singular interaction
potentials. At $\lambda=0$ the convention is to take $W_{0}(x)=\log|x|$ which,
in dimension $N=2$, is the Newtonian potential. The classical
parabolic–elliptic Keller–Segel system [24, 23, 20] for cell movement by
chemotaxis is obtained for $N=2$, $q=1$ and $\lambda=0$.

The time-dependent equation (1) is the (formal) gradient flow of a free energy
functional [18, 1, 34] defined for (regular enough) probability measures
$\mu\in{\mathcal{P}}({\mathbb{R}}^{N})$ by

$\displaystyle{\mathcal{F}}[\mu]$
$\displaystyle:=-\frac{1}{1-q}\int_{{\mathbb{R}}^{N}}\mu_{\rm
ac}(x)^{q}\,{\mathrm{d}}x+\frac{1}{2\lambda}\,\iint_{{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}}|x-y|^{\lambda}\,{\mathrm{d}}\mu(x)\,{\mathrm{d}}\mu(y),$
(2)

where $\mu_{\rm ac}$ is the absolutely continuous part of $\mu$ with respect
to the Lebesgue measure. At $q=1$, the first term is replaced by Boltzmann’s
entropy $\int_{{\mathbb{R}}^{N}}\log\mu_{\rm ac}(x)\,{\mathrm{d}}\mu(x)$. This
formal structure, which can be turned rigorous only for some particular values
of the parameters [19], plays an essential role in the analysis of the
dynamics in (1). In particular, the global minimizers of the free energy
functional ${\mathcal{F}}$ are the best candidates to be locally stable
equilibria for (1) and the large-time asymptotics of time-dependent solutions.

Nonlinear aggregation equations of the form (1) show very challenging
phenomena, both with regard to their time behavior and the properties of their
steady states. They have received lots of attention in the last 20 years, see
for instance [24, 31, 36, 23, 7, 39, 37, 22, 5, 15, 13, 32, 33] and the
references therein. The form of the global minimizers depending on the
parameters is quite rich, but by now it is well understood in the case of
porous medium-like diffusion $q>1$ and linear diffusion $q=1$, see [2, 26, 27,
28, 24, 20, 7, 6, 4, 9, 12, 21, 16, 15] for some references. For all possible
values of $\lambda$ where global minimizers of the free energy exist, they are
given by bounded probability densities.

This work continues the investigation initiated in [14] where J. Dolbeault, F.
Hoffmann and the first three authors of this paper dealt with the much less
understood case of $0<q<1$ and $\lambda>0$. It was shown in [14] that global
minimizers of the free energy do not necessarily exist, even when
${\mathcal{F}}$ is bounded-below, if we insist on requiring that $\mu$ is
absolutely continuous with respect to the Lebesgue measure. However,
minimizers in the form of measures exist when ${\mathcal{F}}$ is bounded-
below. By rearrangement we can, in fact, restrict the minimization of
${\mathcal{F}}$ to radially non-increasing probability distributions $\mu$
modulo translations [15], and those can only have a singularity at the origin
in the form of a Dirac delta. For these probability measures
$\mu=M\delta_{0}+\rho$ with $\rho\in L^{1}({\mathbb{R}}^{N})$ a non-increasing
radial function, the free energy becomes

$\displaystyle\mathcal{F}[\rho,M]\displaystyle=$
$\displaystyle-\frac{1}{1-q}\int_{{\mathbb{R}}^{N}}\rho(x)^{q}\,dx+M\int_{{\mathbb{R}}^{N}}|x|^{\lambda}\rho(x)\,{\mathrm{d}}x$
(3)
$\displaystyle\qquad+\frac{1}{2}\iint_{{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}}\rho(x)|x-y|^{\lambda}\rho(y)\,{\mathrm{d}}x\,{\mathrm{d}}y.$

It was proved in [14] that this free energy is bounded from below if and only
if $N/(N+\lambda)<q<1$, and in this case admits at least one global minimizer
$\mu_{*}=M_{*}\delta_{0}+\rho_{*}$ with

$\rho_{*}\in L^{q}({\mathbb{R}}^{N})\cap L^{1}({\mathbb{R}}^{N})\cap
L^{1}({\mathbb{R}}^{N},|x|^{\lambda}\,{\mathrm{d}}x).$ (4)

The function $\rho_{*}$ is always supported on all of ${\mathbb{R}}^{N}$ and,
if $M_{*}>0$, it diverges at the origin. We note parenthetically that the
minimization of the free energy $\mathcal{F}$ is equivalent to finding the
sharp constant in a reversed Hardy-Littlewood-Sobolev inequality.

Two important questions are whether these minimizers are unique modulo
translations, and whether mass concentration occurs in the sense that
$M_{*}>0$. These questions have been solved in [14] for some values of the
parameters, but only negatively concerning the occurrence of concentration.
For instance, it was shown in [14] that minimizers are always unique for
$2\leq\lambda\leq 4$ and that $M_{*}=0$ for

  * •

all $\lambda>0$ and $\frac{N}{N+\lambda}<q<1$ if $N=1,2$,

  * •

all $0<\lambda\leq 2+\frac{4}{N-2}$ and $\frac{N}{N+\lambda}<q<1$ if $N\geq
3$.

Therefore, concentration never happens in dimensions $N=1,2$, nor in higher
dimensions for too small values of the aggregation parameter $\lambda$. For
example, concentration never happens at $\lambda=2$ in any dimension $N\geq
1$.

We have no intuitive explanation of why concentration cannot happen in
dimensions $N=1,2$ but we notice that the same phenomenon occurs for other
phase transitions in statistical physics, by the Mermin-Wagner theorem [30,
35]. This is for instance the case of Bose-Einstein condensation [38].

Our main goal in this work is to show that concentration does indeed occur in
dimensions $N\geq 3$, for large-enough values of $\lambda$. First, we look at
the quartic case $\lambda=4$ which we can solve completely. We prove in
Theorem 1 below that concentration happens for some (but not all) values of
$q$ in dimension $N\geq 6$ but never occurs in dimensions $N\leq 5$. However,
it does occur in dimensions $N\in\\{3,4,5\\}$ at smaller values of $q$ where
the free energy is unbounded from below, if we allow formal minimizers with an
infinite free energy (see Remark 3). Then, we provide numerical evidence that
concentration does happen for regular minimizers at larger values of
$\lambda$, starting with dimension $N=3$.

Our numerical results detailed in Section 2 suggest some interesting features
of the model, which we are unfortunately not able to prove at the moment. The
numerical evidence indicates that the concentration region satisfies some
monotonicity in terms of our three parameters $N$, $\lambda$ and $q$. More
precisely, if concentration occurs for some $(N,\lambda,q)$ then it should
also occur for all $(N^{\prime},\lambda^{\prime},q^{\prime})$ so that
$N^{\prime}\geq N$, $N^{\prime}/(N^{\prime}+\lambda)<q^{\prime}\leq q$ and
$\lambda^{\prime}\geq\lambda$. Thus, for any fixed $\lambda>0$ and
$N\in{\mathbb{N}}$, the concentration region for $q$ should be an interval
$\big{(}\frac{N}{N+\lambda},q_{N}(\lambda)\big{)}$, with $q_{N}(\lambda)$ a
non-decreasing function. For instance, we think that the interval starts to be
non-empty for $\lambda$ slightly above $8$ in dimension $N=3$. Proving these
observations seems challenging.

