# Existence of weak solutions for Porous medium equation
with a divergence type of drift term

Sukjung Hwang Department of Mathematics Education, Chungbuk National
University, Cheongju, Chungbuk 28644, Republic of Korea,
<EMAIL_ADDRESS>Kyungkeun Kang Department of Mathematics, Yonsei
University, Seoul 03722, Republic of Korea<EMAIL_ADDRESS>Hwa Kil Kim
Department of Mathematics Education, Hannam University, Daejeon 34430,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract

We consider degenerate porous medium equations with a divergence type of drift
terms. We establish existence of nonnegative $L^{q}$-weak solutions
(satisfying energy estimates or even further with moment and speed estimates
in Wasserstein spaces), in case the drift term belongs to a sub-scaling
(including scaling invariant) class depending on $q$ and $m$ caused by
nonlinear structure of diffusion, which is a major difference compared to that
of a linear case. It is noticeable that the classes of drift terms become
wider, if the drift term is divergence-free. Similar conditions of gradients
of drift terms are also provided to ensure the existence of such weak
solutions. Uniqueness results follow under an additional condition on the
gradients of the drift terms with the aid of methods developed in Wasserstein
spaces. One of our main tools is so called the splitting method to construct a
sequence of approximated solutions, which implies, by passing to the limit,
the existence of weak solutions satisfying not only an energy inequality but
also moment and speed estimates. One of crucial points in the construction is
uniform Hölder continuity up to initial time for homogeneous porous medium
equations, which seems to be of independent interest. As an application, we
improve a regularity result for solutions of a repulsive Keller-Segel system
of porous medium type.

2020 AMS Subject Classification : 35A01, 35K55, 35Q84, 92B05

Keywords : Porous medium equation, weak solution, Wasserstein space

###### Contents

  1. 1 Introduction
  2. 2 Main results
    1. 2.1 Existence results
    2. 2.2 Uniqueness
    3. 2.3 An application
  3. 3 Preliminaries
    1. 3.1 Technical lemmas
    2. 3.2 Wasserstein space
    3. 3.3 Flows on $\mathcal{P}_{2}({\mathbb{R}}^{d})$ generated by vector fields
  4. 4 A priori estimates
    1. 4.1  Supplementary estimates of speed and $p$-th moment
    2. 4.2 Energy estimates
    3. 4.3 Estimates of speed and $p$-th moment
    4. 4.4 Estimates of temporal and spatial derivatives.
  5. 5 Existence of regular solutions
    1. 5.1 Splitting method
    2. 5.2 Proof of Proposition 5.3
  6. 6 Existence of weak solutions
    1. 6.1 Existence for case: $\int_{{\mathbb{R}}^{d}}\rho_{0}\log\rho_{0}\,dx<\infty$
    2. 6.2 Existence for case: $\rho_{0}\in L^{q}(\mathbb{R}^{d}),\ q>1$
    3. 6.3 Existence for case: $\nabla\cdot V=0$ and $\rho_{0}\in L^{q}(\mathbb{R}^{d}),\ q>1$
    4. 6.4 Existence for case: $\nabla V$ in sub-scaling classes
  7. 7 Proofs of uniqueness and an application
  8. A Hölder continuity of homogeneous PME
  9. B Figure supplements

##  1 Introduction

In this paper, we consider _porous medium equations_ (PME) with divergence
form of drifts in the form of

$\left\\{\begin{array}[]{ll}\partial_{t}\rho=\nabla\cdot(\nabla\rho^{m}-V\rho),&\text{
in }\ Q_{T}\\\ \rho(\cdot,0)=\rho_{0},&\text{ on }\
\mathbb{R}^{d},\end{array}\right.\quad\text{ for }\ m>1,$ (1.1)

on a cylindrical domain $Q_{T}:=\mathbb{R}^{d}\times(0,T]$, $d\geq 2$,
$0<T<\infty$, where $\rho:Q_{T}\mapsto{\mathbb{R}}$ and a vector field
$V:Q_{T}\to\mathbb{R}^{d}$. We note that neither the maximum principle nor the
gradient flow structure is, in general, available for the equation (1.1) which
has been rarely analyzed. Nevertheless, for given nonnegative initial data, we
establish mainly the existence of nonnegative weak solutions of (1.1) when the
drift belongs to some sub-scaling classes (see Definition 1.1).

First, we point out that
$\Delta\rho^{m}\sim\nabla\cdot\left(\rho^{m-1}\nabla\rho\right)$ which clearly
shows the different tendencies depending on the constant $m$. If $m=1$, then
the diffusion is linear so (1.1) becomes the heat equation with drifts. When
$m>1$, the equation is called _porous medium equations_ (PME) in which we are
interested in this paper. When $0<m<1$,it follows that $\rho^{m-1}$ tends to
infinity as $\rho$ approaches to zero and we call such equations _fast
diffusion equations_. These are important nonlinear evolution equations
describing a number of physical applications: fluid flow, heat radiations in
plasma, mathematical biology, and other fields. For a systematic presentation
of the mathematical theories, we refer to monographs [Vaz06, Vaz07, DK07,
DGV12] and the references therein.

The equations (1.1) carry both nonlinear diffusion and drift that are useful
in numerous places. For example, this type of PDEs represents a certain form
of aggregation-diffusion (reaction-diffusion) equations that arises naturally
in systems of interacting components and is used widely to describe pattern-
formation phenomena and chemotaxis models in biological, chemical and physical
systems. For example, one can consider the porous medium equation with drift
caused by fluid flow,

$\rho_{t}-\Delta\rho^{m}+u\cdot\nabla\rho=0,\quad\text{ in }\
Q_{T}:={\mathbb{R}}^{d}\times(0,T],$

where $u:Q_{T}\rightarrow{\mathbb{R}}^{d}$ is a divergence-free vector field.
Recently, there has been extensive research on so called enhanced dissipation
effects for various PDEs with fluid flows for the linear case $m=1$ (see e.g.
[MR2434887, MR3621820, MR4242626, MR4152270], etc). In particular, there are
recent developments of suppression of blow-up for nonlinear equations such as
Keller-Segel equations with external fluid flows (see e.g. [MR3730537,
MR4156602, MR3544323], etc). Concerning free boundary regularity of (1.1),
there are recent developments of ${\mathcal{C}}^{1,\alpha}$-regularity
assuming enough smoothness on $V$ and $\rho_{0}\in L^{1}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})$ in [KZ18, KZ20-arXiv] and references therein for
relevant works when $V=0$. The large-time asymptotics and regularity of
nonlinear diffusion equations with drifts are also analyzed as well: for
example, the rate of decay to equilibrium of self-similar solutions in
[Carr01], viscosity solutions, equivalent to weak solutions, and their
asymptotic convergence of the free boundary to the equilibrium in [KL10], and
Hölder continuous solutions for aggregation-diffusion equations with singular
interaction kernels in [Zhang20]. Compared to PME without drifts or linear
diffusion-drift equations, fewer results are known for (1.1) because of
difficulties of handling not only nonlinear diffusion but also drift term.

The drift may affect the well-posedness of diffusive parabolic equations.
There are works showing counterexamples of lack of regularities under certain
conditions on the drift, for example, [SVZ13, SSSZ, Z11] for (fractional)
linear diffusion and [KZ18, HZ21] for porous medium equations. From the
perspective of critical conditions on $V$ providing continuity of (1.1) (for
example, [CHKK17, KZ18, HZ21]), we recall

$V\in L^{q_{1},q_{2}}_{x,t}\quad\text{ where
}\quad\frac{d}{q_{1}}+\frac{2}{q_{2}}\leq 1.$ (1.2)

The condition (1.2) is also observed from the scaling invariant classes in
$L^{\infty}$-sense. Moreover, the same conditions are well known from heat
(linear) diffusion-drift equations (see e.g. [LSU, Lie96, SVZ13]) and there is
a connection with so called Prodi-Serrin-Ladyzhenskaya conditions concerning
the existence of smooth solutions of the Navier-Stokes equations (see
[MR0236541, MR0150444]).

One of motivations of this research has started from an observation that the
nonlinear diffusion factor $m$ yields different scaling invariance depending
on conservation of $L^{q}$-norm in spatial variables. Naturally, it follows a
question of searching minimal conditions on $V$ related to $m$, which allows
us to construct weak solutions of (1.1) for a given $L^{q}$-initial data. From
this perspective, we investigate _the scaling invariant classes of $V$ and
$\nabla V$ in $L^{q}$-sense for $q\geq 1$_ which turns out proper conditions
for constructing $L^{q}$-weak solutions (see Definition 2.1). Heuristically,
we look for conditions of $V$ in terms of scaling invariant classes of
solutions for $L^{q}$-norm in spatial variables. It turns out that the
$L^{q}$-norm of $\rho$ is preserved, if the following scaling is taken: for
$q\geq 1$ and $\kappa>0$

$\rho_{\kappa}(x,t)=\kappa^{d/q}\rho(\kappa
x,\kappa^{\beta}t)\quad\text{and}\quad V_{\kappa}(x,t)=\kappa^{\alpha}V(\kappa
x,\kappa^{\beta}t),$

where

$\alpha=q_{m,d}+1,\quad\beta=q_{m,d}+2\quad\text{for}\quad
q_{m,d}:=\frac{d(m-1)}{q}.$

We then see that $V_{\kappa}$ is scaling invariant in a mixed norm
$L^{q_{1},q_{2}}_{x,t}$, i.e. for $1\leq q_{1},q_{2}\leq\infty$, it holds
$\left\|V_{\kappa}(x,t)\right\|_{L^{q_{1},q_{2}}_{x,t}}=\left\|V(x,t)\right\|_{L^{q_{1},q_{2}}_{x,t}}=\left\|\left\|V(\cdot,t)\right\|_{L^{q_{1}}_{x}}\right\|_{L^{q_{2}}_{t}}$
provided that $\frac{d}{q_{1}}+\frac{2+q_{m,d}}{q_{2}}=1+q_{m,d}.$ We also
note that similar condition of the scaling invariance for $\nabla V$ is that
$\nabla V\in L^{\tilde{q}_{1},\tilde{q}_{2}}_{x,t}$, where $\tilde{q}_{1}$ and
$\tilde{q}_{2}$ satisfy
$\frac{d}{\tilde{q}_{1}}+\frac{2+q_{m,d}}{\tilde{q}_{2}}=2+q_{m,d}.$

Now we define the scaling invariant and sub-scaling classes of $V$ and $\nabla
V$ in accordance with $q$ and $m$.

###### Definition 1.1

Let $m,q\geq 1$ and $q_{m,d}:=\frac{d(m-1)}{q}$. For constants
$q_{1},q_{2},\tilde{q}_{1},\tilde{q}_{2}>0$, we define following spaces.

  * (i)

The scaling invariant classes of $V$ and $\nabla V$ are defined as

$\mathcal{S}_{m,q}^{(q_{1},q_{2})}:=\left\\{V:\
\|V\|_{L^{q_{1},q_{2}}_{x,t}}<\infty\ \text{ where }\
\frac{d}{q_{1}}+\frac{2+q_{m,d}}{q_{2}}=1+q_{m,d}\right\\},$ (1.3)
$\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}:=\left\\{V:\
\|\nabla V\|_{L^{\tilde{q}_{1},\tilde{q}_{2}}_{x,t}}<\infty\ \text{ where }\
\frac{d}{\tilde{q}_{1}}+\frac{2+q_{m,d}}{\tilde{q}_{2}}=2+q_{m,d}\right\\}.$
(1.4)

Moreover, let us name $\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}$ and
$\|V\|_{\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}}$, as the
scaling invariant norms corresponding to each spaces.

  * (ii)

The sub-scaling classes are defined as

$\mathfrak{S}_{m,q}^{(q_{1},q_{2})}:=\left\\{V:\
\|V\|_{L^{q_{1},q_{2}}_{x,t}}<\infty\ \text{ where }\
\frac{d}{q_{1}}+\frac{2+q_{m,d}}{q_{2}}\leq 1+q_{m,d}\right\\},$ (1.5)
$\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}:=\left\\{V:\
\|\nabla V\|_{L^{\tilde{q}_{1},\tilde{q}_{2}}_{x,t}}<\infty\ \text{ where }\
\frac{d}{\tilde{q}_{1}}+\frac{2+q_{m,d}}{\tilde{q}_{2}}\leq
2+q_{m,d}\right\\}.$ (1.6)

Let us name $\|V\|_{\mathfrak{S}_{m,q}^{(q_{1},q_{2})}}$ and
$\|V\|_{\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}}$, as the
sub-scaling norms corresponding to each spaces.

We remark that the main existence results are given under certain assumptions
on $V$ in sub-scaling classes. The critical case matters the most and the
strict-subcritical case (the case of strict inequality in (1.5) or (1.6)) is
simpler, and therefore, proofs throughout the paper concern only the case of
scaling invariant classes.

###### Remark 1.2

  1. (i)

In Figure 1, we graph the lines of $(\frac{1}{q_{1}},\frac{1}{q_{2}})$
satisfying $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ defined in (1.3) for $m,q\geq
1$. We easily observe that $q_{m,d}$ becomes $0$ as $m\to 1$ or $q\to\infty$,
which corresponds that $\frac{d}{q_{1}}+\frac{2}{q_{2}}=1$ as in (1.2). The
condition (1.2) appears in the context of understanding heat equations $(m=1)$
or $L^{\infty}$-sense scaling invariant classes of porous medium equations
$(m>1,q=\infty)$ (see e.g. [CHKK17, KZ18, HZ21]). Therefore, our scaling
invariant condition (1.3) is more comprehensive including the linear case,
i.e. $m=1$ or the case of PME for existence of $L^{\infty}$-solutions.

  2. (ii)

In Figure 2, we graph lines of
$(\frac{1}{\tilde{q}_{1}},\frac{1}{\tilde{q}_{2}})$ satisfying
$\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$ defined in (1.4)
for $m,q\geq 1$. In fact, by using embedding property, one can supress the
norm of $V$ by the norm of $\nabla V$; that is,

$\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}:=\|V\|_{L^{q_{1},q_{2}}_{x,t}}\lesssim\|V\|_{\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}}:=\|\nabla
V\|_{L^{\tilde{q}_{1},\tilde{q}_{2}}_{x,t}}$ (1.7)

provided $\tilde{q}_{1}=\frac{dq_{1}}{d+q_{1}}\in(1,d)\ (\text{equivalently
}q_{1}=\frac{d\tilde{q}_{1}}{d-\tilde{q}_{1}}\in(\frac{d}{d-1},\infty)),$ and
$\tilde{q}_{2}=q_{2}$. Hence, the range of $\tilde{q}_{2}$ is

$\begin{cases}\frac{2+q_{m,d}}{1+{q_{m,d}}}<\tilde{q}_{2}\leq\infty,&\text{ if
}1<m<1+\frac{q(d-2)}{d}\vspace{1 mm}\\\
\frac{2+q_{m,d}}{1+{q_{m,d}}}<\tilde{q}_{2}<\frac{2+q_{m,d}}{2-d+q_{m,d}},&\text{
otherwise}.\end{cases}$ (1.8)

In Figure 2, the intersection of lines and shaded region is where (1.8) holds.