Our theoretical and numerical results are both based on the associated (first
order) Euler-Lagrange equation for the absolutely continuous part $\rho_{*}$:

$\frac{q}{1-q}\rho_{*}^{q-1}=\rho_{*}\ast|\cdot|^{\lambda}+M_{*}|x|^{\lambda}-\int_{{\mathbb{R}}^{N}}|y|^{\lambda}\rho_{*}(y)\,{\mathrm{d}}y\;+L,$
(5)

where $\frac{q}{1-q}\rho_{*}^{q-1}(0)=L\geq 0$ is a Lagrange multiplier
associated with the mass constraint

$\int_{{\mathbb{R}}^{N}}\rho(x)\,{\mathrm{d}}x+M=1.$ (6)

A useful fact will be that (5) can be highly simplified when $\lambda$ is an
even integer. In this case we can expand $|x-y|^{2n}$ in terms of a polynomial
in $x$ and $y$ and, using the radial symmetry of $\rho_{*}$, the convolution
$\rho_{*}\ast|\cdot|^{\lambda}$ becomes a simple polynomial in $|x|^{2}$. Thus
(5) can be turned into a simpler equation for these finitely many polynomial
coefficients. This is how we will be able to solve completely the particular
case $\lambda=4$. This simplification will also be used to design a rather
precise numerical algorithm in the case $\lambda\in 2{\mathbb{N}}$, and to
deal with the other values of $\lambda$.

We conclude this introduction with more comments about the meaning of our
findings in light of the time-dependent equation (1). Any minimizer
$\mu_{*}=M_{*}\delta_{0}+\rho_{*}$ for the free energy (2) is a stationary
solution of (1), in the sense that

$\Delta\rho_{*}^{q}+\,\nabla\cdot\left(\mu_{*}\,\nabla
W_{\lambda}\ast\mu_{*}\right)=0$

as distributions, due to the properties (4) proved in [14]. We point out that
the range $0<q<1$ is usually called “fast diffusion” corresponding to the
faster diffusion (than the heat equation) for small values of the density
while the diffusion is in fact slower for large values of the density, where
concentration happens. Therefore, when $q$ gets smaller, the diffusion of a
Dirac delta at the origin is weaker. Even more, as proved in [8] by Brézis and
Friedman, an initial Dirac delta is in fact a kind of stationary solution to
the fast diffusion equation for $0<q<\frac{N-2}{N}$. More precisely, they
showed that approximating an initial Dirac delta at the origin by mollifiers
and sending the regularization parameter to zero was not leading to a source-
type strong $L^{1}$-solution (Barenblatt-type solution) as opposed to the case
$q>\frac{N-2}{N}$. In fact, they proved that the Dirac delta stays “stable”
for all times [8, Thm. 8]. It is an open problem to give sense to the
evolution problem (1) with general probability measures as initial datum. The
Wasserstein gradient flow of the suitably defined extension of the free energy
(2) to measures as in [14, Sect. 5] is the natural candidate.

Our results show that the combined effect of the small diffusion at large
densities with the attraction due to the potential $W_{\lambda}$ can lead to a
partial concentration of mass at the origin in the stationary case.
Understanding the well-posedness and long time behavior of the evolution
equation (1) is essentially open in this regime of parameters. The possible
concentration of mass happening in finite or infinite time for its evolution
is just one of the many challenging open questions. The long time behavior has
recently been investigated in the preprint [10], but for $q$ so close to 1
that no concentration can happen.

The paper is organized as follows. The next section is devoted to the special
case $\lambda=4$ whereas in Section 2 we present our numerical simulations for
other values of $\lambda$, looking first at the special case of even integers
and then general values.

##  1 The quartic case $\lambda=4$

In the case $\lambda=4$, we are reduced to the regime

$\frac{N}{N+4}<q<\frac{N}{N+2}\,,$

since it was already proved in [14, Prop. 14] that concentration does not
happen for larger values of $q$ and that the free energy is not bounded below
for smaller values. We know from [14, Prop. 20] that there is a minimizer
$(\rho_{*},M_{*})$ and by [14, Thm. 27] that this minimizer is unique up to
translations. The question is whether $M_{*}=0$ or not. We define

$\boxed{q_{N}(4):=\frac{N-2}{N+2}\left(1+\frac{4}{3N}\right)}$

which will be proved to be the critical exponent at $\lambda=4$ in the next
statement. For all $N\geq 6$, one has

$\frac{N}{N+4}<q_{N}(4)<\frac{N-2}{N}<\frac{N}{N+2},$

whereas $q_{N}(4)<\frac{N}{N+4}$ for $N\leq 5$. Note that $q_{N}(4)$ is
increasing with the dimension $N$.

######  Theorem 1.

Let $\lambda=4$ and $\frac{N}{N+4}<q<\frac{N}{N+2}$. Then ${\mathcal{F}}$ in
(3) admits a unique minimizer $(\rho_{*},M_{*})$ with $\rho_{*}$ a radial
function satisfying (4).

  1. (a)

If $N\leq 5$, then $M_{*}=0$.

  2. (b)

If $N\geq 6$ and $q<q_{N}(4)$, then $\displaystyle
M_{*}=\frac{3}{2}\,\frac{q_{N}(4)-q}{\frac{N-2}{N}-q}>0$.

  3. (c)

If $N\geq 6$ and $q\geq q_{N}(4)$, then $M_{*}=0$.

In dimensions $N\geq 6$ with $q\leq q_{N}(4)$, we are able to compute the
unique minimizer explicitly. It is given by

$\rho_{*}(x)=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}\left(|x|^{4}+B_{*}|x|^{2}\right)^{-\frac{1}{1-q}}$
(7)

for a constant $B_{*}>0$ given in the proof of Lemma 2. Note that $\rho_{*}$
diverges at the origin, as it should [14]. Note also that the concentrated
mass $M_{*}$ is decreasing with respect to $q$. One could also compute
$\mathcal{F}[\rho_{*},M_{*}]$ explicitly for $q\leq q_{N}(4)$. See Remark 3
below for an interpretation of what is happening in dimensions
$N\in\\{3,4,5\\}$.

### Proof of Theorem 1

First, we recall that any minimizer of the form
$\mu_{*}=M_{*}\delta_{0}+\rho_{*}$ solves the nonlinear equation (5). Using

$|x-y|^{4}=|x|^{4}+|y|^{4}+4(x\cdot y)^{2}+2|x|^{2}|y|^{2}-4|x|^{2}x\cdot
y-4|y|^{2}x\cdot y$

and the fact that $\rho_{*}$ is radial, we can express the convolution in the
form

$\displaystyle\int_{{\mathbb{R}}^{N}}|x-y|^{4}\rho_{*}(y)\,{\mathrm{d}}y=$
$\displaystyle\,|x|^{4}\int_{{\mathbb{R}}^{N}}\rho_{*}(y)\,{\mathrm{d}}y+\int_{{\mathbb{R}}^{N}}|y|^{4}\rho_{*}(y)\,{\mathrm{d}}y$
$\displaystyle\quad+\\!\left(2+\frac{4}{N}\right)|x|^{2}\\!\int_{{\mathbb{R}}^{N}}|y|^{2}\rho_{*}(y)\,{\mathrm{d}}y\,.$
(8)

Thus, using the mass constraint (6), the Euler-Lagrange equation (5) for
$\rho_{*}$ can be rewritten in the form

$\begin{cases}\displaystyle\frac{q}{1-q}\rho_{*}(x)^{q-1}=|x|^{4}+B|x|^{2}+L,\\\\[8.53581pt]
\displaystyle
B=\left(2+\frac{4}{N}\right)|x|^{2}\int_{{\mathbb{R}}^{N}}|y|^{2}\rho_{*}(y)\,{\mathrm{d}}y,\end{cases}$
(9)

where we recall that $L\geq 0$ is an unknown Lagrange multiplier. Our goal is
to find the values of $L$ and $B$ in (9). Of course we then have
$M_{*}=1-\int_{{\mathbb{R}}^{N}}\rho_{*}$.

The idea of the proof is to look at all the possible solutions of (9)
parametrized by $L$. Plugging the first formula into the second, we obtain a
simple nonlinear equation for $B$ which we show admits a _unique solution_
$B(L)$ for any $L\geq 0$. A scaling argument will also give us an exact
expression of $B(0)$. Next, we show that the mass $m(L)$ of this solution is
strictly decreasing with $L$. Thus, we have two possibilities either $m(0)\geq
1$ or $m(0)<1$. If $m(0)\geq 1$, then there exists a unique $L_{*}\geq 0$ such
that $m(L_{*})=1$, and the minimizer is given by $(\rho_{*},0)$, where
$\rho_{*}$ is the solution of (9) with this $L_{*}$. This conclusion follows
from the uniqueness of critical points, which is shown by a convexity
argument. Alternatively, if $m(0)<1$, then there exist no solution of (9) with
mass $1$ and thus there must be concentration. From [14, Prop. 14] we then
know that $L=0$, that is, $\rho_{*}$ is given by the unique solution with
$L=0$ and $B=B(0)$ and we have $M_{*}=1-m(0)>0$.

The following contains the main properties of solutions of (9) that we need
for the proof.

######  Lemma 2.

Let $\lambda=4$ and $\max\left(0,\frac{N-2}{N+2}\right)<q<\frac{N}{N+2}$.

  1. (a)

There is a unique differentiable function $B:[0,\infty)\to(0,\infty)$ such
that for each $L\geq 0$,

$\rho_{L}(x):=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}\left(|x|^{4}+B(L)|x|^{2}+L\right)^{-\frac{1}{1-q}}$

satisfies

$\left(2+\frac{4}{N}\right)\int_{{\mathbb{R}}^{N}}|y|^{2}\rho_{L}(y)\,{\mathrm{d}}y=B(L).$

  2. (b)

The mass
$m(L):=\int_{{\mathbb{R}}^{N}}\rho_{L}(x)\,{\mathrm{d}}x\in(0,+\infty]$ is
continuous and strictly decreasing with respect to $L\geq 0$. It converges to
0 when $L\to\infty$.

  3. (c)

At $L=0$ we have

$m(0)=\begin{cases}\frac{1}{2}\,\frac{q-\frac{N-2}{N+2}}{\frac{N-2}{N}-q}&\text{if
$q<\frac{N-2}{N}$,}\\\\[8.53581pt] +\infty&\text{otherwise.}\end{cases}$ (10)

Thus, if $q<\frac{N-2}{N}$, then $m(0)\leq 1$ if and only if $q\leq q_{N}(4)$.

###### Proof of Lemma 2.

Part (a). Let us define

$\rho_{B,L}(x)=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}\left(|x|^{4}+B|x|^{2}+L\right)^{-\frac{1}{1-q}}$
(11)

for all $B,L\geq 0$ and consider the function $F_{L}(B)$ given by

$\displaystyle
F_{L}(B):=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}|{\mathbb{S}}^{N-1}|\,\int_{0}^{\infty}\frac{r^{N+1}{\mathrm{d}}r}{\left(r^{4}+Br^{2}+L\right)^{\frac{1}{1-q}}}=\int_{{\mathbb{R}}^{N}}|y|^{2}\rho_{B,L}(y)\;{\mathrm{d}}y.$
(12)

This function is well defined for all $L\geq 0$ and $B>0$ under our
assumptions on $q$. It is strictly decreasing as a function of $B$ and $L$
separately. Moreover, for every given $L\geq 0$ the limit as $B\to 0^{+}$ is
positive (infinite if $L=0$) and the limit $B\to\infty$ is zero. Therefore,
for every $L\geq 0$, there exists a unique $B(L)>0$ such that

$\displaystyle B(L)=\kappa\,F_{L}\big{(}B(L)\big{)}\qquad\mbox{with
}\kappa:=2+\frac{4}{N}.$ (13)

From the monotonicity of $F_{L}$, we have that $B$ is decreasing with $L$.
From the implicit function theorem, $B$ is in fact a smooth function of $L$.
Using $r^{4}+B(L)r^{2}+L\geq r^{4}+L$ and scaling out $L$ we find

$\displaystyle B(L)$
$\displaystyle\leq\kappa\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}|{\mathbb{S}}^{N-1}|\,\int_{0}^{\infty}\frac{r^{N+1}{\mathrm{d}}r}{\left(r^{4}+L\right)^{\frac{1}{1-q}}}$
$\displaystyle=\kappa\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}|{\mathbb{S}}^{N-1}|L^{\frac{N+2}{4}-\frac{1}{1-q}}\,\int_{0}^{\infty}\frac{r^{N+1}{\mathrm{d}}r}{\left(r^{4}+1\right)^{\frac{1}{1-q}}}.$

Under our assumption on $q$ we have $\frac{N+2}{4}<\frac{1}{1-q}$ and thus
$B(L)\to 0$ when $L\to\infty$. We will compute the exact value of $B(0)$
below.

Part (b). Now we show that the mass $m(L)$ of $\rho_{L}$ is decreasing in $L$.
Similarly as above, we write the integral in radial coordinates to obtain

$m(L)=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}\\!\\!\\!|{\mathbb{S}}^{N-1}|\int_{0}^{\infty}\\!\\!\\!\frac{r^{N-1}{\mathrm{d}}r}{\left(r^{4}+B(L)r^{2}+L\right)^{\frac{1}{1-q}}}.$
(14)

The integral converges for all $L>0$, but not necessarily for $L=0$. The same
estimate as for $B(L)$ provides

$m(L)\leq\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}|{\mathbb{S}}^{N-1}|L^{\frac{N}{4}-\frac{1}{1-q}}\int_{0}^{\infty}\frac{r^{N-1}{\mathrm{d}}r}{\left(r^{4}+1\right)^{\frac{1}{1-q}}}$

and shows that $m(L)\to 0$ when $L\to\infty$. Then, taking a derivative we
obtain

$m^{\prime}(L)=-\int_{0}^{\infty}(1+B^{\prime}(L)r^{2})\varphi(r)\,{\mathrm{d}}r,$

where we have introduced

$\varphi(r):=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}\frac{|{\mathbb{S}}^{N-1}|}{1-q}r^{d-1}\left(r^{4}+B(L)r^{2}+L\right)^{-\frac{1}{1-q}-1}$

to simplify the notation. On the other hand, we have from (13)

$B^{\prime}(L)=-\kappa\int_{0}^{\infty}r^{2}(1+B^{\prime}(L)r^{2})\varphi(r)\,{\mathrm{d}}r.$

Hence

$B^{\prime}(L)=-\frac{\kappa\int_{0}^{\infty}r^{2}\varphi(r)\,{\mathrm{d}}r}{1+\kappa\int_{0}^{\infty}r^{4}\varphi(r)\,{\mathrm{d}}r}$

and thus,

$\displaystyle
m^{\prime}(L)=-\int_{0}^{\infty}\varphi(r)\,{\mathrm{d}}r+\frac{\kappa\left(\int_{0}^{\infty}r^{2}\varphi(r)\,{\mathrm{d}}r\right)^{2}}{1+\kappa\int_{0}^{\infty}r^{4}\varphi(r)\,{\mathrm{d}}r}\leq-\frac{\int_{0}^{\infty}\varphi(r)\,{\mathrm{d}}r}{1+\kappa\int_{0}^{\infty}r^{4}\varphi(r)\,{\mathrm{d}}r}<0,$

since

$\left(\int_{0}^{\infty}r^{2}\varphi(r)\,{\mathrm{d}}r\right)^{2}\leq\int_{0}^{\infty}\varphi(r)\,{\mathrm{d}}r\int_{0}^{\infty}r^{4}\varphi(r)\,{\mathrm{d}}r$

by the Cauchy-Schwarz inequality. This proves our claim that the mass is a
decreasing function of $L>0$.