  3. (iii)

Let $1<q<q^{L}$ and let $(q_{1}^{L},q_{2}^{L})$ be a pair satisfying
$\mathcal{S}_{m,q^{L}}^{(q_{1}^{L},q_{2}^{L})}$. Concerning
$\mathcal{S}_{m,q}^{(q_{1}^{L},q_{2})}$ and
$\mathcal{S}_{m,q^{L}}^{(q_{1}^{L},q_{2}^{L})}$, we observe that
$q_{2}^{L}>q_{2}$ illustrated in Figure 1. Moreover, the Hölder inequality in
$t\in[0,T]$ yields

$\|V\|_{\mathcal{S}_{m,q}^{(q_{1}^{L},q_{2})}}\leq
T^{\frac{1}{q_{2}}-\frac{1}{q_{2}^{L}}}\|V\|_{\mathcal{S}_{m,q^{L}}^{(q_{1}^{L},q_{2}^{L})}}.$
(1.9)

Figure 1. $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ in Definition 1.1.
$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$$+$$\frac{1}{2}$$\times$$\frac{1}{d}$$\mathcal{S}$$\times$$\frac{1}{{\lambda_{1}}}$$\times$$\frac{1+d(m-1)}{d}$$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$$\times$$\frac{1+q_{m,d}}{2+q_{m,d}}$$\times$$\frac{1+q_{m,d}}{d}$$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$$\times$$\times$$*$$\frac{1}{q_{1}^{L}}$
$*$$\frac{1}{q_{2}^{L}}$$*$$*$$\frac{1}{q_{2}}$$*$$\mathcal{S}=\mathcal{S}_{1,q}^{(q_{1},q_{2})}$
or $\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$
$\mathcal{S}_{m,q^{L}}^{(q_{1},q_{2})}$ (dotted line) for $1<q<q^{L}<\infty$

Figure 2. $\tilde{\mathcal{S}}_{m,q}^{(q_{1},q_{2})}$ in Definition 1.1.
$\frac{1}{\tilde{q}_{1}}$$\frac{1}{\tilde{q}_{2}}$$+$$1$$\bullet$$\frac{2}{d}$$\tilde{\mathcal{S}}$$\times$$\frac{2+d(m-1)}{d}$$\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$$\times$$\bullet$$\frac{2+q_{m,d}}{d}$$\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$$\tilde{\mathcal{S}}=\tilde{\mathcal{S}}_{1,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$
or
$\tilde{\mathcal{S}}_{m,\infty}^{(\tilde{q}_{1},\tilde{q}_{2})}$$*$$\frac{1}{d}$$\tilde{q}_{1}=d$$+$$\frac{1}{2}$$*$$\frac{1+q_{m,d}}{2+q_{m,d}}$$*$$\frac{1+d(m-1)}{2+d(m-1)}$$*$$1$$\tilde{q}_{1}=1$$*$$\frac{2-d+q_{m,d}}{2+q_{m,d}}$$*$$\frac{2+d(m-2)}{2+d(m-1)}$

Our main objective is to establish the existence of $L^{q}$-weak solutions of
(1.1) subject to $\rho_{0}\in L^{q}({\mathbb{R}}^{d})$ for $1\leq q<\infty$.
More specifically, we provide sufficient conditions of drift term $V$ or
$\nabla V$ to ensure the existence of solutions for (1.1) with $L^{q}$-initial
data (the collection of $V$ or $\nabla V$ is defined in (1.5) or (1.6)). An
interesting feature is that such conditions of $V$ or $\nabla V$ depend on $m$
and $q$, which is one of major differences compared to linear equation with
drift term, since (1.2) of the linear case is independent of $q$ (compare to
[KK-arxiv]).

As far as authors understand, weak solutions of PME with general drift terms
have been barely studied, although weak solutions of homogeneous PME or
coupled system with PME type with gradient flow structures have been
extensively studied so far. Since the PME with drift terms under our
considerations, in general, does not have either maximum principle or gradient
flow structure, existence of weak solutions does not seem to be established by
the existing methods of proofs based on such arguments. In fact, our main tool
in constructing weak solutions is, so called splitting method, which solves
the PME with drift term by solving homogeneous PME and transport equations via
flow maps separately, which enables us to obtain aprroximated solutions and
convergence to weak solutions. We are positive that there are probably other
ways of proving existence of weak soltuions, but they must be independent of
maximum principle and gradient flow structure, and thus we think that our
proof is a theoretical way of new approach. We expect that our method of the
proof and consequential results could be applicable to broader classes of
equations.

We make several comments, for simplicity, we deal with only the case $d\geq 3$
in the tables below (two dimensional case is also treated in similar ways but
there are little differences such as range of parameters’ values, compared to
higher dimensions).

Table 1: Guide of existence results of $L^{1}$-weak solution in Wasserstein space, $q=1$ and $d\geq 3$. Range of $m$ | Intial data | Conditions on $V$ in critial class | References
---|---|---|---
$1<m\leq 2$ | $1<p\leq{\lambda_{1}}:=1+\frac{1}{d(m-1)+1}$ |  $V\in\mathfrak{S}_{m,1}^{(q_{1},q_{2})}$ for $2\leq q_{2}\leq\frac{m}{m-1}$ |  Theorem 2.3, Figure 3 (dark)
| $\rho_{0}\in\mathcal{P}_{p}(\mathbb{R}^{d})$, $\int_{\mathbb{R}^{d}}\rho_{0}\log\rho_{0}\,dx<\infty$ |  $\nabla\cdot V=0$ & $V\in\mathfrak{S}_{m,1}^{(q_{1},q_{2})}$ for ${\lambda_{1}}\leq q_{2}\leq\frac{{\lambda_{1}}m}{m-1}$ |  Theorem 2.4, Figure 3 (light)
|  $V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$ for ${\lambda_{1}}<\tilde{q}_{2}\leq\frac{m}{m-1}$ |  Theorem 2.5, Figure 9-$(i)$ (line $\overline{\textit{{DE}}}$ )

Firstly, we distinguish the case $q=1$ and $q>1$. For the case $q=1$ (in Table
1), in general, we need to restrict $m\in(1,2]$, since a priori entropy
estimate is not clear for the case $m>2$, unless $V$ is divergence-free or
$\nabla V$ belongs sub-scaling classes. Compactness arguments also give the
same restriction on $m$, even for the cases of $\nabla\cdot V=0$ or
$V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$.

In addition, initial data requires more than just $L^{1}$, and more precisely
it is necessary that $\int_{{\mathbb{R}}^{d}}\rho_{0}\log\rho_{0}dx<\infty$
due to entropy estimates. In this case, we suppose that
$V\in\mathfrak{S}_{m,1}^{(q_{1},q_{2})}$ with $2\leq q_{2}\leq\frac{m}{m-1}$.
Furthermore, if $\rho_{0}\in{\mathcal{P}}_{p}({\mathbb{R}}^{d})$ with
$1<p\leq{\lambda_{1}}:=1+\frac{1}{d(m-1)+1}$, we can obtain the $p$-th moment
estimates and the _speed_ (see Remark 3.12) estimates, which assure the
existence of weak solutions in Wasserstein spaces (see Theorem 2.3).

Similar analysis can be performed for the case
$V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$ (refer
Theorem 2.5) or $V$ is divergence-free (see Theorem 2.4). In divergence-free
case, we emphasize that the range of subscaling class becomes wider, i.e.
$V\in\mathfrak{S}_{m,1}^{(q_{1},q_{2})}$ with ${\lambda_{1}}\leq
q_{2}\leq\frac{{\lambda_{1}}m}{m-1}$, mainly because the drift term does not
show up in the energy estimate via integration by parts. We refer Section 6
for the details of proofs.

Table 2: Guide of existence results of $L^{q}$-weak solutions, $m>1$ and $d\geq 3$. Range of $q$ | Intial data | Conditions on $V$ in critical class | References
---|---|---|---
$q>1$ and $q\geq m-1$ | $\rho_{0}\in L^{q}({\mathbb{R}}^{d})\cap{\mathcal{P}}({\mathbb{R}}^{d})$ |  $V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}$ for $2\leq q_{2}\leq\frac{q+m-1}{m-1}$ |  Theorem 2.7, Figure 4, 5
$q>\max\\{1,\frac{m}{2}\\}$ |  $\nabla\cdot V=0$ & $V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}$ holding (2.17) |  Theorem 2.11, Figure 6-$(i,ii)$
|  $V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$ for $\frac{2+q_{m,d}}{1+q_{m,d}}<\tilde{q}_{2}\leq\frac{q+m-1}{m-1}$ |  Theorem 2.15, Figure 9-$(i,ii)$

On the other hand (see Table 2), in case $q>1$, when usual $L^{q}$-energy
estimates are performed, it is necessary to assmue that $\rho_{0}\in
L^{q}({\mathbb{R}}^{d})$. Here we need to consider $1<m\leq 2$ and $m>2$
separately. In case that $m>2$, an additional restriction, $q\geq m-1$ is
necessary for the existence of weak solutions (see Theorem 2.7). Indeed, when
$L^{q}$-energy estimates and compactness arguments are conducted, we should
control the quantity, $\rho$ to the power $q-m+1$, which requres nonnegative
sign that is equivalent to $q\geq m-1$. This restriction might be technical
but we do not know how to handle the case of $1<q<m-1$, and thus we leave such
case as an open question.

In case $m>2$, the restriction on $q\geq m-1$ can be relaxed to
$q>\frac{m}{2}$ if either $V$ is divergence-free or $\nabla V$ is under
consideration (refer Theorem 2.11 and 2.15). This is because the restriction
$q>\frac{m}{2}$ for $m>2$ is required for compactness arguments while a priori
$L^{q}$-energy estimates are obtained for all $m,q>1$.

Table 3: Guide of existence results of $L^{q}$-weak solutions in Wasserstein spaces, $m>1$ and $d\geq 3$. Range of $q$ | Intial data | Conditions on $V$ in critical class | References
---|---|---|---
$q>1$ and $q\geq m-1$ |  |  $V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}$ holding (2.13) |  Theorem 2.9 $(i)$ , Figure 4-($i$-$iii$), 5-$(i,ii)$
$q>1$ and $q\geq\frac{m}{2}$ | $\rho_{0}\in\mathcal{P}_{p}(\mathbb{R}^{d})\cap L^{q}(\mathbb{R}^{d})$ |  $\nabla\cdot V=0$ & $V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}$ holding (2.19) |  Theorem 2.13 $(i)$, Figure 7-($i$-$iii$), 8-$(i,ii)$
| where $1<p\leq{\lambda_{q}}$ |  $V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$ holding (2.22) |  Theorem 2.16 $(i)$, Figure 9-$(i,ii)$
Embedding results | ${\lambda_{q}}:=\min\\{2,1+\frac{d(q-1)+q}{d(m-1)+q}\\}$ |  Theorem 2.9 $(ii)$ , Figure 4-$(e)$, 5-$(e)$
|  |  Theorem 2.13 $(ii)$, Figure 7-$(e)$, 8-$(e)$
|  |  Theorem 2.16 $(ii)$, Figure 9-$(e)$

The same restrictions on $m$ and $q$ are required, when we look for weak
solutions in Wasserstein space (see Table 3). In such case, the subscaling
class of $V$ or $\nabla V$ becomes narrower because the $p$-th moment and the
speed estimates are shown to be valid for more restrictive values of
parameters (see Theorem 2.9 $(i)$, Theorem 2.13 $(i)$ and Theorem 2.16 $(i)$
for the details).

Therefore, it could happen that there is a weak solution satisfying only
energy estimates but we do not know if it belongs to Wasserstein spaces,
because subscaling class of $V$ corresponding to energy space turns out to be
wider than that of $V$ relevant to Wasserstein spaces. Nevertheless, using
temporal embedding, some cases guarantee that weak solutions with energy
estimates belong to Wasserstein spaces as well (see Theorem 2.9 $(ii)$,
Theorem 2.13 $(ii)$ and Theorem 2.16 $(ii)$ for the details).

As a main tool, we adopt the splitting method to establish approximated
solutions, which converges to a $L^{q}$-weak solution (see section 5 and
section 6). In the process of constructing the approximated solutions, we use
that Hölder continuity of PME in the absense of drift term is preserved
uniformly up to initial time, which seems to be of independt interest (see
Appendix A).

On the other hand, uniqueness for solutions of (1.1) does not seem to be
obvious, since the maximum principle, in general, is not true due to nonlinear
diffusion with drift term. Nevertheless, we obtain uniqueness result of
$L^{m}$-weak solutions under additional conditions for gradient of drift and
initial data, exploiting mass transportation theory (see Theorem 2.18 and
Corollary 2.19, and refer Section 7 for proofs).

As an application, we consider a repulsive Keller-Segel system of porous
medium type. With the aid of existence results, if the power $m$ of degenerate
diffusion is beyond a certain positive number, we show that solutions become
bounded, which turns out to be an improvement, compared to known results (see
Theorem 2.21 and refer Section 7 for its proof). Existence results developed
in this paper are expected to be applicable to other situations related to
equations of PME type as well.

Our paper is organized as follows: In Section 2, we state all main results and
several remarks and figures are provided to help readers understand. Some
preliminaries are prepared in Section 3. Section 4 is devoted to giving a
priori estimates for regular solutions (see Definition 4.1). In Section 5, the
existence of regular solutions of PME is established by the splitting method.
Section 6 is prepared for proofs of existence results stated in Section 2.
Finally, we provide the proof of uniqueness results and we treat a repulsive
Keller-Segel system of porous medium type in Section 7. We present the proof
of uniform Hölder continuity of homogeneous PME up to initial time in Appendix
A. In Appendix B, supplemetary figures are additionally added.

##  2 Main results

In this section, we describe our main results and make relevant remarks for
each of them. First, existence of weak solutions for (1.1) are categorized
according to hypotheses of given initial data and drifts. Uniqueness of weak
solutions are discussed as well. Finally, as an application, we study a
repulsive type Keller-Segel equations to improve previously known results by
taking advantage of develped main results.

Before stating main theorems, for convenience, we introduce some notations and
make a few remarks.

  * •

Let us denote by $\mathcal{P}(\mathbb{R}^{d})$ the set of all Borel
probability measures on $\mathbb{R}^{d}$. Furthermore, denote
$\mathcal{P}_{p}(\mathbb{R}^{d})$ as the space $\mathcal{P}(\mathbb{R}^{d})$
with a finite $p$-th moment, that is $\int_{\mathbb{R}^{d}}\langle
x\rangle^{p}\,d\mu<\infty$ if $\mu\in\mathcal{P}_{p}(\mathbb{R}^{d})$. We
refer Section 3.2 for definitions and related properties of the Wasserstein
distance denoted by $W_{p}$, the Wasserstein space, and AC (absolutely
continuous) curves.

  * •

The $p$-th power is given as

$\langle x\rangle^{p}=\left(1+|x|^{2}\right)^{\frac{p}{2}},\ \quad\text{ for
}\ 1\leq p<\infty\ \text{ and }\ x\in{\mathbb{R}}^{d},\ d\geq 2.$

  * •

Let $A\subset\mathbb{R}$ be a measurable set. We denote by $\delta_{A}$ the
indicator function of $A$.

  * •

For any $m>1$ and $q\geq 1$, let us define the following constant

$\lambda_{q}:=\min\left\\{2,1+\frac{d(q-1)+q}{d(m-1)+q}\right\\},$ (2.1)

that can be rewritten as
${\lambda_{q}}=\left\\{1+\frac{d(q-1)+q}{d(m-1)+q}\right\\}\cdot\delta_{\\{1\leq
q\leq m\\}}+2\cdot\delta_{\\{q>m\\}}$. The constant ${\lambda_{q}}\in(1,2]$ in
(2.1) naturally comes from evaluating the $p$-th moment and the speed
estimates which are essential to play in Wasserstein spaces (refer Section
4.1). A straightforward observation gives that $1+\frac{d(q-1)+q}{d(m-1)+q}=2$
if $q=m$. Also, for any $q\geq 1$, note that ${\lambda_{q}}=2$ if $m=1$.

  * •

The letters $c$ and $C$ are used for generic constants. Also, the letter
$\theta$ is a generic constant which varies with arguments of interpolation.
Throughout the paper, we omit the dependence on $m$, $q$, $d$, $T$, and
$\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{d})}$, because we regard $m>1$, $q\geq 1$,
$d\geq 2$, $T>0$ are given constants and
$\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{d})}=1$ in either
$\mathcal{P}(\mathbb{R}^{d})$ or $\mathcal{P}_{p}(\mathbb{R}^{d})$.

  * •

In figures, the notation $\cal{R}(\textbf{Ab\ldots Yz})$ is used to indicate a
polygon with vertices $\textbf{A},\textbf{b},\ldots,\textbf{Y},\textbf{z}$.
Also the notation $\overline{\textbf{Ab\ldots Yz}}$ is used for the piecewise
line segments connecting A, b, $\ldots$, Y, z.

Here we introduce the notion of weak solutions of (1.1).

###### Definition 2.1

Let $q\in[1,\infty)$ and $V$ be a measurable vector field. We say that a
nonnegative measurable function $\rho$ is a $L^{q}$-weak solution of (1.1)
with $\rho_{0}\in L^{1}(\mathbb{R}^{d})\cap L^{q}(\mathbb{R}^{d})$ if the
followings are satisfied:

  * (i)

It holds that

$\rho\in L^{\infty}\left(0,T;\left(L^{1}\cap
L^{q}\right)(\mathbb{R}^{d})\right)\cap
L^{m}(Q_{T}),\quad\nabla\rho^{\frac{m+q-1}{2}}\in
L^{2}(Q_{T}),\quad\text{and}\quad\rho V\in L^{1}(Q_{T}).$

  * (ii)

For any function
$\varphi\in{\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d}\times[0,T))$ it holds
that

$\iint_{Q_{T}}\left\\{\rho\varphi_{t}+\rho^{m}\Delta\varphi+\rho
V\cdot\nabla\varphi\right\\}\,dxdt=-\int_{\mathbb{R}^{d}}\rho_{0}(\cdot)\varphi(\cdot,0)\,dx.$
(2.2)

We remind the property of _mass conservation_ for nonnegative solutions of
(1.1), i.e.
$\|\rho(\cdot,t)\|_{L^{1}(\mathbb{R}^{d})}=\|\rho_{0}\|_{L^{1}(\mathbb{R}^{d})}$
for a.e. $t\in[0,T]$. This can be derived from the weak formulation (see e.g.
[Vaz06, Section 9.5.1]).

###### Remark 2.2

Without loss of generality, we assume that
$\rho_{0}\in\mathcal{P}(\mathbb{R}^{d})$, that is,
$\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{d})}=1$. We note that the mass conservation
property in time implies that $\|\rho(\cdot,t)\|_{L^{1}(\mathbb{R}^{d})}=1$
a.e. $t\in[0,T]$.

If $\|\rho_{0}\|_{L^{1}({\mathbb{R}}^{d})}=c>0$, then we replace $\rho$ by
$\tilde{\rho}=\rho/c$. Thus we have
$\tilde{\rho}_{0}\in\mathcal{P}({\mathbb{R}}^{d})$ and

$\partial_{t}\rho+\Delta\rho^{m}=\nabla\cdot(V\rho)\quad\Longleftrightarrow\quad\partial_{t}\tilde{\rho}+c^{m-1}\Delta(\tilde{\rho})^{m}=\nabla\cdot(V\tilde{\rho}).$

In fact, the diffusive coefficient $c^{m-1}$ of the equation for
$\tilde{\rho}$ does not play important role in our analysis, and thus, for
simplicity, we assume that $c=1$.

###  2.1 Existence results

In this subsection, we state all existence results and related remarks and
figures are presented.

####  2.1.1 Existence for case:
$\int_{{\mathbb{R}}^{d}}\rho_{0}\log\rho_{0}\,dx<\infty$.

In this subsection, we assume that the initial data satisfy

$\rho_{0}\in\mathcal{P}_{p}({\mathbb{R}}^{d})\quad\text{and}\quad\int_{{\mathbb{R}}^{d}}\rho_{0}\log\rho_{0}\,dx<\infty,$
(2.3)

where $p\in(1,2]$ will be specified later. We remark that it is not enough to
assume merely $\rho_{0}\in L^{1}(\mathbb{R}^{d})$ for showing the existence of
$L^{1}$-weak solutions. Instead, with slightly better initial data satisfying
(2.3), we compute a priori estimates in Proposition 4.5 and Proposition 4.9,
which enable us to obtain existence of weak solutions via suitable
approximations. Here we consider separately three different cases depending on
the condition of $V$. Firstly, Theorem 2.3 is when $V$ belongs to a certain
subscaling classses. Moreover, Theorem 2.4 deals with the case that $V$ is a
divergence-free vector and Theorem 2.5 is for the case $\nabla V$ belongs a
sub-scaling classes, respectively. The restriction of $1<m\leq 2$ appears in
general because of compactness arguments (see Remark 2.6 $(ii)$).

For the first case, our result reads as follows:

###### Theorem 2.3

Let $1<m\leq 2$ and $p>1$. Suppose that (2.3) and

$V\in\mathfrak{S}_{m,1}^{(q_{1},q_{2})}\quad\text{for}\quad\begin{cases}2\leq
q_{2}\leq\frac{m}{m-1},&\text{ if }d>2,\vspace{1 mm}\\\ 2\leq
q_{2}<\frac{m}{m-1},&\text{ if }d=2.\end{cases}$ (2.4)

  * (i)

For $1<p\leq{\lambda_{1}}:=1+\frac{1}{d(m-1)+1}$, then there exists a
nonnegative $L^{1}$-weak solution of (1.1) in Definition 2.1 such that
$\rho\in AC(0,T;\mathcal{P}_{p}(\mathbb{R}^{d}))$ with
$\rho(\cdot,0)=\rho_{0}$. Furthermore, $\rho$ satisfies

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}\times\\{t\\}}\rho\left(\left|\log\rho\right|+\langle
x\rangle^{p}\right)\,dx+\iint_{Q_{T}}\\{\left|\nabla\rho^{\frac{m}{2}}\right|^{2}+\left(\left|\frac{\nabla\rho^{m}}{\rho}\right|^{p}+|V|^{p}\right)\rho\\}\,dx\,dt\leq
C,$ (2.5)

and

$W_{p}(\rho(t),\rho(s))\leq C(t-s)^{\frac{p-1}{p}},\qquad\forall~{}~{}0\leq
s\leq t\leq T,$ (2.6)

where the constant
$C=C(\|V\|_{\mathfrak{S}_{m,1}^{(q_{1},q_{2})}},\,\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx)$.

  * (ii)

For $p>{\lambda_{1}}$, the same conclusions hold as in $(i)$ except that $p$
is replaced by ${\lambda_{1}}$. That is, there exists a nonnegative
$L^{1}$-weak solution of (1.1) in Definition 2.1 such that $\rho\in
AC(0,T;\mathcal{P}_{{\lambda_{1}}}(\mathbb{R}^{d}))$ with
$\rho(\cdot,0)=\rho_{0}$. Furthermore, $\rho$ satisfies

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}\times\\{t\\}}\rho\left(\left|\log\rho\right|+\langle
x\rangle^{{\lambda_{1}}}\right)\,dx+\iint_{Q_{T}}\\{\left|\nabla\rho^{\frac{m}{2}}\right|^{2}+\left(\left|\frac{\nabla\rho^{m}}{\rho}\right|^{{\lambda_{1}}}+|V|^{{\lambda_{1}}}\right)\rho\\}\,dx\,dt\leq
C,$ (2.7)

and

$W_{{\lambda_{1}}}(\rho(t),\rho(s))\leq
C(t-s)^{\frac{{\lambda_{1}}-1}{{\lambda_{1}}}},\qquad\forall~{}~{}0\leq s\leq
t\leq T,$ (2.8)

where the constant
$C=C(\|V\|_{\mathfrak{S}_{m,1}^{(q_{1},q_{2})}},\,\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx)$.

In the next theorem, $V$ is assumed to be divergence-free. The scaling
invariant class of $V$ is larger than that of Theorem 2.3 (See Remark 2.6
$(iii)$ and Fig. 3 for a more definte comparison).

###### Theorem 2.4

Let $1<m\leq 2$ and $p>1$. Assume (2.3) and the divergence-free $V$ (i.e.,
$\nabla\cdot V=0$) satisfying

$V\in\mathfrak{S}_{m,1}^{(q_{1},q_{2})}\quad\text{for}\quad\begin{cases}{\lambda_{1}}\leq
q_{2}\leq\frac{{\lambda_{1}}m}{m-1},&\text{ if }d>2,\vspace{1 mm}\\\
{\lambda_{1}}\leq q_{2}<\frac{{\lambda_{1}}m}{m-1},&\text{ if
}d=2,\end{cases}$ (2.9)

where ${\lambda_{1}}$ is the number defined in Theorem 2.3. Then the same
conclusions hold as in Theorem 2.3.

The last theorem is concerned about the case that $\nabla V$ belongs to a sub-
scaling class (see Remark 2.6 $(iv)$).

###### Theorem 2.5

Let $1<m\leq 2$ and let $p>1$. Suppose that (2.3) and

$V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}\ \text{ for }\
\begin{cases}\frac{2+d(m-1)}{1+d(m-1)}<\tilde{q}_{2}\leq\frac{m}{m-1},&\text{
if }d>2,\vspace{1 mm}\\\
\frac{2+d(m-1)}{1+d(m-1)}<\tilde{q}_{2}<\frac{m}{m-1},&\text{ if
}d=2.\end{cases}$ (2.10)

Then the same conclusions hold as in Theorem 2.3 with
$C=C(\|V\|_{\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}},\,\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx)$.

Let us provide some remarks regarding three theorems in this subsection.

###### Remark 2.6

  * (i)

Results in Theorem 2.3 are also true for linear case $(m=1)$ for $1\leq p\leq
2$ (see [KK-arxiv]). We do not know whether or not the moment estimate (2.5)
can be extended to $p>{\lambda_{1}}$, and thus we leave it as an open
question.

  * (ii)

There is restriction on $m$ that $1<m\leq 2$ in Theorem 2.4 and 2.5 for cases
$\nabla\cdot V=0$ or
$V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$, though a
priori estimates are obtained for $m>1$ in Proposition 4.5 (ii), (iii). Such a
restriction is necessary because the compactness argument in Proposition 4.14
only works for $1<m\leq 2$. Therefore, if one can resolve the compactness
argument in cases $\nabla\cdot V=0$ or
$V\in\tilde{\mathfrak{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$ for $m>2$,
then the results in Theorem 2.4 and 2.5 become valid for any $m>1$.

  * (iii)

In the following figure, we illustrate the range of $(q_{1},q_{2})$ satisfying
(2.4) and (2.9) in particular for scaling invariant class
$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$ as dark-shaded region
$\mathcal{R}\left(\textbf{ABC}\right)$ and as lightly-shaded region
$\mathcal{R}\left(\textbf{ABDE}\right)$, respectively. When $d=2$, the line
$\overline{\textbf{BC}}$ and $\overline{\textbf{BD}}$ are excluded from
$\mathcal{R}\left(\textbf{ABC}\right)$ and
$\mathcal{R}\left(\textbf{ABDE}\right)$, respectively.