Part (c). By scaling out $B(0)$ in (13) via the change of variables
$r=\sqrt{B(0)}\bar{r}$ we obtain

$B(0)=\left(2+\frac{4}{N}\right)c_{N,q}\,B(0)^{\frac{N+2}{2}-\frac{2}{1-q}}\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}$

or equivalently

$B(0)^{\frac{2}{1-q}-\frac{N}{2}}=\left(2+\frac{4}{N}\right)c_{N,q}\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}},$
(15)

with

$c_{N,q}=|\mathbb{S}^{N-1}|\int_{0}^{\infty}\frac{r^{N+1}}{(r^{4}+r^{2})^{\frac{1}{1-q}}}\,{\mathrm{d}}r\,,$

which is finite under our assumptions on $q$. Next we turn to the mass. Since
$B(L)\to B(0)>0$ we have

$\lim_{L\to
0^{+}}m(L)=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}|{\mathbb{S}}^{N-1}|\int_{0}^{\infty}\frac{r^{N-1}{\mathrm{d}}r}{\left(r^{4}+B(0)r^{2}\right)^{\frac{1}{1-q}}}.$

This is infinite for $q\geq\frac{N-2}{N}$ due to the singularity at the
origin. For $q<\frac{N-2}{N}$ we can compute the explicit value of $m(0)$ in
(10), using the formula (15) of $B(0)$. By scaling we have, this time,

$\displaystyle m(0)=\int_{{\mathbb{R}}^{N}}\rho_{0}(y)\,{\mathrm{d}}y$
$\displaystyle=c_{N,q}^{\prime}\,B(0)^{-\frac{N}{2}+\frac{2}{1-q}}\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}$

with

$c_{N,q}^{\prime}=|\mathbb{S}^{N-1}|\int_{0}^{\infty}\frac{r^{N-1}}{(r^{4}+r^{2})^{\frac{1}{1-q}}}\,{\mathrm{d}}r.$

Inserting (15) into this expression, we obtain

$m(0)=\frac{c_{N,q}^{\prime}}{c_{N,q}}\,\frac{N}{2(N+2)}\,.$

With the change of variables $t=r^{2}$ we can write

$\displaystyle c_{N,q}$
$\displaystyle=\frac{1}{2}\,|\mathbb{S}^{N-1}|\int_{0}^{\infty}t^{\frac{N}{2}-\frac{1}{1-q}}(1+t)^{-\frac{1}{1-q}}\,{\mathrm{d}}t$

and

$c_{N,q}^{\prime}=\frac{1}{2}\,|\mathbb{S}^{N-1}|\int_{0}^{\infty}t^{\frac{N-2}{2}-\frac{1}{1-q}}(1+t)^{-\frac{1}{1-q}}\,{\mathrm{d}}t.$

Thus, by beta and gamma function identities,

$\frac{c_{N,q}^{\prime}}{c_{N,q}}=\frac{\Gamma(\tfrac{N}{2}-\tfrac{1}{1-q})\,\Gamma(\tfrac{2}{1-q}-\tfrac{N}{2})}{\Gamma(\tfrac{N+2}{2}-\tfrac{1}{1-q})\,\Gamma(\tfrac{2}{1-q}-\tfrac{N+2}{2})}=\frac{N+2}{N}\,\frac{q-\frac{N-2}{N+2}}{\frac{N-2}{N}-q},$

which proves (10) and concludes the proof of the Lemma 2. ∎

Note that Lemma 2 covers a larger range for the exponent $q$ since
$\frac{N-2}{N+2}<\frac{N}{N+4}$, see Remark 3. However, the free energy
${\mathcal{F}}$ is bounded-below only for $q>\frac{N}{N+4}$, which we assume
from now on. For the convenience of the reader we provide a self-contained
proof of Theorem 1 using Lemma 2, which does not use any material from [14]
and solely relies on the convexity of ${\mathcal{F}}$ in the cone of radial
probability measures, in the spirit of Lopes’ work [29].

There are two situations. When $q\geq q_{N}(4)$, Lemma 2 implies that there is
a unique $L_{*}\geq 0$ so that $m(L_{*})=1$. In this case we define
$(\rho_{*},M_{*}):=(\rho_{L_{*}},0)$. On the other hand, when $q<q_{N}(4)$
(which can only happen in dimensions $N\geq 6$), then the equation $m(L)=1$
admits no solution. In this case we choose $L_{*}=0$ and set
$(\rho_{*},M_{*}):=(\rho_{0},1-\int_{{\mathbb{R}}^{N}}\rho_{0})$, which
satisfies $M_{*}>0$. The choice $L_{*}=0$ is dictated by [14, Prop. 14] but we
will see below that this follows from the first order Euler-Lagrange condition
on $M_{*}$, which we have not yet used in the argument.

We claim that the so-defined $(\rho_{*},M_{*})$ is the unique minimizer of
${\mathcal{F}}$, which will conclude the proof. To prove this claim, we notice
that for $\mu=M\delta_{0}+\rho$ with $\rho$ a radial function, we can use (8)
to write

${\mathcal{F}}[\rho,M]=\int_{{\mathbb{R}}^{N}}|x|^{4}\rho(x)\,{\mathrm{d}}x+\left(1+\frac{2}{N}\right)\left(\int_{{\mathbb{R}}^{N}}|x|^{2}\rho(x)\,{\mathrm{d}}x\right)^{2}-\frac{1}{1-q}\int_{{\mathbb{R}}^{N}}\rho(x)^{q}{\mathrm{d}}x,$
(16)

From this, it is apparent that ${\mathcal{F}}$ is strictly convex in $\rho$.
Let us now prove that ${\mathcal{F}}[\rho,M]\geq{\mathcal{F}}[\rho_{*},M_{*}]$
for every $\int_{{\mathbb{R}}^{N}}\rho+M=1$, with equality if and only if
$(\rho,M)=(\rho_{*},M_{*})$. Using (16) and the definition (11) of $\rho_{*}$
we obtain after a calculation

$\displaystyle{\mathcal{F}}[\rho,M]-{\mathcal{F}}[\rho_{*},M_{*}]=$
$\displaystyle
L_{*}\int_{{\mathbb{R}}^{N}}(\rho_{*}-\rho)(x)\,{\mathrm{d}}x+\left(1+\frac{2}{N}\right)\left(\int_{{\mathbb{R}}^{N}}|x|^{2}(\rho-\rho_{*})(x)\,{\mathrm{d}}x\right)^{2}$
$\displaystyle\quad-\frac{1}{1-q}\int_{{\mathbb{R}}^{N}}\left(\rho^{q}-q\rho_{*}^{q-1}(\rho-\rho_{*})-\rho_{*}^{q}\right)(x)\,{\mathrm{d}}x.$
(17)

Let us distinguish cases here. If $M_{*}=0$, then
$\int_{{\mathbb{R}}^{N}}\rho_{*}=1$ and the first term on the right side is
certainly non-negative. If $M_{*}>0$ then the integral can have either sign,
but the term vanishes since we have chosen $L_{*}=0$. Hence in both cases, the
first term is non-negative. This is how the two conditions $L_{*}=0$ for
$\int_{{\mathbb{R}}^{N}}\rho_{*}<1$ and $\int_{{\mathbb{R}}^{N}}\rho_{*}=1$
for $L_{*}>0$ appear to be necessary for minimizers. The other solutions
$(\rho_{L},1-\int_{{\mathbb{R}}^{N}}\rho_{L})$ cannot be minimizers. On the
other hand, by concavity of $a\mapsto a^{q}$ on ${\mathbb{R}}_{+}$ we have
$a^{q}-qb^{q-1}(a-b)-b^{q}\leq 0$ with equality if and only if $a=b$. Thus we
have proved, as we claimed, that
${\mathcal{F}}[\rho,M]\geq{\mathcal{F}}[\rho_{*},M_{*}]$ with equality if and
only if $\rho=\rho_{*}$. This implies $M=M_{*}$ due to the mass constraint and
concludes the proof of Theorem 1.∎