Figure 3. Theorem 2.3 & 2.5 for $1<m\leq 2$,
$d>2$.$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$$+$$1$$\bullet$A$+$$\frac{1}{2}$$\bullet$C$\bullet$$\bullet$D$\bullet$$\frac{1+d}{2+d}$E$\times$$\frac{1+d}{d}$$\mathcal{S}_{2,1}^{(q_{1},q_{2})}$B$\bullet$$\frac{1}{{\lambda_{1}}}$f$\times$$\frac{1+d(m-1)}{d}$$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$$*$$\bullet$g$*$$\bullet$h$*$$*$$\bullet$i$*$$*$$*$$\overline{\textbf{AB}}=\mathcal{S}_{1,1}^{(q_{1},q_{2})}$,
$\textbf{A}=(0,\frac{1}{2})$,
$\textbf{B}=(\frac{1}{d},0)$$\mathcal{R}\left(\textbf{ABC}\right)$: scaling
invarant case of (2.4) $\mathcal{R}\left(\textbf{ABDE}\right)$: scaling
invariant case of (2.9) $\textbf{C}=(\frac{1}{2},\frac{1}{2})$,
$\textbf{D}=(\frac{1+d}{2d},\frac{1+d}{2(2+d)})$,
$\textbf{E}=(0,\frac{1+d}{2+q})$ f, g, h,i: on
$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$ for
$1<m<2$$\textbf{g}=(\frac{m-1}{2},\frac{1}{2})$,
$\textbf{h}=(\frac{(2-m)+d(m-1)}{md},\frac{m-1}{m})$
$\textbf{f}=(0,\frac{1}{{\lambda_{1}}})$,
$\textbf{i}=(\frac{1+d(m-1)}{md},\frac{m-1}{{\lambda_{1}}m})$

  * (iv)

The range of $(\tilde{q}_{1},\tilde{q}_{2})$ for $V$ satisfying (2.10) in
$\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$ is illustrated as
the line $\overline{\textbf{DE}}$ on Figure 9-$(i)$ in Section 2.1.4
(including D but excluding E if $d>2$ and excluding both if $d=2$).

####  2.1.2 Existence for case: $\rho_{0}\in L^{q}(\mathbb{R}^{d}),\ q>1$

We first state results regarding the existence of a $L^{q}$-weak solution
satisfying an energy inequality. In case that $1<m\leq 2$, there is no
restriction on $q$ that $q>1$. It is, however, not clear, for the case $m>2$
and, in fact, according to our analysis, it is requred that $q\geq m-1$. We
also construct a $L^{q}$-weak solution satisfying moments and speed estimates
as well as an energy inequality, and the restriction $q\geq m-1$ is necessary
as similarly, if $m>2$ (Theorem 2.9).

Now we start with $L^{q}$-weak solutions satisfying energy inequality.

###### Theorem 2.7

Let $m>1$. Also let $q>1$ and $q\geq m-1$. Assume that
$\rho_{0}\in\mathcal{P}({\mathbb{R}}^{d})\cap L^{q}({\mathbb{R}}^{d})$ and

$V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}\ \text{ for }\ \begin{cases}2\leq
q_{2}\leq\frac{q+m-1}{m-1},&\text{ if }d>2\vspace{1 mm}\\\ 2\leq
q_{2}<\frac{q+m-1}{m-1},&\text{ if }d=2.\end{cases}$ (2.11)

Then, there exists a nonnegative $L^{q}$-weak solution of (1.1) in Definition
2.1 that holds

$\operatorname*{ess\ sup}_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\rho^{q}(\cdot,t)\,dx+\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt\leq
C,$ (2.12)

with
$C=C(\|V\|_{\mathfrak{S}_{m,q}^{(q_{1},q_{2})}},\|\rho_{0}\|_{L^{q}({\mathbb{R}}^{d})})$.

We make a few remarks regarding Theorem 2.7.

###### Remark 2.8

  1. (i)

Figure 4 (for cases: $1<m\leq 2$, $q>1$) is to visualize the pairs of
$(q_{1},q_{2})$ satisfying (2.11) in $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ for
$d>2$ as the shaded region $\mathcal{R}(\textbf{ABCD})$. If $d=2$, then the
line $\overline{\textbf{BC}}$ does not included to the shaded region. The
points $\textbf{e}\to\textbf{A}$ and $\textbf{f}\to\textbf{B}$ in both cases
either $m\to 1$ or $q\to\infty$. Moreover, we observe that
$\textbf{C}\to\textbf{B}$ and $\textbf{D}\to\textbf{A}$ as $m\to 1$. Also,
when $m=2$, then $\textbf{C}=\textbf{D}=(1/2,1/2)$.

  2. (ii)

Figure 5 (for cases: $m>2$, $q\geq m-1$) is illustrating the pairs of
$(q_{1},q_{2})$ satisfying (2.11) in $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ as
the shaded region $\mathcal{R}(\textbf{ABC})$ for $d>2$. If $d=2$, then the
line $\overline{\textbf{BC}}$ does not included to the shaded region. The
points $\textbf{e}\to\textbf{A}$ and $\textbf{f}\to\textbf{B}$ as
$q\to\infty$.

Figure 4: Theorem 2.7 for $1<m\leq 2$, $q>1$.
$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$$+$DCAB$\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$$\times$$\frac{1}{{\lambda_{1}}}$$\times$$\frac{1+d(m-1)}{d}$$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$$\times$$\frac{1+q_{m,d}}{2+q_{m,d}}$$\times$$\frac{1+q_{m,d}}{d}$$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$$\bullet$f$\bullet$e$\overline{\textbf{AB}}=\mathcal{S}_{m,\infty}^{(q_{1},q_{2})},\,\textbf{A}=(0,\frac{1}{2}),\,\textbf{B}=(\frac{1}{d},0)$$\mathcal{R}(\textbf{ABCD})$
: scaling invariant class of (2.11)
$\textbf{C}=(\frac{(2-m)+d(m-1)}{md},\frac{m-1}{m})$,
$\textbf{D}=(\frac{m-1}{2},\frac{1}{2})$e, f: on
$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ for $1<q<\infty$ coordinates are given in
Fig. 5

Figure 5: Theorem 2.7 for $m>2$, $q\geq m-1$.
$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$$+$AB$\bullet$C$\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$$\times$$\frac{1+d}{2+d}$$\times$$\frac{1+d}{d}$$\mathcal{S}_{m,m-1}^{(q_{1},q_{2})}$$\times$$\frac{1+q_{m,d}}{2+q_{m,d}}$$\times$$\frac{1+q_{m,d}}{d}$$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$$\bullet$f$\bullet$e$\overline{\textbf{AB}}=\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$,
$\textbf{C}=(\frac{1}{2},\frac{1}{2})$, A,B: in Fig.
4$\mathcal{R}(\textbf{ABC})$: scaling invariant class of (2.11)e, f: on
$S_{m,q}^{(q_{1},q_{2})}$ for
$m-1<q<\infty$$\textbf{e}=(\frac{m-1}{2q},\frac{1}{2})$,
$\textbf{f}=(\frac{(q-m+1)+d(m-1)}{d(q+m-1)},\frac{m-1}{q+m-1})$

In the next theorem, we express the existence of a $L^{q}$-weak solution as an
absolutely continuous curve in Wasserstein space for
$\rho_{0}\in\mathcal{P}_{p}({\mathbb{R}}^{d})$. More precisely, $p$-th moment
and speed estimates are additionally added, compared to weak solutions in
Theorem 2.7. Due to constraint of handling speed and $p$-th moment estimates
for $p\leq 2$, the condition on $V$ is subject to restrain when $q>m$ that
(2.13) gives smaller region compared to (2.11) (cf. Figure 4 Vs. 4-$(i)$,
Figure 5 Vs. 5-$(i)$). We refer Section 4.3 for more details. Meanwhile,
because the temporal domain is bounded, we are able to extend conditions on
$V$ (meaningful when $q>m$) by applying embedding arguments in Proposition
4.11.

###### Theorem 2.9

Let $m>1$ and $p>1$. Also let $q>1$ and $q\geq m-1$. Suppose that
$\rho_{0}\in\mathcal{P}_{p}({\mathbb{R}}^{d})\cap L^{q}({\mathbb{R}}^{d})$.

  * (i)

Assume that

$V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}\ \text{ for }\
\begin{cases}q_{1}\leq\frac{2m}{m-1},\ 2\leq
q_{2}\leq\frac{q+m-1}{m-1}\,\delta_{\\{1<q\leq
m\\}}+\frac{2m-1}{m-1}\,\delta_{\\{q>m\\}},&\text{ if }d>2,\vspace{1 mm}\\\
q_{1}\leq\frac{2m}{m-1},\ 2\leq q_{2}<\frac{q+m-1}{m-1}\,\delta_{\\{1<q\leq
m\\}}+\frac{2m-1}{m-1}\,\delta_{\\{q>m\\}},&\text{ if }d=2.\end{cases}$ (2.13)

    * (a)

Then, for
$1<p\leq{\lambda_{q}}:=\min\left\\{2,1+\frac{d(q-1)+q}{d(m-1)+q}\right\\}$,
there exists a nonnegative $L^{q}$-weak solution of (1.1) in Definition 2.1
such that $\rho\in AC(0,T;\mathcal{P}_{p}(\mathbb{R}^{d}))$ with
$\rho(\cdot,0)=\rho_{0}$. In addition, $\rho$ satisfies (2.6) and

$\sup_{0\leq t\leq T}\int_{\mathbb{R}^{d}}\left(\rho^{q}+\rho\langle
x\rangle^{p}\right)\,dx+\iint_{Q_{T}}\\{\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}+\left(\left|\frac{\nabla\rho^{m}}{\rho}\right|^{p}+|V|^{p}\right)\rho\\}\,dx\,dt\leq
C,$ (2.14)

where
$C=C(\|V\|_{\mathfrak{S}_{m,q}^{(q_{1},q_{2})}},\int_{\mathbb{R}^{d}}\left(\rho_{0}^{q}+\rho_{0}\langle
x\rangle^{p}\right)\,dx)$.

    * (b)

For $p>{\lambda_{q}}$, the same conclusions hold as in (a) except that $p$ is
replaced by ${\lambda_{q}}$.

  * (ii)

(Embedding) Let $q>m$. Assume that

$V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}\ \text{ for }\
\begin{cases}q_{1}\leq\frac{2m}{m-1},\
\frac{2m-1}{m-1}<q_{2}\leq\frac{q+m-1}{m-1},&\text{ if }d>2,\vspace{1 mm}\\\
q_{1}\leq\frac{2m}{m-1},\ \frac{2m-1}{m-1}\leq q_{2}<\frac{q+m-1}{m-1},&\text{
if }d=2.\end{cases}$ (2.15)

For $(q_{1},q_{2})$ in (2.15), there exists a constant
$q_{2}^{\ast}=q_{2}^{\ast}(q_{1})\in[2,q_{2})$ such that
$V\in\mathfrak{S}_{m,q^{\ast}}^{(q_{1},q_{2}^{\ast})}$ satisfying (2.13).
Furthermore, the same conclusions hold as in $(i)$ except that ${\lambda_{q}}$
is replaced by ${\lambda_{q^{\ast}}}$.

Here are a few remarks about Theorem 2.9.

###### Remark 2.10

  * (i)

Results in Theorem 2.9 are also true for the heat equation ($m=1$) with drift
term for $1<p\leq 2$ (see [KK-arxiv]). We do not know whether or not the
moment estimate (2.14) can be extended to $p>{\lambda_{q}}$, and thus we leave
those as open questions.

  * (ii)

When $1<q\leq m$, the condition (2.13) is the same as (2.11). When $q>m$, we
use speed and $p$-th moment estimates for $q=m$, therefore the condition (4.2)
for $q=m$ is the same as (2.13) for $q>m$ (refer Proposition 4.10 $(i)$).

  * (iii)

Let $1<m\leq 2$ and $q>1$. The following figure shows the pairs
$(q_{1},q_{2})$ satisfying (2.13) and (2.15) in
$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$. There are Figure 4-(ii) and 4-(iii) in
Appendix B for cases $(ii)$
$\max\\{2,\frac{2m}{(2m-1)(m-1)}\\}<d\leq\frac{2m}{m-1}$ and $(iii)$
$d>\frac{2m}{m-1}$. When $d=2$, $\overline{\textbf{BCD}}$ is excluded from
$\mathcal{R}(\textbf{ABCDEF})$.

Figure 4-(i). Theorem 2.9 for $1<m\leq 2$, $q>1$,
$2<d\leq\max\\{2,\frac{2m}{(2m-1)(m-1)}\\}$.$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$$+$EDB$\bullet$CA$\bullet$F$\times$ab$\times$$\frac{1}{{\lambda_{1}}}$$\times$$\frac{1+d(m-1)}{d}$$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$$\times$$\frac{m+d(m-1)}{2m+d(m-1)}$$\times$c$\mathcal{S}_{m,m}^{(q_{1},q_{2})}$$*$$\frac{m-1}{2m}$$*$$*$$\frac{m-1}{2m-1}$$*$$*$$*$$*$$*$$\overline{\textbf{ab}}=\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$,
$\textbf{a}=(0,\frac{1}{2})$, $\textbf{b}=(\frac{1}{d},0)$
$\mathcal{R}(\textbf{ABCDEF})$: scaling invariant class of (2.13).
$\mathcal{R}(\textbf{bCB})$: scaling invariant class of (2.15).
$\textbf{A}=(\frac{m-1}{2m},\frac{d+m(2-d)}{4m})$,
$\textbf{B}=(\frac{1}{d(2m-1)},\frac{m-1}{2m-1})$
$\textbf{C}=(\frac{1+d(m-1)}{d(2m-1)},\frac{m-1}{2m-1})$,
$\textbf{c}=(\frac{m+d(m-1)}{md},0)$
$\textbf{D}=(\frac{(2-m)+d(m-1)}{md},\frac{m-1}{m})$$\textbf{E}=(\frac{m-1}{2},\frac{1}{2})$,
$\textbf{F}=(\frac{m-1}{2m},\frac{1}{2})$

  * (iv)

Figure 5-(i) is illustrating the pairs $(q_{1},q_{2})$ satisfying (2.13) and
(2.15) in $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ when $m>2$ and $q\geq m-1$.
There is Figure 5-(ii) in Appendix B for case $d>\frac{2m}{m-1}$. When $d=2$,
$\overline{\textbf{GCD}}$ is excluded from $\mathcal{R}(\textbf{GCDE})$.