######  Remark 3 (Concentration for $3\leq N\leq 5$).

For $\frac{N-2}{N+2}<q\leq\frac{N}{N+4}$ we can still define
$(\rho_{*},M_{*})$ by the same procedure. The right side of (17) makes sense
and is non-negative. Thus $(\rho_{*},M_{*})$ is a formal minimizer of the free
energy, but the corresponding value is infinite:
${\mathcal{F}}[\rho_{*},M_{*}]=-\infty$. This is due to the first and third
integrals in (16) which diverge at large $x$. With this formal definition of a
minimizer, we see that concentration indeed happens at $\lambda=4$ for all
$N\geq 3$, since $q_{N}(4)>\frac{N-2}{N+2}$ in this case. This fact can allow
one to use relative free energy quantities as a tool to quantify the basin of
attraction to equilibria similarly to the case of the Barenblatt solutions in
the very fast diffusion range even if their second moment or their mass
becomes infinite leading to infinite free energy, see [17, 25, 3].

######  Remark 4 (Concentration is independent of mass).

Notice that we have chosen to work with probability measures. Assuming a
different mass normalization $\mu({\mathbb{R}}^{N})=m$, the minimizer
$\mu_{*,m}$ of the free energy (2) is given by

$\begin{cases}\mu_{*,m}=\gamma_{1}\mu_{*}(\gamma_{2}x)\\\
\gamma_{1}\gamma_{2}^{-N}=m\\\
\gamma_{1}^{3-q}\gamma_{2}^{-\lambda-2N}=m,\end{cases}$

with $\mu_{*}$ the minimizer of the free energy (2) of mass 1.

##  2 Numerical results for other values of $\lambda$

For $\lambda=4$, we have shown in Lemma 2 that the mass $m(L)$ of the unique
solution $\rho_{L}$ to the simplified Euler-Lagrange equation (9) is monotone
decreasing with respect to the parameter $L$. Therefore, when $m(0)<1$ we can
conclude that concentration occurs. We think that the same strategy applies to
other values of $\lambda$. Namely, the idea is to look for a radial solution
of the Euler-Lagrange equation (5) _imposing_ $L=0$, and compute its mass
$m(0)$. Should the latter be less than 1, we would have found a stationary
state with concentration, which is a good candidate for being a global
minimizer. If the mass is monotone with $L$ there exist in fact no solution of
the Euler-Lagrange equation with mass one. If on the contrary $m(0)>1$, this
proves that a minimizer has to have $L>0$ and thus it cannot display
concentration, by [14, Prop. 14].

To make this strategy work, we would have to show the existence and uniqueness
of solutions to (5) for every $L\geq 0$ and prove that the corresponding mass
$m(L)$ is decreasing. We are not able to show this fact analytically. However,
we have numerically investigated this question thoroughly for a large range of
$\lambda$. The numerical evidence supports the conjecture that these claims
hold. We present here our numerical results, dealing first with the simpler
case of even integers. The python scripts and their outputs can be found in
the GitHub repository ‘Concentration’ [11] together with a list of the cases
investigated.

###  2.1 The case of even integers

When $\lambda=2n$ is an even integer, we can expand $|x-y|^{2n}$ in terms of a
polynomial in $x$ and $y$. Using the radial symmetry of $\rho_{*}$, we obtain
that only the even terms contribute to the convolution
$\rho_{*}\ast|\cdot|^{\lambda}$ in the Euler-Lagrange equation (5). In
addition, the coefficient of $|x|^{\lambda}$ simplifies due to the mass
constraint and (5) can be rewritten in the form

$\frac{q}{1-q}\rho_{*}^{q-1}\\!=\\!|x|^{\lambda}\\!+\\!\sum_{i=1}^{n-1}c^{\lambda}_{i}\left(\int_{{\mathbb{R}}^{N}}|y|^{2n-2i}\rho_{*}(y)\,{\mathrm{d}}y\right)\\!|x|^{2i}+L,$
(18)

where $c_{i}^{\lambda}$ is the positive coefficient of
$|x|^{2i}\int_{{\mathbb{R}}^{N}}|y|^{2n-2i}\rho(y)\,{\mathrm{d}}y$ in the
polynomial $\int_{{\mathbb{R}}^{N}}|x-y|^{2n}\rho(y)\,{\mathrm{d}}y$, for any
radial function $\rho$. The values of $c_{i}^{\lambda}$ are provided in the
appendix for $\lambda\in\\{2,4,6,8,10\\}$ for completeness. For instance, we
have $c^{4}_{1}=2+4/N$, $c^{6}_{1}=c^{6}_{2}=3+12/N$, etc. Note that the
unknown mass $M_{*}$ has disappeared from the equation but it of course still
appears in the mass constraint (6).

The new form (18) of our Euler-Lagrange equation implies that we can restrict
our attention to densities

$\rho_{\beta,L}(x)=\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}\left(|x|^{\lambda}+\sum_{i=1}^{n-1}\beta_{i}|x|^{2i}+L\right)^{-\frac{1}{1-q}}\\!\\!\\!\\!,$
(19)

with $\beta_{i},L\geq 0$. The equation (18) then reduces to a system of $n-1$
nonlinear equations for the parameters $\beta_{i}$:

$\beta_{i}=c_{i}^{\lambda}\int_{{\mathbb{R}}^{N}}|y|^{2n-2i}\rho_{\beta,L}(y)\;{\mathrm{d}}y=:F_{i,L}(\beta_{1},...,\beta_{n-1}),$
(20)

with $i=1,\dots,n-1$. The gradient of $F_{i,L}$ equals

$\frac{\partial}{\partial\beta_{j}}F_{i,L}(\beta_{1},...,\beta_{n-1})=-\frac{c_{i}^{\lambda}}{q}\int_{{\mathbb{R}}^{N}}|y|^{2n-2i+2j}\rho_{\beta,L}(y)^{2-q}\,{\mathrm{d}}y.$

We have implemented in Python an algorithm which solves the nonlinear
equations (20). We used the BFGS algorithm of the library _SciPy_ [40] on the
auxiliary functional

$I(\beta_{1},...,\beta_{n-1}):=\sum_{i=1}^{n-1}\big{(}\beta_{i}-F_{i,L}(\beta_{1},...,\beta_{n-1})\big{)}^{2},$

with the integrals computed by a quadrature method.

In Figure 1 we provide the result of the computation of the mass $m(0)$ for
$\lambda\in\\{4,6,8,10\\}$ as a function of the parameter

$\alpha:=\frac{2N-q(2N+\lambda)}{N(1-q)}.$ (21)

This parameter (already used in [14]) is useful since the critical value
$q=\frac{N}{N+\lambda}$ simply becomes $\alpha=1$ and thus does not depend on
$N$ and $\lambda$. The interval of interest is then $\alpha\in(0,1)$ but,
similarly as in Lemma 2, one can in fact go up to
$\alpha=2-\frac{\lambda}{4}+\frac{\lambda}{2N}$ which corresponds to
$q=\frac{N-2}{N+2}$. By Remark 3, for $\alpha\geq 1$ the found solution is a
formal minimizer with infinite free energy. From the figure we see that the
mass $m(0)$ is smaller than $1$ for some $\alpha<1$ in all dimensions $N\geq
4$ for $\lambda\in\\{6,8,10\\}$. We also recover the results of Theorem 1 at
$\lambda=4$. In dimension $N=3$ the mass barely misses 1 at $\lambda=8$ but
concentration happens at $\lambda=10$. In Table 1 we provide an approximation
of the corresponding critical $q_{N}(d)$ below which concentration occurs,
which clearly illustrates the monotonicity in $N$ and $\lambda$.