Figure 5-(i). Theorem 2.9 for $m>2$, $q\geq m-1$,
$2<d\leq\frac{2m}{m-1}$.$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$$+$a$\times$ab$\bullet$D$\bullet$EAB$\bullet$C$\bullet$G$\times$$\frac{1+d}{2+d}$$\times$$\frac{1+d}{d}$$\mathcal{S}_{m,m-1}^{(q_{1},q_{2})}$$\times$$\frac{m+d(m-1)}{md}$$\times$$\frac{m+d(m-1)}{2m+d(m-1)}$$\mathcal{S}_{m,m}^{(q_{1},q_{2})}$$\overline{\textbf{ab}}=\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$,
$\textbf{a}=(0,\frac{1}{2})$, $\textbf{b}=(\frac{1}{d},0)$
$\mathcal{R}(\textbf{GCDE})$: scaling invariant class of (2.13)
$\mathcal{R}(\textbf{GAbC}))$: scaling invariant class of (2.15)
$\textbf{A}=(\frac{m-1}{2m},\frac{d+m(2-d)}{4m})$,
$\textbf{B}=(\frac{1}{d(2m-1)},\frac{m-1}{2m-1})$$\textbf{C}=(\frac{1+d(m-1)}{d(2m-1)},\frac{m-1}{2m-1})$,
$\textbf{D}=(\frac{1}{2},\frac{1}{2})$
$\textbf{E}=(\frac{m-1}{2m},\frac{1}{2})$,
$\textbf{G}=(\frac{m-1}{2m},\frac{m-1}{2m-1})$

####  2.1.3 Existence for case: $\nabla\cdot V=0$ and $\rho_{0}\in
L^{q}(\mathbb{R}^{d}),\ q>1$.

Here we assume that $V$ is a divergence-free vector field. Compared to Section
2.1.2, theorems in this subsection work for less restricted range for $q$ and
less restrictions on $V$ in sub-scaling class.

Let us define

$q_{2}^{M}:=\left\\{\begin{array}[]{cl}\left(\frac{d(q+m-1)+q}{d(q+m-1)+2q}-\frac{q}{q+m-1}\right)^{-1},&\text{if
}1<q<q^{\ast},\vspace{2 mm}\\\ \infty,&\text{if }q\geq
q^{\ast},\end{array}\right.$

where

$q^{\ast}:=\frac{2d(m-1)}{\sqrt{d^{2}+6d+1}-(d+1)}.$ (2.16)

In this case, the existence result is stated for $L^{q}$-weak solutions with
an energy inequality as follows:

###### Theorem 2.11

Let $m>1$. Also let $q>\max\\{1,\frac{m}{2}\\}$. Suppose that
$\rho_{0}\in\mathcal{P}({\mathbb{R}}^{d})\cap L^{q}({\mathbb{R}}^{d})$.
Furthermore, assume that $V$ is divergence-free (that is, $\nabla\cdot V=0$)
and

$V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}\quad\text{for}\quad\begin{cases}\frac{2+q_{m,d}}{1+q_{m,d}}\leq
q_{2}\leq q_{2}^{M},&\text{if }d>2,\vspace{2 mm}\\\
\frac{2+q_{m,d}}{1+q_{m,d}}\leq q_{2}<q_{2}^{M}\cdot\delta_{\\{1<q\leq
q^{\ast}\\}},&\text{if }d=2,\vspace{2 mm}\\\ \frac{2+q_{m,d}}{1+q_{m,d}}\leq
q_{2}\leq q_{2}^{M}\cdot\delta_{\\{q>q^{\ast}\\}},&\text{if }d=2.\end{cases}$
(2.17)

Then, there exists a nonnegative $L^{q}$-weak solution of (1.1) in Definition
2.1 that holds

$\displaystyle\operatorname*{ess\ sup}_{0\leq t\leq T}$
$\displaystyle\int_{\mathbb{R}^{d}}\rho^{q}(\cdot,t)dx+\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt\leq
C,$ (2.18)

where $C=C(\|\rho_{0}\|_{L^{q}({\mathbb{R}}^{d})})$.

We make a few remarks regarding the above theorem.

###### Remark 2.12

  * (i)

The restriction on (2.17) is directly from Lemma 4.17 and Proposition 4.19
$(c)$ for carrying compactness arguments which is essential to show the
convergence of approximated solutions, although a priori estimate is
calculated for any $m,q>1$ in Proposition 4.6 (iii). Moreover, we note that
(2.17) is weaker than (2.11) or (2.19).

  * (ii)

The range of $q>\max\\{1,\frac{m}{2}\\}$ is sorted into two cases: either
$1<m\leq 2$ and $q>1$, or $m>2$ and $q>\frac{m}{2}$. However, the compactness
arguments in Proposition 4.19 $(ii)$ works for $q>\frac{m}{2}$ (which is
strictly less than $m-1$ for $m>2$). In fact, when $m>2$, the range $q\geq
m-1$ in Proposition 4.19 $(i)$ is weakened to $q>\frac{m}{2}$ thanks to
$\nabla\cdot V=0$.

  * (iii)

Here we plot the region of pairs $(q_{1},q_{2})$ satisfying (2.17) in
$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ for $1<m\leq 2$, $q>1$, $d>2$. Note that
$\textbf{E},\frac{1+q_{m,d}}{2+q_{m,d}},\textbf{f}\to\textbf{A}$ and
$\textbf{C},\frac{1+q_{m,d}}{d}\to\textbf{B}$ as either $m\to 1$ or
$q\to\infty$. When $d=2$, the line $\overline{\textbf{CD}}$ is excluded from
the region $\mathcal{R}(\textbf{ABCDE})$. There is Figure 6-$(ii)$ for $m>2$
and $q>\frac{m}{2}$ in Appendix B.

Figure 6-(i). Theorem 2.11 for $1<m\leq 2$, $q>1$.
$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$A$\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$B$\frac{1}{{\lambda_{1}}}$E$\times$$\frac{1+d(m-1)}{d}$$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$$\bullet$f$\bullet$C$\mathcal{S}_{m,q^{\ast}}^{(q_{1},q_{2})}$$\bullet$$\frac{1+q_{m,d}}{2+q_{m,d}}$
$\times$$\frac{1+q_{m,d}}{d}$$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$$\bullet$gD$\overline{\textbf{AB}}=\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$,
$\textbf{A}=(0,\frac{1}{2})$, $\textbf{B}=(\frac{1}{d},0)$
$\mathcal{R}(\textbf{ABCDE})$: scaling invariant class of
(2.17)$\overline{\textbf{fC}}=\mathcal{S}_{m,q^{\ast}}^{(q_{1},q_{2})}$,
$\textbf{f}=(0,\frac{1+q^{\ast}_{m,d}}{2+q^{\ast}_{m,d}})$,
$\textbf{C}=(\frac{1+q^{\ast}_{m,d}}{d},0)$ $q^{\ast}$ is defined in (2.16)
and
$q^{\ast}_{m,d}=\frac{d(m-1)}{q^{\ast}}$$\textbf{g}=(\frac{d(q+m-1)+q}{d(q+m-1)+2q}-\frac{q(d-2)}{d(q+m-1)},\frac{d(q+m-1)+q}{d(q+m-1)+2q}-\frac{q}{q+m-1})$g
is on $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ for
$1<q<q^{\ast}$$\textbf{D}=(\frac{md+1}{md+2}-\frac{d-2}{md},\frac{md+1}{md+2}-\frac{1}{m})$
on $\mathcal{S}_{m,1}^{(q_{1},q_{2})}$

Next, we show the existence of a $L^{q}$-weak solution in Wasserstein space in
which speed and moment estimates play importantly. The second part of the
following theorem is obtained by embedding in time variable. Unlike result in
Theorem 2.9 $(ii)$, one can make observation that the embedding argument is
meaningful for all $q>1$.

###### Theorem 2.13

Let $m>1$ and $p>1$. Also let $q>1$ and $q\geq\frac{m}{2}$. Suppose that
$\rho_{0}\in\mathcal{P}_{p}({\mathbb{R}}^{d})\cap L^{q}({\mathbb{R}}^{d})$ and
$\nabla\cdot V=0$.

  * (i)

Suppose that

$V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}\ \text{ for }\
\begin{cases}{\lambda_{q}}\leq
q_{2}\leq\frac{{\lambda_{q}}(q+m-1)}{q+m-2},&\text{ if }d>2,\ 1<q\leq
m,\vspace{1 mm}\\\ q_{1}\leq\frac{2m}{m-1},\ 2\leq
q_{2}\leq\frac{2m-1}{m-1},&\text{ if }d>2,\ q>m,\vspace{1 mm}\\\
{\lambda_{q}}\leq q_{2}<\frac{{\lambda_{q}}(q+m-1)}{q+m-2},&\text{ if }d=2,\
1<q\leq m,\vspace{1 mm}\\\ q_{1}\leq\frac{2m}{m-1},\ 2\leq
q_{2}<\frac{2m-1}{m-1},&\text{ if }d=2,\ q>m.\end{cases}$ (2.19)

    * (a)

Then, for
$1<p\leq{\lambda_{q}}:=\min\left\\{2,1+\frac{d(q-1)+q}{d(m-1)+q}\right\\}$,
there exists a nonnegative $L^{q}$-weak solution of (1.1) in Definition 2.1
such that $\rho\in AC(0,T;\mathcal{P}_{p}(\mathbb{R}^{d}))$ with
$\rho(\cdot,0)=\rho_{0}$. Furthermore, $\rho$ satisfies (2.18), (2.14), and
(2.6) with
$C=C(\left\|V\right\|_{\mathfrak{S}_{m,q}^{(q_{1},q_{2})}},\int_{\mathbb{R}^{d}}\left(\rho_{0}^{q}+\rho_{0}\langle
x\rangle^{p}\right)\,dx)$.

    * (b)

For $p>{\lambda_{q}}$, the same conclusions hold as in (a) except that $p$ is
replace by ${\lambda_{q}}$.

  * (ii)

(Embedding) Suppose that

$V\in\mathfrak{S}_{m,q}^{(q_{1},q_{2})}\ \text{ for }\
\begin{cases}\frac{2+q_{m,d}}{1+q_{m,d}}<q_{2}<\frac{{\lambda_{1}}m}{m-1},&\text{
if }1<q\leq\frac{md}{d-1},\vspace{1 mm}\\\
\frac{2+q_{m,d}}{1+q_{m,d}}<q_{2}\leq\infty,&\text{ if
}q>\frac{md}{d-1}.\end{cases}$ (2.20)

For given $(q_{1},q_{2})$ in (2.20), there exists constants
$q_{2}^{\ast}=q_{2}^{\ast}(q_{1})\in[2,q_{2}]$ such that
$V\in\mathfrak{S}_{m,q^{\ast}}^{(q_{1},q_{2}^{\ast})}$ satisfying (2.19).
Furthermore, the same conclusion holds as in $(i)$ except that ${\lambda_{q}}$
is replaced by ${\lambda_{q^{\ast}}}$.

Now we explain more on the condition (2.19) in Theorem 2.13.

###### Remark 2.14

  * (i)

The condition (2.19) is distinguished for $1<q\leq m$ and $q>m$ because the
structure of ${\lambda_{q}}$ in (2.1) is fixed as $2$ if $q>m$ (it is hard to
obtain speed and moment estimates for $p>2$). Details of getting a priori
estimate under (2.19) are in Proposition 4.10 $(iii)$. The limiting case
$q=\frac{m}{2}$ is allowed because
$\rho_{0}\in\mathcal{P}_{p}({\mathbb{R}}^{d})$ and $V$ satisfying (2.19) by
Remark 4.20 $(ii)$.

  * (ii)

In the following figure, we plot pairs $(q_{1},q_{2})$ satisfying (2.19) and
(2.20) in $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ for $1<m\leq 2$ and $q>1$. The
irregular shape is mainly because of (4.2) for estimating speed and $p$-th
moment estimates which separates two cases either $1<q\leq m$ or $q>m$.
Moreover, Figure 7-(i) is when $2<d\leq\max\\{2,\frac{2m}{(2m-1)(m-1)}\\}$.
For cases $\max\\{2,\frac{2m}{(2m-1)(m-1)}\\}<d\leq\frac{2m}{m-1}$ and
$d>\frac{2m}{m-1}$, there are Figure 7-$(ii)$, $(iii)$ in Appendix B.
Additionally, Figure 8-$(i)$, $(ii)$ in Appendix B are when $m>2$ and
$q\geq\frac{m}{2}$. When $d=2$, $\overline{\textbf{BCD}}$ is excluded from
$\mathcal{R}(\textbf{ABCDEF})$.

Figure 7-(i). Theorem 2.13 for $1<m\leq 2$, $q>1$,
$2<d\leq\max\\{2,\frac{2m}{(2m-1)(m-1)}\\}$.$0$$\frac{1}{q_{1}}$$\frac{1}{q_{2}}$$+$$1$$+$aDB$\bullet$Cb$\frac{1}{{\lambda_{1}}}$Eg$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$$\times$$\frac{m+d(m-1)}{2m+d(m-1)}$$\times$i$\mathcal{S}_{m,m}^{(q_{1},q_{2})}$A$*$$\bullet$F$*$$*$$*$$*$$*$h$*$$\overline{\textbf{ab}}=\mathcal{S}_{m,\infty}^{(q_{1},q_{2})}$,
$\textbf{a}=(0,\frac{1}{2})$, $\textbf{b}=(\frac{1}{d},0)$
$\mathcal{R}(\textbf{ABCDEF})$: scaling invariant class of (2.19).
$\mathcal{R}(\textbf{abhDE})$: scaling invariant class of (2.20).
$\textbf{A}=(\frac{m-1}{2m},\frac{d+m(2-d)}{4m})$,
$\textbf{B}=(\frac{1}{d(2m-1)},\frac{m-1}{2m-1})$
$\textbf{C}=(\frac{1+d(m-1)}{d(2m-1)},\frac{m-1}{2m-1})$,
$\textbf{D}=(\frac{1+d(m-1)}{md},\frac{m-1}{{\lambda_{1}}m})$
$\textbf{E}=(0,\frac{1}{{\lambda_{1}}})$,
$\textbf{F}=(\frac{m-1}{2m},\frac{1}{2})$
$\textbf{g}=(\frac{1+d(m-1)}{d},0)$$\textbf{h}=(\frac{1+d(m-1)}{md},0)$,
$\textbf{i}=(\frac{m+d(m-1)}{md},0)$

####  2.1.4 Existence for case: $\nabla V$ in sub-scaling classes

Now we assume that $\nabla V$ is satisfying sub-scaling classes, that is,
$V\in\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$. We then see
that there are corresponding existence results and collect them in this
subsection when $\rho_{0}\in L^{q}(\mathbb{R}^{d})$.

The first result is concerned about $L^{q}$-weak solutions with an energy
inequality.

###### Theorem 2.15

Let $m>1$. Also let $q>\max\\{1,\frac{m}{2}\\}$. Suppose that
$\rho_{0}\in\mathcal{P}({\mathbb{R}}^{d})\cap L^{q}({\mathbb{R}}^{d})$ and

$V\in\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}\ \text{ for }\
\begin{cases}\frac{2+q_{m,d}}{1+q_{m,d}}<\tilde{q}_{2}\leq\frac{q+m-1}{m-1},&\text{
if }d>2,\vspace{1 mm}\\\ \
\frac{2+q_{m,2}}{1+q_{m,2}}<\tilde{q}_{2}<\frac{q+m-1}{m-1},&\text{ if
}d=2.\end{cases}$ (2.21)

Then, there exists a nonnegative $L^{q}$-weak solution of (1.1) in Definition
2.1 satisfying (2.12) with
$C=C(\left\|V\right\|_{\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}},\|\rho_{0}\|_{L^{q}(\mathbb{R}^{d})})$.

The second result is the existence of a $L^{q}$-weak solution as an absolutely
continuous curve in Wasserstein space.

###### Theorem 2.16

Let $m>1$ and $p>1$. Also let $q>1$ and $q\geq\frac{m}{2}$. Suppose that
$\rho_{0}\in\mathcal{P}_{p}({\mathbb{R}}^{d})\cap L^{q}({\mathbb{R}}^{d})$.

  * (i)

Assume that

$V\in\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}\ \text{ for }\
\begin{cases}\frac{2+q_{m,d}}{1+q_{m,d}}<\tilde{q}_{2}\leq\frac{q+m-1}{m-1}\,\delta_{\\{1<q\leq
m\\}}+\frac{2m-1}{m-1}\,\delta_{\\{q>m\\}},&\text{ if }d>2,\vspace{1 mm}\\\
\frac{2+q_{m,2}}{1+q_{m,2}}<\tilde{q}_{2}<\frac{q+m-1}{m-1}\,\delta_{\\{1<q\leq
m\\}}+\frac{2m-1}{m-1}\,\delta_{\\{q>m\\}},&\text{ if }d=2.\end{cases}$ (2.22)

    * (a)

Then, for
$1<p\leq{\lambda_{q}}:=\min\left\\{2,1+\frac{d(q-1)+q}{d(m-1)+q}\right\\}$,
there exists a nonnegative $L^{q}$-weak solution of (1.1) in Definition 2.1
such that $\rho\in AC(0,T;\mathcal{P}_{p}(\mathbb{R}^{d}))$ with
$\rho(\cdot,0)=\rho_{0}$. Furthermore, $\rho$ satisfies (2.12) with
$C=C(\left\|V\right\|_{\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}},\|\rho_{0}\|_{L^{q}({\mathbb{R}}^{d})})$.
Also, (2.14) and (2.6) hold with
$C=C(\left\|V\right\|_{\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}},\int_{\mathbb{R}^{d}}\left(\rho_{0}^{q}+\rho_{0}\langle
x\rangle^{p}\right)\,dx)$.

    * (b)

For $p>{\lambda_{q}}$, the same conclusions hold as in (a) except that $p$ is
replaced by ${\lambda_{q}}$.

  * (ii)

(Embedding) Let $q>m$. Assume that

$V\in\tilde{\mathfrak{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}\ \text{ for }\
\begin{cases}\frac{2m-1}{m-1}<\tilde{q}_{2}\leq\frac{q+m-1}{m-1},&\text{ if
}d>2,\vspace{1 mm}\\\
\frac{2m-1}{m-1}\leq\tilde{q}_{2}<\frac{q+m-1}{m-1},&\text{ if
}d=2.\end{cases}$ (2.23)

For given $(\tilde{q}_{1},\tilde{q}_{2})$ in (2.23), there exists a constant
$\tilde{q}_{2}^{\ast}=\tilde{q}_{2}^{\ast}(\tilde{q}_{1})\in(\frac{2+q^{\ast}_{m,d}}{1+q^{\ast}_{m,d}},\tilde{q}_{2}]$
such that
$V\in\tilde{\mathcal{S}}_{m,q^{\ast}}^{(\tilde{q}_{1},\tilde{q}_{2}^{\ast})}$
satisfying (2.22). Furthermore, the same conclusion holds as in Theorem 2.16
$(i)$ except that ${\lambda_{q}}$ is replaced by ${\lambda_{q^{\ast}}}$.

Here we make a few remarks of Theorem 2.15 and Theorem 2.16.

###### Remark 2.17

  * (i)

When $V\in\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$, a priori
estimate is calculated for any $m,q>1$ in Proposition 4.6 (ii), but the
restriction $q>\frac{m}{2}$ is required because of the compactness arguments
in Proposition 4.19 $(ii)$. In Theorem 2.15, we allow the limiting case
$q=\frac{m}{2}$ if $\rho_{0}\in\mathcal{P}_{p}({\mathbb{R}}^{d})$ with $C$
depending also on $\int_{{\mathbb{R}}^{d}}\rho_{0}\langle x\rangle^{p}\,dx$
under the same condition on $V$ satisfying (2.21).

  * (ii)

In Figure 9-$(i)$, we plot the pair $(\tilde{q}_{1},\tilde{q}_{2})$ satisfying
$\tilde{S}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$ for $1<m\leq 2$, $q>1$. In
particular, the region $\mathcal{R}\left(\textbf{ABCDE}\right)$ (dark shaded)
to visualize (2.22), and the region $\mathcal{R}\left(\textbf{bCB}\right)$
(lightly shaded) is for (2.23). The restriction $\tilde{q}_{1}\in(1,d)$ is
because of the relation of $\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ and
$\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$ in Remark 1.2
(ii). There is Figure 9-$(ii)$ in Appendix B for $m>2$ and $q\geq\frac{m}{2}$.
When $d=2$, the line $\overline{\textbf{BCD}}$ is excluded from
$\mathcal{R}(\textbf{ABCDE})$.

Firgure 9-(i). Theorem 2.15 and 2.16 for $1<m\leq 2$,
$q>1$.$\frac{1}{\tilde{q}_{1}}$$\frac{1}{\tilde{q}_{2}}$$\times$a$\times$$\frac{2+d(m-1)}{d}$$\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$$\times$$\frac{2m+d(m-1)}{md}$$\tilde{\mathcal{S}}_{m,m}^{(\tilde{q}_{1},\tilde{q}_{2})}$b$*$$\frac{1}{d}$$\tilde{q}_{1}=d$AfEDB$\bullet$C$+$$\frac{1}{2}$$*$$*$$*$$\frac{1+d(m-1)}{2+d(m-1)}$$+$$1$$\tilde{q}_{1}=1$$*$g$*$$*$$*$$*$$\frac{2+d(m-2)}{2+d(m-1)}$$\overline{\textbf{ab}}=\tilde{\mathcal{S}}_{m,\infty}^{(\tilde{q}_{1},\tilde{q}_{2})}$,
$\textbf{a}=(0,1)$, $\textbf{b}=(\frac{2}{d},0)$$\mathcal{R}(\textbf{AbDE})$:
scaling invariant class of (2.21). $\mathcal{R}(\textbf{ABCDE})$: scaling
invariant class of (2.22). $\mathcal{R}(\textbf{bCB})$: scaling invariant
class of (2.23). $\textbf{A}=(\frac{1}{d},\frac{1}{2})$,
$\textbf{B}=(\frac{2m}{d(2m-1)},\frac{m-1}{2m-1})$
$\textbf{C}=(\frac{2m+d(m-1)}{d(2m-1)},\frac{m-1}{2m-1})$
$\textbf{D}=(\frac{2+d(m-1)}{md},\frac{m-1}{m})$,
$\textbf{E}=(\frac{1}{d},\frac{1}{{\lambda_{1}}})$
$\textbf{f}=(\frac{1}{d},\frac{m+d(m-1)}{2m+d(m-1)})$,
$\textbf{g}=(1,\frac{2+d(m-2)}{2+d(m-1)})$

  1. (iii)

By camparing Figures 4-(i) and 9-(i), we may capture similiarities of
$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$ and
$\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$. The points E and
D in Figure 9-(i) correspond to $(0,\frac{1}{{\lambda_{1}}})$ and C in Figure
4-(i), respectively. Moreover, f, C, and A in Figure 9-(i) match to
$(0,\frac{m+d(m-1)}{2m+d(m-1)})$, C, and a in Fig. 4-(i), respectively.

###  2.2 Uniqueness

The uniqueness of $L^{q}$-weak solution of (1.1) is not obvious because, in
general, the maximal principle fails for nonlinear diffusive equations with
drifts. However, with additional assumptions on $\nabla V$ and restriction on
$q$, we are able to obtain uniqueness results. Here, we state the uniqueness
of $L^{m}$-weak solution of (1.1).

###### Theorem 2.18

Let $\rho_{0}\in
L^{m}({\mathbb{R}}^{d})\cap\mathcal{P}_{2}({\mathbb{R}}^{d})$. Assume $V$
satisfy

$V\in\mathcal{S}_{m,m}^{(q_{1},q_{2})}\ \text{ for }\ \begin{cases}2\leq
q_{2}\leq\frac{2m-1}{m-1},&\text{ if }\ d>2,\vspace{1 mm}\\\ 2\leq
q_{2}<\frac{2m-1}{m-1},&\text{ if }\ d=2.\end{cases}$ (2.24)

Assume further $\nabla V\in L_{x,t}^{\infty,1}(Q_{T})$. Suppose that $\rho$ is
a $L^{m}$-weak solution of (1.1) such that
$\rho:[0,T]\mapsto\mathcal{P}({\mathbb{R}}^{d})$ is narrowly continuous. Then
it is unique.

Next corollary is an immediate consequence of Theorem 2.18, which reads as
follows:

###### Corollary 2.19

Assume $\rho_{0}\in
L^{q}({\mathbb{R}}^{d})\cap\mathcal{P}_{p}({\mathbb{R}}^{d})$ for $p\geq 2$
and $q\geq m$. Let $V$ satisfy the hypothesis in one of Theorems 2.9, 2.13 and
2.16, and let $\nabla V\in L_{x,t}^{\infty,1}(Q_{T})$. Suppose that $\rho$ is
narrowly continuous $L^{m}$-weak solution of (1.1). Then it is unique.

###### Remark 2.20

  * (i)

The notion of narrowly continuous is the continuity in the weak* topology of
$\left(\mathcal{C}_{b}({\mathbb{R}}^{d})\right)^{\prime}$ (refer (3.7) in
Section 3.2).

  * (ii)

Assume $\rho_{0}\in
L^{q}({\mathbb{R}}^{d})\cap\mathcal{P}_{p}({\mathbb{R}}^{d})$ for $p\geq 2$
and $q\geq m$. In Theorems 2.9, 2.13 and 2.16, for each $1<\tilde{p}\leq p$,
we show the existence of $L^{q}$-weak solution $\rho_{\tilde{p}}$ satisfying
$\rho_{\tilde{p}}\in AC(0,T;\mathcal{P}_{\tilde{p}}({\mathbb{R}}^{d}))$.
Especially, this implies that
$\rho_{\tilde{p}}:[0,T]\mapsto\mathcal{P}({\mathbb{R}}^{d})$ is narrowly
continuous. We note that $\rho_{\tilde{p}}$ is also $L^{m}$-weak solution.
Hence, if we further assume $\nabla V\in L_{x,t}^{\infty,1}(Q_{T})$, then all
these solutions $\rho_{\tilde{p}}$ are identical.

  * (iii)

We do not know the uniqueness of $L^{m}$-weak solutions in the absence of the
hypothesis $\nabla V\in L_{x,t}^{\infty,1}(Q_{T})$. Furthermore, even with
$\nabla V\in L_{x,t}^{\infty,1}(Q_{T})$, uniqueness is not clear if $q<m$ or
$p<2$. Therefore, those questions remain open.

###  2.3 An application

In this subsection, we provide an application to which our results can be
properly referred. We consider an repulsion type of Keller-Segel equations,
which is of the form

$\rho_{t}-\Delta\rho^{m}=\nabla\cdot\left(\rho\nabla c\right),\qquad
c_{t}-\Delta c=\rho,\quad\text{in }\ Q_{T}:={\mathbb{R}}^{d}\times(0,T],$
(2.25)

where $d\geq 2$ and $0<T<\infty$. If (2.25) is considered in a bounded domain
$\Omega\subset{\mathbb{R}}^{d}$, no flux boundary condition is typically
assgined, that is

$\frac{\partial\rho}{\partial\nu}=\frac{\partial c}{\partial\nu}=0\qquad\mbox{
on }\,\,\partial\Omega.$ (2.26)

One can compare to the porous medium type of classical Keller-Segel equations,
which is a coupled system of the form

$\rho_{t}-\Delta\rho^{m}=-\nabla\cdot\left(\rho\nabla c\right),\qquad
c_{t}-\Delta c=\rho,\quad\text{in }\ Q_{T}.$ (2.27)

Contrast to (2.25), it shows aggregation effect due to the opposite sign of
reaction-diffusion term for the equation $\rho$, and it turns out that
solutions globally exist if $m>\frac{2(d-1)}{d}$ or blow-ups can occur in a
finite time if $1<m\leq\frac{2(d-1)}{d}$ (see e.g. [HashiraSachikoYokota],
[Sug] and reference therein). We remark that, in general, the linear case
($m=1$) with large data yields blow-up in a finite time except for one
dimensional case (see e.g. [JL92, Herrero-Velazquez, HorWin, Win, Win12,
Win13] and reference therein).

In the case that $m=\frac{2(d-1)}{d}$ and equation of $c$ in (2.27) is of
elliptic type, asymptotic behaviors of solutions were classified in
[BlaCarLau] by using the gradient flow structure of the equation for each case
of subcritical, critical or supercritical mass. The free energy functional was
analyzed in [CarHitVolYao] to show that global minimizers are radial and
compactly supported, and, in particular, the minimizer is unique in two
dimensions and solutions of PME Keller-Segel equations asymptotically
converges to the minimizer (see also the survey paper [CarCraYao]). Meanwhile,
the existence of global $L^{q}$-weak solutions, $q>1$, was established in
[BiaLiu], provided that $L^{p}$-norm, $p=\frac{d(2-m)}{2}$, of initial data is
sufficiently small.