We have also found numerically that the mass $m(L)$ of the solution was always
decreasing with $L$ as pointed out earlier. Should this be confirmed, then we
would have a clear picture of the situation for $\lambda$ an even integer. The
final conclusion would be that concentration occurs for all $N\geq 3$ starting
at $\lambda=10$ and for all $N\geq 4$ for $\lambda\in\\{6,8\\}$. We have
already proved in Theorem 1 that it happens for all $N\geq 6$ for $\lambda=4$,
and it was shown in [14] that there is never concentration for $\lambda=2$. Of
course we expect the true critical values of $\lambda$ to be in between these
special cases of even integers. For instance, we think that concentration
happens in $N=3$ for $\lambda$ just slightly above $\lambda=8$, as is
suggested by Figure 1.

$\lambda=4$ | $\lambda=6$
---|---
|
$\lambda=8$ | $\lambda=10$
|
Figure 1: Mass $m(0)$ of the solution of (20) found at $L=0$, in terms of the parameter $\alpha$ in (21), for the indicated values of $\lambda$.  | $N=3$ | $N=4$ | $N=5$ | $N=6$
---|---|---|---|---
$\lambda=4$ | X | X | X | $\frac{11}{18}\simeq 0.61\;(0.95)$
$\lambda=6$ | X | $0.42\;(0.90)$ | $0.52\;(0.72)$ | $0.58\;(0.62)$
$\lambda=8$ | X | $0.40\;(0.66)$ | $0.48\;(0.50)$ | $0.54\;(0.41)$
$\lambda=10$ | $0.26\;(0.81)$ | $0.38\;(0.49)$ | $0.45\;(0.35)$ | $0.51\;(0.29)$
Table 1: Numerical value of the critical diffusion exponent $q_{N}(\lambda)$
below which concentration occurs. The corresponding $\alpha_{N}(\lambda)$ in
(21) is provided in parenthesis (these are the points at which the curves of
Figure 1 cross the horizontal axis $m=1$). The value of $q_{N}(\lambda)$ for
$\lambda=4$ and $N=6$ is from Theorem 1. An ‘X’ means that concentration does
not occur for minimizers of the free energy in this combination of parameters.

###  2.2 The general case

For $\lambda\notin 2{\mathbb{N}}$, the convolution $\rho\ast|\cdot|^{\lambda}$
is not a polynomial in $|x|^{2}$, hence the Euler-Lagrange equation cannot be
reduced to a nonlinear equation involving finitely many parameters as in (20).
However, we can still restrict our attention to radial functions and write the
convolution in the form

$\rho\ast|\cdot|^{\lambda}(x)=|{\mathbb{S}}^{N-1}|\int_{0}^{\infty}K_{N,\lambda}(|x|,s)\rho(s)s^{d-1}{\mathrm{d}}s,$

with the slight abuse of notation $\rho(|x|):=\rho(x)$, where the kernel
$K_{N,\lambda}(r,s)$ is the spherical average

$K_{N,\lambda}(r,s):=\frac{1}{|{\mathbb{S}}^{N-1}|^{2}}\iint_{{\mathbb{S}}^{N-1}\times{\mathbb{S}}^{N-1}}|r\omega-s\omega^{\prime}|^{\lambda}\,{\mathrm{d}}\omega\,{\mathrm{d}}\omega^{\prime}.$

Its explicit formula in terms of Hypergeometric functions is provided later in
the appendix for completeness. Our goal is to solve the nonlinear equation (5)
at $L=0$. A Taylor expansion as in [14, Lemma 18] gives

$\rho(r)=\begin{cases}Cr^{-\frac{2}{1-q}}(1+o(1))&\text{for $r\to 0$,}\\\
\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}r^{-\frac{\lambda}{1-q}}(1+o(1))&\text{for
$r\to\infty$}.\end{cases}$

This prompts us to make the ansatz

$\rho_{f}(r)^{q-1}=r^{2}(1+r)^{\lambda-2}f(r),$ (22)

where $f$ is a positive continuous function on ${\mathbb{R}}_{+}$ tending to a
positive limit at 0 and infinity. The Euler-Lagrange equation (5) can be
expressed in terms of $f$ as

$\displaystyle f(r)=$
$\displaystyle\,\Phi(f)(r):=\frac{1-q}{q}\bigg{\\{}|{\mathbb{S}}^{N-1}|\int_{0}^{\infty}\frac{C_{N,\lambda,q}(r,s)}{f(s)^{\frac{1}{1-q}}}{\mathrm{d}}s$
(23)
$\displaystyle\qquad+\left(1-|{\mathbb{S}}^{N-1}|\int_{0}^{\infty}\frac{s^{N-1-\frac{2}{1-q}}}{(1+s)^{\frac{\lambda-2}{1-q}}f(s)^{\frac{1}{1-q}}}{\mathrm{d}}s\right)\frac{r^{\lambda-2}}{(1+r)^{\lambda-2}}\bigg{\\}},$
(24)

where

$C_{N,\lambda,q}(r,s)=\frac{K_{N,\lambda}(r,s)-s^{\lambda}}{r^{2}(1+r)^{\lambda-2}}\frac{s^{N-1-\frac{2}{1-q}}}{(1+s)^{\frac{\lambda-2}{1-q}}}.$

We used again the mass condition (6) to remove $M$ from the equation. Our goal
is to find a numerical approximation of the solution $f$ to (24). We
approximate $f$ by

$f(s)=\left|P\left(\frac{r}{1+r}\right)\right|^{2}$ (25)

where $P$ is a (complex) polynomial of finite degree $d$. This choice is
motivated by the fact that when $\lambda=2n\in 2{\mathbb{N}}$, the exact
solution takes exactly this form with $d=n-1$. For $\lambda\notin
2{\mathbb{N}}$ there will probably be no solution to the nonlinear equation
$f=\Phi(f)$ in the class (25) and we rather aim at minimizing a certain
distance between $f$ and $\Phi(f)$. In our algorithm we minimized the $L^{2}$
(square) distance

$\int_{0}^{\infty}\frac{s^{N-1-\frac{2}{1-q}}}{(1+s)^{\frac{\lambda-2}{1-q}}}\left(f(s)^{-\frac{1}{1-q}}-\Phi(f)(s)^{-\frac{1}{1-q}}\right)^{2}{\mathrm{d}}s$
(26)

with respect to the coefficients of the polynomial $P$. By Jensen’s
inequality, the $L^{2}$ norm in (26) controls the $L^{1}$ norm of the
corresponding densities in (22)

$\displaystyle\left|\\!\left|\rho_{f}-\rho_{\Phi(f)}\right|\\!\right|_{L^{1}({\mathbb{R}}^{N})}=$
$\displaystyle\left(\frac{q}{1-q}\right)^{\frac{1}{1-q}}|{\mathbb{S}}^{N-1}|$
(27)
$\displaystyle\times\int_{0}^{\infty}\frac{s^{N-1-\frac{2}{1-q}}}{(1+s)^{\frac{\lambda-2}{1-q}}}\left|f(s)^{-\frac{1}{1-q}}-\Phi(f)(s)^{-\frac{1}{1-q}}\right|{\mathrm{d}}s$
(28)

and it is a smoother function of the polynomial coefficients.