Since the equation (1.1) is quite general, our results do not seem to be
directly applicable to those mentioned above for specific system (2.27). Our
analysis, however, provide a heuristic evidence why $p=\frac{d(2-m)}{2}$
appears in [BiaLiu]. Suppose that a priori $L^{q}$-energy estimate is
available. Then, $\rho\in L^{\frac{d(m+q-1)}{d-2},m+q-1}_{x,t}$, so is $\Delta
c$ due to $-\Delta c=\rho$. Applying the subscaling invariant condition for
$\nabla V$ (in this case $\nabla V=\Delta c$), we can conclude that
$L^{q}$-weak solutions exist, provided that $q$ satisfies

$\frac{d+q_{m,d}}{m+q-1}\leq 2+q_{m,d}\quad\Longrightarrow\quad
q\geq\frac{d(2-m)}{2}.$

On the other hand, it was also shown in [BiaLiu] that, if no smallness
restriction is assumed, blow-up in a finite time may occur. More precisely,
$\lim_{t\to T^{\ast}}\left\|\nabla c(t)\right\|_{L^{2}_{x}}=\lim_{t\to
T^{\ast}}\left\|\rho(t)\right\|_{L^{r}_{x}}=\infty$ for any $r>p$, where
$T^{\ast}$ is a finite blow-up time. Our results can be also used to see that
blow-up properties are relevant to each other. Indeed, suppose $\left\|\nabla
c(t)\right\|_{L^{2}_{x}}$ is finite until the blow-up time. Then, via the
subscaling invariant condition, we can have global $L^{q}$-weak solution wtih
$q$ satisfying $q\leq\frac{2d(m-1)}{d-2}$. If $m\geq\frac{2d}{d+2}$, then
there, however, exists $q\in(\frac{d(2-m)}{2},\frac{2d(m-1)}{d-2}]$ so that
$L^{q}$-weak solution exists globally in time, which is contrary to the fact
that $L^{r}$-norm of $\rho$ for $r>p$ becomes infinite at time $T^{\ast}$.
Therefore, $\left\|\nabla c(t)\right\|_{L^{2}_{x}}$ must be unbounded if
$\lim_{t\to T^{\ast}}\|\rho(t)\|_{L^{r}_{x}}=\infty$ for $r>p$. Our results do
not give any proof of the results in [BiaLiu], but it could be used to conduct
heuristic observations.

Now we return to the system (2.25) of repulsive type. Due to repusive effect
of reaction diffusion term in (2.25), it is, however, unknown whether or not
blow-up can occur in a finite time. More precisely, the repulsive effect
deduces via cancelation property the following estimate:

$\frac{d}{dt}\left(\int_{{\mathbb{R}}^{d}}\rho\log\rho+\frac{1}{2}\int_{{\mathbb{R}}^{d}}\left|\nabla
c\right|^{2}\right)+\frac{4}{m^{2}}\int_{{\mathbb{R}}^{d}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}+\int_{{\mathbb{R}}^{d}}\left|\Delta
c\right|^{2}=0,$ (2.28)

which is not available to the case of classical Keller-Segel equations (2.27).

It is not difficult to see that the a priori estimate (2.28) yields existence
of entropy weak solutions. However, it is not simple to show whether or not
such weak solutions are regular. Similar phenomena can be found in the Nernst-
Planck-Navier-Stokes system, which has a type of repulsive structure as in
(2.25). We remark that [ConIgn] established global well-posedness of regular
solutions in two dimensions and [LiuWan] proved the existence of weak
solutions for two-ionic species by using free energy dissipation via gradient
flow structure in three dimensions. In this directions, lots of improvements
have recently been accomplished and we do not make a list of all relavent
results here.

Our aim in this subsection is to improve the result of [Freitag] with the aid
of our main results. As far as authors know, the best known results in
regularity theory for (2.25)-(2.26) is that if $m>1+\frac{(d-1)(d-2)}{d^{2}}$,
$d\geq 2$, then solutions become bounded globally in time (see [Freitag]). We
remark that note that $1+\frac{(d-1)(d-2)}{d^{2}}<\frac{2(d-1)}{d}$. Our main
objective is to show that solutions for (2.25) become globally bounded for the
case $m>\max\\{\frac{2d-3}{d},\,\frac{2d}{d+2}\\}$. We note, due to previous
result in [Freitag], that it suffices to consider the case
$\max\\{\frac{2d-3}{d},\,\frac{2d}{d+2}\\}<m\leq 1+\frac{(d-1)(d-2)}{d^{2}}$
for $d>2$. With the aid of our main results, we obtain the following. The
proof will be given in Section 7.

###### Theorem 2.21

Let $m>\max\\{\frac{2d-3}{d},\,\frac{2d}{d+2}\\}$, $d>2$, and
$p\in(1,\frac{md-d+2}{d-1}]$. Suppose that
$\rho_{0}\in{\mathcal{P}}_{p}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})$ and $c_{0}\in L^{1}({\mathbb{R}}^{d})\cap
W^{2,\infty}({\mathbb{R}}^{d})$. Then, a pair of solution $(\rho,c)$ for the
equations (2.25) becomes bounded globally in time.

###### Remark 2.22

  * (i)

We remark that $m>\frac{2d-3}{d}$ is equivalent to $\frac{md-d+2}{d-1}>1$.
Moreover, we note that the condition
$\max\\{\frac{2d-3}{d},\,\frac{2d}{d+2}\\}$ can be rewritten as
$\frac{2d}{d+2}\cdot\delta_{\\{d\leq
6\\}}+\frac{2d-3}{d}\cdot\delta_{\\{d>6\\}}$.

  * (ii)

If $\rho_{0}\in{\mathcal{P}}_{p}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})$ is assumed to be Hölder continuous, then so is
$\rho$ for $t\in(0,T]$. Since its proof is rather straightforward once
solutions become bounded (see e.g. [CHKK17]), we omit the details. The
uniqueness of solutions, however, remains open, since the results in Section
2.2 do not seem to be applicable to the system (2.25).

##  3 Preliminaries

In this section, we introduce preliminaries that are used throughout the
paper.

For a function $f:\Omega\times[0,T]\to\mathbb{R}$,
$\Omega\subset\mathbb{R}^{d},d\geq 2$ and constants $q_{1},q_{2}>1$, we define

$\|f\|_{L^{q_{1},q_{2}}_{x,t}}:=\left(\int_{0}^{T}\left[\int_{\Omega}\left|f(x,t)\right|^{q_{1}}\,dx\right]^{q_{2}/q_{1}}\,dt\right)^{\frac{1}{q_{2}}}.$

For simplicity, let $\|f\|_{L^{q}_{x,t}}=\|f\|_{L^{q,q}_{x,t}}$ for some
$q>1$. Also we denote for $\Omega\subseteq{\mathbb{R}}^{d}$ and
$\Omega_{T}\subseteq Q_{T}$ that

$\begin{gathered}\|f(\cdot,t)\|_{{\mathcal{C}}^{\alpha}(\Omega)}:=\sup_{x,y\in\Omega,\,x\neq
y}\frac{|f(x,t)-f(y,t)|}{|x-y|^{\alpha}},\\\
\|f\|_{{\mathcal{C}}^{\alpha}(\Omega_{T})}:=\sup_{(x,t),(y,s)\in\Omega_{T},\,(x,t)\neq(y,s)}\frac{|f(x,t)-f(y,s)|}{|x-y|^{\alpha}+|t-s|^{\alpha/2}}.\end{gathered}$

###  3.1 Technical lemmas

We introduce well known Sobolev type inequalities. Throughout this subsection,
let $\Omega\subseteq{\mathbb{R}}^{d}$.

###### Lemma 3.1

[GT, Theorem 7.10] Let $\rho\in W_{o}^{1,p}(\Omega)$ and assume $1\leq p<d$.
There exists a constant $c=c(d,p)$ such that

$\|\rho\|_{L_{x}^{\frac{dp}{d-p}}(\Omega)}\leq
c\|\nabla\rho\|_{L^{p}_{x}(\Omega)}.$

###### Lemma 3.2

[DB93, Proposition I.3.1] Let $\rho\in L^{\infty}(0,T;L^{q}(\Omega))\cap
L^{p}(0,T;W_{o}^{1,p}(\Omega))$ for some $1\leq p<d$, and
$0<q<\frac{dp}{d-p}$. Then there exists a constant $c=c(d,p,q)$ such that

$\iint_{\Omega_{T}}|\rho(x,t)|^{\frac{p(d+q)}{d}}\,dx\,dt\leq
c\left(\sup_{0\leq t\leq
T}\int_{\Omega}|\rho(x,t)|^{q}\,dx\right)^{\frac{p}{d}}\iint_{\Omega_{T}}|\nabla\rho(x,t)|^{p}\,dx\,dt.$

The following lemma is the Garilardo-Niremberg multiplicative embedding
inequality.

###### Lemma 3.3

[DB93, Theorem I.2.1] Let $\rho\in W_{o}^{1,p}(\Omega)$ for $p\geq 1$. For
every fixed number $s\geq 1$, there exists a constant $c=c(d,p,s)$ such that

$\|\rho\|_{L_{x}^{q}(\Omega)}\leq
c\|\nabla\rho\|_{L^{p}_{x}(\Omega)}^{\alpha}\|\rho\|_{L^{s}_{x}(\Omega)}^{1-\alpha},$

where $\alpha\in[0,1]$, $p,q\geq 1$, are linked by
$\alpha=\left(\frac{1}{s}-\frac{1}{q}\right)\left(\frac{1}{d}-\frac{1}{p}+\frac{1}{s}\right)^{-1}$,
and their admissible range is:

$\begin{cases}s\leq q\leq\infty,\ 0\leq\alpha\leq\frac{p}{p+s(p-1)},&\text{ if
}\ d=1,\\\ s\leq q\leq\frac{dp}{d-p},\ 0\leq\alpha\leq 1,&\text{ if }\ 1\leq
p<d,\ s\leq\frac{dp}{d-p},\\\ \frac{dp}{d-p}\leq q\leq s,\ 0\leq\alpha\leq
1,&\text{ if }\ 1\leq p<d,\ s\geq\frac{dp}{d-p},\\\ s\leq q<\infty,\
0\leq\alpha<\frac{dp}{dp+s(p-d)},&\text{ if }\ p\geq d>1.\end{cases}$

Now we derive the following lemma which is useful to obtain a priori
estimates.

###### Lemma 3.4

Let $m>1$ and $q\geq 1$. Suppose that

$\rho\in
L^{\infty}(0,T;L^{q}(\Omega))\quad\text{and}\quad\rho^{\frac{q+m-1}{2}}\in
L^{2}(0,T;W^{1,2}_{0}(\Omega)).$ (3.1)

Then $\rho\in L^{r_{1},r_{2}}_{x,t}(\Omega\times[0,T])$ such that

$\begin{gathered}\frac{d}{r_{1}}+\frac{2+q_{m,d}}{r_{2}}=\frac{d}{q},\quad\text{for}\quad
q_{m,d}:=\frac{d(m-1)}{q},\\\ \text{for}\ \ \begin{cases}q\leq
r_{1}\leq\frac{d(q+m-1)}{d-2},\quad q+m-1\leq r_{2}\leq\infty,&\text{ if }\
d>2,\\\ q\leq r_{1}<\infty,\quad q+m-1<r_{2}\leq\infty,&\text{ if }\ d=2,\\\
q\leq r_{1}\leq\infty,\quad q+m-1\leq r_{2}\leq\infty,&\text{ if }\
d=1.\end{cases}\end{gathered}$ (3.2)

Moreover, there exists a constant $c=c(d)$ such that, for
$(d-2)_{+}=\max\\{0,d-2\\}$,

$\|\rho\|_{L^{r_{1},r_{2}}_{x,t}}\leq c\left(\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\rho^{q}(\cdot,t)\,dx\right)^{\frac{1}{r_{1}}\left[1-\frac{(r_{1}-q)(d-2)_{+}}{d(m-1)+2q}\right]}\,\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x,t}}^{\frac{2}{r_{2}}}.$
(3.3)

Proof. First, let $d>2$. By applying Lemma 3.1 with (3.1), we let

$q\leq
r_{1}\leq\frac{d(q+m-1)}{d-2},\quad\frac{1}{r_{1}}=\frac{\theta}{q}+\frac{(1-\theta)(d-2)}{(q+m-1)d}\quad\Longrightarrow\quad\theta=\frac{q}{r_{1}}\left(1-\frac{(r_{1}-q)(d-2)}{2q+d(m-1)}\right).$
(3.4)

Then we carry the following computations

$\left\|\rho\right\|_{L^{r_{1}}_{x}}\leq\left\|\rho\right\|^{\theta}_{L^{q}_{x}}\left\|\rho\right\|^{1-\theta}_{L^{\frac{(q+m-1)d}{d-2}}_{x}}\leq\left\|\rho\right\|^{\theta}_{L^{q}_{x}}\left\|\rho^{\frac{q+m-1}{2}}\right\|^{\frac{2(1-\theta)}{q+m-1}}_{L^{\frac{2d}{d-2}}_{x}}\leq
c\left\|\rho\right\|^{\theta}_{L^{q}_{x}}\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|^{\frac{2(1-\theta)}{q+m-1}}_{L^{2}_{x}}.$
(3.5)

By taking $L^{r_{2}}_{t}$, we obtain

$\|\rho\|_{L^{r_{1},r_{2}}_{x,t}}\leq
c\left\|\rho\right\|^{\theta}_{L^{q,\infty}_{x,t}}\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|^{\frac{2}{r_{2}}}_{L^{2}_{x,t}},$

provided

$\frac{(1-\theta)r_{2}}{q+m-1}=1\quad\iff\quad\frac{d}{r_{1}}+\frac{2+q_{m,d}}{r_{2}}=\frac{d}{q}.$

Now let $1\leq d\leq 2$, then Lemma 3.3 with $v=\rho^{\frac{q+m-1}{2}}$ and
$s=\frac{2q}{q+m-1}$ gives that

$\left(\int_{\mathbb{R}^{d}}\rho^{\gamma\frac{q+m-1}{2}}\,dx\right)^{\frac{1}{\gamma}}\leq
c\left(\int_{\mathbb{R}^{d}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\right)^{\frac{\alpha}{2}}\left(\int_{\mathbb{R}^{d}}\rho^{q}\,dx\right)^{\frac{1-\alpha}{s}},$

for $\alpha=\frac{\gamma-s}{\gamma}$ and $\frac{2q}{q+m-1}\leq\gamma<\infty$.
Then we set

$r_{1}=q\theta+\gamma\frac{q+m-1}{2}(1-\theta),\quad\text{for}\quad\theta=1-\frac{2(r_{1}-q)}{\gamma(q+m-1)-q}\in[0,1].$

Therefore, the combination of Hölder inequality for $\theta+(1-\theta)=1$
yield

$\int_{\mathbb{R}^{d}}\rho^{r_{1}}\,dx\leq
c\left(\int_{\mathbb{R}^{d}}\rho^{q}\,dx\right)^{\theta+\frac{\theta\gamma(1-\alpha)}{s}}\left(\int_{\mathbb{R}^{d}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\right)^{(1-\theta)\frac{\alpha\gamma}{2}}.$

Then by taking power by $\frac{r_{2}}{r_{1}}$ for $r_{2}>1$, we set
$(1-\alpha)\frac{\alpha\gamma r_{2}}{2r_{1}}=1$ which gives (3.2) and complete
the proof.