We used again the BFGS method of _SciPy_ to minimize (26) with respect to the
coefficients of the polynomial $P$ in (25). This provides an approximate
solution to (24). We then computed the mass of this solution in terms of $q$
and obtain curves which look very much like those of Figure 1. The point at
which the mass crosses 1 provides an approximation of the critical exponent
$q_{N}(\lambda)$ below which condensation happens.

We provide in Figure 2 the result of the calculation of $q_{N}(\lambda)$ in
dimension $N=5$. The curve suggests that condensation starts to appear for
$\lambda$ slightly above $4$. This curve was obtained by taking for $P$ a
complex polynomial of degree 10 and discretizing the integrals in (26) and
(24) by a simple Riemann sum with $1000$ regularly spaced points on $[0,20]$.
With this choice of discretization parameters, the $L^{1}$ error (28) was
found to be at most of the order $10^{-5}$ for all the points of the curve.

Figure 2: Numerical value of $q_{5}(\lambda)$ below which condensation occurs.
The dashed line is the curve $\lambda\mapsto\frac{N}{N+\lambda}$ which is a
lower bound to $q$. The dots are the points computed with the algorithm from
Section 2.1 when $\lambda$ is an even integer. This suggests that
concentration happens in dimension $N=5$ for $\lambda$ slightly above $4$.

##  3 Discussion

We have studied free energy minimizers of the nonlinear aggregation-diffusion
equation (1) and found that these can have part of their mass concentrated in
a Dirac delta at the origin. We rigorously proved this property in the case of
the quartic attraction $\lambda=4$ in all space dimensions $N\geq 6$ and
provided numerical results which suggest the same phenomenon in all space
dimensions $N\geq 3$ for $\lambda=10$. Our analysis is based on the study of
the integral $m(L)$ of (the regular part of) solutions to the Euler-Lagrange
equation with Lagrange multiplier $L$, and mass concentration means that
$m(0)<1$. Numerical evidence supports that $m(L)$ is decreasing with respect
to $L$. It is an interesting problem to show that these candidates are the
unique minimizers of the free energy (2) for $\lambda>4$. Unlike the results
of [8], which proved that the Dirac delta is an unstable equilibrium for the
fast diffusion equation with $0<q<\frac{N-2}{N}$, we expect that the
minimizers exhibiting concentration are attractors of the dynamics (1). It is
an interesting open problem if an appropriate notion of solution to (1) can
develop singularities in finite or infinite time.

## Appendix: Computation of $K_{N,\lambda}(r,s)$

Passing to radial coordinates, we obtain

$\displaystyle K_{N,\lambda}(r,s)$
$\displaystyle=\frac{|{\mathbb{S}}^{N-2}|}{|{\mathbb{S}}^{N-1}|}\int_{0}^{\pi}\\!\\!\left(r^{2}+s^{2}-2rs\cos\varphi\right)^{\frac{\lambda}{2}}\sin^{N-2}(\varphi)\,{\mathrm{d}}\varphi$
$\displaystyle=\frac{|{\mathbb{S}}^{N-2}|}{|{\mathbb{S}}^{N-1}|}\int_{-1}^{1}\left(r^{2}+s^{2}-2rst\right)^{\frac{\lambda}{2}}(1-t^{2})^{\frac{N-3}{2}}\,{\mathrm{d}}t.$

In dimension $N=3$ we have the explicit formula

$K_{3,\lambda}(r,s)=\frac{1}{2(\lambda+2)}\frac{(r+s)^{\lambda+2}-|r-s|^{\lambda+2}}{rs}$

while, in other dimensions, this can be expressed as

$K_{N,\lambda}(r,s)=\left(r^{2}+s^{2}\right)^{\frac{\lambda}{2}}\,_{2}F_{1}\left(\frac{2-\lambda}{4},-\frac{\lambda}{4},\frac{N}{2},\frac{4r^{2}s^{2}}{(r^{2}+s^{2})^{2}}\right)$

where ${}_{2}F_{1}$ is the Hypergeometric function. In the case of even
integers we find

$\displaystyle K_{N,2}(r,s)=$ $\displaystyle r^{2}+s^{2},$ $\displaystyle
K_{N,4}(r,s)=$ $\displaystyle
r^{4}+s^{4}+\left(2+\frac{4}{N}\right)r^{2}s^{2},$ $\displaystyle
K_{N,6}(r,s)=$ $\displaystyle
r^{6}+s^{6}+\left(3+\frac{12}{N}\right)(r^{4}s^{2}+r^{2}s^{4}),$
$\displaystyle K_{N,8}(r,s)=$ $\displaystyle
r^{8}+s^{8}+\frac{32+48/N+4N}{2+N}(r^{6}s^{2}+s^{6}r^{2})$
$\displaystyle\qquad+\frac{60+144/N+6N}{2+N}r^{4}s^{4},$ $\displaystyle
K_{N,10}(r,s)=$ $\displaystyle
r^{10}+s^{10}+\frac{50+80/N+5N}{2+N}(r^{8}s^{2}+s^{8}r^{2})$
$\displaystyle\qquad+\frac{140+480/N+10N}{2+N}(r^{6}s^{4}+s^{6}r^{4}).$

This provides the coefficients $c_{i}^{\lambda}$ appearing in (20).

## Acknowledgements

The authors would like to thank Jean Dolbeault, David Gómez-Castro and Juan
Luis Vázquez for pointing us out reference [8] and fruitful comments. This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Advanced
Grant Nonlocal-CPD 883363 of J.A.C. and Consolidator Grant MDFT 725528 of
M.L.). M.G.D. was partially supported by CNPq-Brazil (#308800/2019-2) and
Instituto Serrapilheira. R.L.F. was partially supported by the U.S. National
Science Foundation through grants DMS-1363432 and DMS-1954995.

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J.A. Carrillo

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

E-mail<EMAIL_ADDRESS>
M.G. Delgadino

Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

E-mail<EMAIL_ADDRESS>
Pontifical Catholic University of Rio de Janeiro, 38097 RJ, Brazil

E-mail<EMAIL_ADDRESS>(On Leave)

R.L. Frank

Department of Mathematics, California Institute of Technology, Pasadena, CA
91125, USA

E-mail address<EMAIL_ADDRESS>
Mathematisches Institut, Ludwig-Maximilans Universität München, Theresinstr.
39, 80333 München, Germany

M. Lewin

CNRS & CEREMADE, University Paris-Dauphine, PSL University, 75 016 Paris,
France

E-mail address<EMAIL_ADDRESS>


Smooth homomorphisms admit noetherian reductions

Nitin Nitsure

###### Abstract

We give a short proof that any smooth (means formally smooth and finitely
presented) homomorphism of rings can be obtained by base change from a smooth
homomorphism of noetherian rings. Together with the elegant short proof by J.
Conde-Lago that smooth homomorphisms of noetherian rings are flat, this gives
a short and elementary proof of the theorem of Grothendieck that all smooth
homomorphisms are flat.

MSC-2020: 14B25, 13B40.

A fundamental theorem of Grothendieck [2] says that any smooth (means formally
smooth and finitely presented) homomorphism of commutative rings is flat. The
proof given in [2] is quite intricate. Recently, J. Conde-Lago [1] gave a
short and simple proof that a smooth homomorphism of noetherian rings is flat.
The noetherian hypothesis is crucial to the proof of Conde-Lago, as it enables
the use of the standard machinery of completions of noetherian rings.

The main result of this note is the following, which allows us to reduce
arbitrary smooth homomorphisms to smooth homomorphisms between noetherian
rings.

Proposition 1. Let $A\to B$ be a ring homomorphism that is smooth, that is,
finitely presented and formally smooth. Then there exist finite-type
$\mathbb{Z}$-algebras $A_{0}$ and $B_{0}$, a smooth homomorphism $A_{0}\to
B_{0}$ and a homomorphism $A_{0}\to A$, such that $A\otimes_{A_{0}}B_{0}$ is
isomorphic to $B$ as an $A$-algebra.

Moreover, we can choose $A_{0}\to A$ to be injective, in which case the
induced homomorphism $B_{0}\to B$ is also injective.

Proof. As $B$ is finitely presented over $A$, we can assume that $B=P/I$ where
$P$ is a polynomial ring $A[x_{1},\ldots,x_{n}]$ in finitely many variables,
and $I\subset P$ is a finitely generated ideal $(f_{1},\ldots,f_{m})$. Let
$p:P\to P/I$, $p^{\prime}:P\to P/I^{2}$, and $q:P/I^{2}\to P/I$ denote the
quotient maps. Let $b_{i}=p(x_{i})\in P/I=B$. As by assumption $B$ is formally
smooth over $A$, the quotient homomorphism $q:P/I^{2}\to P/I=B$ admits a
section $s:B\to P/I^{2}$ over $A$. As $s$ is a section, there exist elements
$g_{1},\ldots,g_{n}\in I$ such that $s(b_{i})=p^{\prime}(x_{i}+g_{i})\in
P/I^{2}$ for all $i$. As $g_{i}\in I$, there exist $u_{ij}\in P$ for $1\leq
i\leq n$ and $1\leq j\leq m$ such that $g_{i}=\sum_{j}\,u_{ij}f_{j}$. As
$0=p(f_{j})=f_{j}(b_{1},\ldots,b_{n})\in P/I$, applying $s$ we have
$0=sp(f_{j})=sf_{j}(b_{1},\ldots,b_{n})=f_{j}(s(b_{1}),\ldots,s(b_{n}))=f_{j}(p^{\prime}(x_{1}+g_{1}),\ldots,p^{\prime}(x_{n}+g_{n}))=p^{\prime}f_{j}(x_{1}+g_{1},\ldots,x_{n}+g_{n})\in
P/I^{2}$, so $f_{j}(x_{1}+g_{1},\ldots,x_{n}+g_{n})\in{\mathop{\rm
ker}\nolimits}(p^{\prime})=I^{2}$. Hence there exist elements $h_{jk\ell}\in
P$ for $1\leq j,k,\ell\leq m$ such that
$f_{j}(x_{1}+g_{1},\ldots,x_{n}+g_{n})=\sum_{k\ell}\,h_{jk\ell}f_{k}f_{\ell}$.

Now let $A_{0}$ be the subring (sub-$\mathbb{Z}$-algebra) of $A$ generated by
all coefficients of the polynomials $f_{j}$, $u_{ij}$ and $h_{jk\ell}$, with
inclusion homomorphism $A_{0}\hookrightarrow A$. This ring is noetherian by
the Hilbert basis theorem. Let $P_{0}=A_{0}[x_{1},\ldots,x_{n}]\subset P$,
which is a polynomial ring over $A_{0}$ in the variables $x_{i}$’s, as they
remain algebraically independent over $A_{0}$. Note that by definition of
$A_{0}$, the elements $f_{j}$, $u_{ij}$ and $h_{jk\ell}$ of $P$ lie in
$P_{0}$. The ideal $I_{0}=\sum_{j}\,P_{0}f_{j}\subset P_{0}$ is contained in
$I$, and $I=PI_{0}$. Let $B_{0}=P_{0}/I_{0}$, which is a finitely presented
$A_{0}$-algebra, and let $B_{0}\to B$ be the $A_{0}$-algebra homomorphism
induced by the inclusion $P_{0}\hookrightarrow P$. Let $p_{0}:P_{0}\to
P_{0}/I_{0}$, $p^{\prime}_{0}:P_{0}\to P_{0}/I_{0}^{2}$, and
$q_{0}:P_{0}/I_{0}^{2}\to P_{0}/I_{0}$ denote the quotient maps. This induces
an $A_{0}$-algebra homomorphism $A\otimes_{A_{0}}B_{0}\to B$, which as an
isomorphism because the natural maps $A\otimes_{A_{0}}P_{0}\to P$ and
$P\otimes_{P_{0}}(P_{0}/I_{0})\to P/I$ are isomorphisms.

We now define a section $s_{0}:B_{0}=P_{0}/I_{0}\to P_{0}/I_{0}^{2}$ of the
quotient $q_{0}:P_{0}/I_{0}^{2}\to P_{0}/I_{0}=B_{0}$ over $A_{0}$. As
$u_{ij}\in P_{0}$, we have $g_{i}=\sum_{j}\,u_{ij}f_{j}\in I_{0}$. We have a
unique $A_{0}$-homomorphism $\sigma:P_{0}\to P_{0}$ defined by
$\sigma(x_{i})=x_{i}+g_{i}$. As $f_{j},\,g_{i},\,h_{jk\ell}\in P_{0}$, both
sides of the equality
$f_{j}(x_{1}+g_{1},\ldots,x_{n}+g_{n})=\sum_{k\ell}\,h_{jk\ell}f_{k}f_{\ell}$
are in $P_{0}$. As this equality is satisfied in $P$ it is satisfied in
$P_{0}\subset P$. Hence
$\sigma(f_{j})=f_{j}(x_{1}+g_{1},\ldots,x_{n}+g_{n})\in I_{0}^{2}$. This shows
that $\sigma:P\to P$ satisfies $\sigma(I_{0})\subset I_{0}^{2}$, and so
$\sigma$ descends to an $A_{0}$-algebra homomorphism
$s_{0}:B_{0}=P_{0}/I_{0}\to P_{0}/I_{0}^{2}$.

By definition, $s_{0}p_{0}(x_{i})=p_{0}^{\prime}(x_{i}+g_{i})\in
P_{0}/I_{0}^{2}$, and $q_{0}\circ p^{\prime}_{0}=p_{0}$. Hence
$q_{0}s_{0}p_{0}(x_{i})=q_{0}p_{0}^{\prime}(x_{i}+g_{i})=p_{0}(x_{i}+g_{i})=p_{0}(x_{i})$
as $p_{0}(g_{i})=0$. This shows that $s_{0}:B_{0}\to P_{0}/I_{0}^{2}$ is a
section of $q_{0}:P_{0}/I_{0}^{2}\to P_{0}/I_{0}=B_{0}$. As $P_{0}$ is
formally smooth over $A_{0}$, this implies that $A_{0}\to B_{0}$ is formally
smooth, and hence smooth, being finitely presented.

By our choice, $A_{0}$ is a subring of $A$, with $A_{0}\to A$ the inclusion.
By the elementary proof by Conde-Lago [1], any smooth homomorphism $A_{0}\to
B_{0}$ of noetherian rings is flat. The homomorphism $B_{0}\to B$ is the base-
change of $A_{0}\to A$ under the flat homomorphism $A_{0}\to B_{0}$, and so it
is again injective if $A_{0}\to A$ is injective. $\square$

Corollary 2. Any smooth (means formally smooth and finitely presented)
homomorphism of rings $A\to B$ is flat.

Proof. With notation as in the above proposition, we have shown that $A\to B$
is the base change of a smooth homomorphism $A_{0}\to B_{0}$ of noetherian
rings. By the elementary proof by Conde-Lago [1], $A_{0}\to B_{0}$ is flat.
Hence its base change $A\to B$ is flat. $\square$

References

[1] J. Conde-Lago: A short proof of smooth implies flat. Communications in
Algebra 2017, vol 45, 774-775.

[2] A. Grothendieck and J. Dieudonné: Éléments de Géométrie Algébrique IV.
Publ. Math. IHES, vols. 20, 24, 28, 32 (1964-67).

Nitin Nitsure
Retired Professor, School of Mathematics
Tata Institute of Fundamental Research
Mumbai 400 005, India
email<EMAIL_ADDRESS>