Now we introduce the Aubin-Lions lemma.

###### Lemma 3.5

[Show97, Proposition III.1.3], [Sim87] Let $X_{0}$, $X$ and $X_{1}$ be Banach
spaces with $X_{0}\subset X\subset X_{1}$. Suppose that $X_{0}$ and $X_{1}$
are reflexive, and $X_{0}\hookrightarrow X$ is compact, and $X\hookrightarrow
X_{1}$ is continuous. For $1\leq p,q\leq\infty$, let us define $W=\left\\{u\in
L^{p}\left(0,T;X_{0}\right),\ u_{t}\in L^{q}\left(0,T;X_{1}\right)\right\\}$.
If $p<\infty$, then the inclusion $W\hookrightarrow L^{p}(0,T;X)$ is compact.
If $p=\infty$ and $q>1$, then the embedding of $W$ into ${\mathcal{C}}(0,T;X)$
is compact.

###  3.2 Wasserstein space

In this subsection, we introduce the Wasserstein space and its properties. For
more detail, we refer [ags:book, V].

###### Definition 3.6

Let $\mu$ be a probability measure on $\mathbb{R}^{d}$. Suppose there is a
measurable map $\mathcal{T}:\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$, then
$\mathcal{T}$ induces a probability measure $\nu$ on $\mathbb{R}^{d}$ which is
defined as

$\int_{\mathbb{R}^{d}}\varphi(y)\,d\nu(y)=\int_{\mathbb{R}^{d}}\varphi(\mathcal{T}(x))\,d\mu(x),\qquad\forall~{}\varphi\in
C(\mathbb{R}^{d}).$

We denote $\nu:=\mathcal{T}_{\\#}\mu$ and say that $\nu$ is the push-forward
of $\mu$ by $\mathcal{T}$.

Let us denote by $\mathcal{P}_{p}(\mathbb{R}^{d})$ the set of all Borel
probability measures on $\mathbb{R}^{d}$ with a finite $p$-th moment. For
$\mu,\nu\in\mathcal{P}_{p}(\mathbb{R}^{d})$, we consider

$W_{p}(\mu,\nu):=\left(\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{p}\,d\gamma(x,y)\right)^{\frac{1}{p}},$
(3.6)

where $\Gamma(\mu,\nu)$ denotes the set of all Borel probability measures on
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ which has $\mu$ and $\nu$ as marginals;

$\gamma(A\times\mathbb{R}^{d})=\mu(A)\quad\text{and}\quad\gamma(\mathbb{R}^{d}\times
A)=\nu(A)$

for every Borel set $A\subset\mathbb{R}^{d}.$ Equation (3.6) defines a
distance on $\mathcal{P}_{p}(\mathbb{R}^{d})$ which is called the Wasserstein
distance. Equipped with the Wasserstein distance,
$\mathcal{P}_{p}(\mathbb{R}^{d})$ is called the Wasserstein space. It is known
that the infimum in the right hand side of Equation (3.6) always achieved.

We will denote by $\Gamma_{o}(\mu,\nu)$ the set of all $\gamma$ which minimize
the expression. If $\mu$ is absolutely continuous with respect to the Lebesgue
measure then there exists a map
$\mathcal{T}:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ such that
$\gamma:=(Id\times\mathcal{T})_{\\#}\mu$ is the unique element of
$\Gamma_{o}(\mu,\nu)$, that is,
$\Gamma_{o}(\mu,\nu)=\\{(Id\times\mathcal{T})_{\\#}\mu\\}$.

We say that a sequence $\\{\mu_{n}\\}\subset\mathcal{P}(\mathbb{R}^{d})$ is
narrowly convergent to $\mu\in\mathcal{P}(\mathbb{R}^{d})$ as
$n\rightarrow\infty$ if

$\lim_{n\rightarrow\infty}\int_{{\mathbb{R}}^{d}}\varphi(x)\,d\mu_{n}(x)=\int_{{\mathbb{R}}^{d}}\varphi(x)\,d\mu(x)$
(3.7)

for every function $\varphi\in{\mathcal{C}}_{b}({\mathbb{R}}^{d})$, the space
of continuous and bounded real functions defined on ${\mathbb{R}}^{d}$. Then
we recall

$\lim_{n\rightarrow\infty}W_{p}(\mu_{n},\mu)=0\quad\Longleftrightarrow\quad\begin{cases}\mu_{n}~{}\mbox{narrowly
converges to }\mu~{}\mbox{ in }\mathcal{P}(\mathbb{R}^{d})\\\
\lim_{n\rightarrow\infty}\int_{{\mathbb{R}}^{d}}|x|^{p}\,d\mu_{n}(x)=\int_{{\mathbb{R}}^{d}}|x|^{p}\,d\mu(x).\end{cases}$

Following [AG, ags:book], we give a notion of a differential and a definition
of convex functions on $\mathcal{P}_{p}({\mathbb{R}}^{d})$.

###### Definition 3.7

Let $\phi:\mathcal{P}_{p}(\mathbb{R}^{d})\mapsto(-\infty,\infty]$. We say that
$\phi$ is geodesically convex in $\mathcal{P}_{p}(\mathbb{R}^{d})$ if for
every couple $\mu_{1},\mu_{2}\in\mathcal{P}_{p}(\mathbb{R}^{d})$ there exists
an optimal plan $\gamma\in\Gamma_{o}(\mu_{1},\mu_{2})$ such that

$\phi(\mu_{t}^{1\rightarrow
2})\leq(1-t)\phi(\mu_{1})+t\phi(\mu_{2}),\qquad\forall~{}t\in[0,1],$

where $\mu_{t}^{1\rightarrow 2}$ is a constant speed geodesic between
$\mu^{1}$ and $\mu^{2}$ defined as

$\mu_{t}^{1\rightarrow 2}:=((1-t)\pi^{1}+t\pi^{2})_{\\#}\gamma.$

Here, $\pi^{1}:\mathbb{R}^{d}\times\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ and
$\pi^{2}:\mathbb{R}^{d}\times\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ are the
first and second projections of $\mathbb{R}^{d}\times\mathbb{R}^{d}$ onto
$\mathbb{R}^{d}$ defined by

$\pi^{1}(x,y)=x,\quad\pi^{2}(x,y)=y,\qquad\forall~{}x,y\in\mathbb{R}^{d}.$

###### Definition 3.8

Let $\phi:\mathcal{P}_{p}({\mathbb{R}}^{d})\rightarrow(-\infty,\infty]$ be a
proper function on $\mathcal{P}_{p}({\mathbb{R}}^{d})$, i.e. the effective
domain of $\phi$ defined by
$D(\phi):=\\{\mu\in\mathcal{P}_{p}({\mathbb{R}}^{d}):\phi(\mu)<\infty\\}$ is
not empty. We say that $\xi\in L^{\frac{p}{p-1}}(\mu)$ belongs to the
subdifferential $\partial_{-}\phi(\mu)$ if

$\phi(\nu)\geq\phi(\mu)+\sup_{\gamma\in\Gamma_{o}(\mu,\nu)}\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\langle\xi(x),y-x\rangle
d\gamma(x,y)+o(W_{p}(\mu,\nu)),$

as $\nu\rightarrow\mu$ in $\mathcal{P}_{p}({\mathbb{R}}^{d})$.

###### Lemma 3.9

[AG, Proposition 4.2] Let
$\phi:\mathcal{P}_{p}({\mathbb{R}}^{d})\rightarrow(-\infty,\infty]$ be lower
semicontinuous and geodesically convex and let $\xi\in\partial_{-}\phi(\mu)$.
Then we have

$\phi(\nu)\geq\phi(\mu)+\sup_{\gamma\in\Gamma_{o}(\mu,\nu)}\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\langle\xi(x),y-x\rangle
d\gamma(x,y),\qquad\forall~{}\nu\in\mathcal{P}_{p}({\mathbb{R}}^{d}).$

Now, we introduce the notion of absolutely continuous curve and its relation
with the continuity equation.

###### Definition 3.10

Let $\sigma:[0,T]\mapsto\mathcal{P}_{p}(\mathbb{R}^{d})$ be a curve. We say
that $\sigma$ is absolutely continuous and denote it by $\sigma\in
AC(0,T;\mathcal{P}_{p}(\mathbb{R}^{d}))$, if there exists $l\in L^{1}([0,T])$
such that

$W_{p}(\sigma(s),\sigma(t))\leq\int_{s}^{t}l(r)dr,\qquad\forall~{}0\leq s\leq
t\leq T.$ (3.8)

If $\sigma\in AC(0,T;\mathcal{P}_{p}(\mathbb{R}^{d}))$, then the limit

$|\sigma^{\prime}|(t):=\lim_{s\rightarrow
t}\frac{W_{p}(\sigma(s),\sigma(t))}{|s-t|},$

exists for $L^{1}$-a.e $t\in[0,T]$. Moreover, the function $|\sigma^{\prime}|$
belongs to $L^{1}(0,T)$ and satisfies

$|\sigma^{\prime}|(t)\leq
l(t)\qquad\mbox{for}~{}L^{1}\mbox{-a.e.}~{}t\in[0,T],$

for any $l$ satisfying (3.8). We call $|\sigma^{\prime}|$ by the metric
derivative of $\sigma$.

###### Lemma 3.11

[ags:book, Theorem 8.3.1] If $\sigma\in
AC(0,T;\mathcal{P}_{p}(\mathbb{R}^{d}))$ then there exists a Borel vector
field $w:{\mathbb{R}}^{d}\times(0,T)\mapsto{\mathbb{R}}^{d}$ such that

$w(t)\in
L^{p}(\sigma(t)),\quad\|w(t)\|_{L^{p}(\sigma(t))}\leq|\sigma^{\prime}|(t),\quad\mbox{
for}~{}~{}L^{1}\text{-a.e}~{}~{}t\in[0,T],$

and the continuity equation

$\partial_{t}\sigma+\nabla\cdot(w\,\sigma)=0,$

holds in the sense of distribution. Conversely, if a narrowly continuous curve
$\sigma:[0,T]\mapsto\mathcal{P}_{p}(\mathbb{R}^{d})$ satisfies the continuity
equation for some Borel vector field $w$ with $\|w(t)\|_{L^{p}(\sigma(t))}\in
L^{1}(0,T)$, then $\sigma:[0,T]\mapsto\mathcal{P}_{p}(\mathbb{R}^{d})$ is
absolutely continuous and
$|\sigma^{\prime}|(t)\leq\|w(t)\|_{L^{p}(\sigma(t))}$ for $L^{1}$-a.e
$t\in[0,T]$.

###### Remark 3.12

We call $\|w(t)\|_{L^{p}(\sigma(t))}$ an auxiliary speed of curve $\sigma(t)$
in Wasserstein space $\mathcal{P}_{p}(\mathbb{R}^{d})$. We use ‘speed’ instead
of ‘auxiliary speed’ unless any confusion is to be expected.

###### Lemma 3.13

[ags:book, Proposition 3.3.1] Let $K\subset\mathcal{P}_{p}({\mathbb{R}}^{d})$
be a sequentially compact set w.r.t the narrow topology. Let
$\sigma_{n}:[0,T]\rightarrow\mathcal{P}_{p}(\mathbb{R}^{d})$ be curves such
that

$\displaystyle\sigma_{n}(t)\in K,\quad\forall~{}n$
$\displaystyle\in\mathbb{N},~{}t\in[0,T],$ $\displaystyle
W_{p}(\sigma_{n}(s),\sigma_{n}(t))$
$\displaystyle\leq\omega(s,t),\qquad\forall~{}0\leq s\leq t\leq
T,~{}n\in\mathbb{N},$

for a continuous function $\omega:[0,T]\times[0,T]\rightarrow[0,\infty)$ such
that

$\omega(t,t)=0,~{}~{}\forall~{}t\in[0,T].$

Then there exists a subsequnece $\sigma_{n_{k}}$ and a limit curve
$\sigma:[0,T]\rightarrow{P}_{p}(\mathbb{R}^{d})$ such that

$\sigma_{n_{k}}(t)\mbox{ narrowly converges to}~{}~{}\sigma(t),\qquad\text{for
all}~{}~{}t\in[0,T].$

###  3.3 Flows on $\mathcal{P}_{2}({\mathbb{R}}^{d})$ generated by vector
fields

For a given $T>0$, let $V\in
L^{1}(0,T;W^{1,\infty}({\mathbb{R}}^{d};{\mathbb{R}}^{d}))$. For any
$s,~{}t\in[0,T]$, let
$\psi:[0,T]\times[0,T]\times\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ be the flow
map of the vector field $V$. More precisely, $\psi$ solves the following ODE

$\begin{cases}\frac{d}{dt}\psi(t;s,x)=V(\psi(t;s,x),t),&\text{for
}s,t\in[0,T]\vspace{1 mm}\\\ \psi(s;s,x)=x,&\text{for
}x\in{\mathbb{R}}^{d}.\end{cases}$ (3.9)

Using the flow map $\psi$, we define a flow
$\Psi:[0,T]\times[0,T]\times\mathcal{P}_{2}({\mathbb{R}}^{d})\mapsto\mathcal{P}_{2}({\mathbb{R}}^{d})$
through the push forward operation as follows

$\Psi(t;s,\mu):={\psi}(t;s,\cdot)_{\\#}\mu,\qquad\forall~{}\mu\in\mathcal{P}_{2}({\mathbb{R}}^{d}).$
(3.10)

In this subsection, we remind two basic results on the flow map $\psi$.

###### Lemma 3.14

[KK-SIMA, Lemma 3.1] Let $s\in[0,T]$ and $\psi$ be defined as in (3.9). Then,
for any $t\in[s,T]$, the map
$\psi(t;s,\cdot):\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ satisfies

$e^{-\int_{s}^{t}\mbox{Lip}(V(\tau))\,d\tau}|x-y|\leq|\psi(t;s,x)-\psi(t;s,y)|\leq
e^{\int_{s}^{t}\mbox{Lip}(V(\tau))\,d\tau}|x-y|,\qquad\forall~{}x,y\in\mathbb{R}^{d},$

where

$\mbox{Lip}(V(\tau)):=\sup_{x,y\in{\mathbb{R}}^{d},\,x\neq
y}\frac{|V(x,\tau)-V(y,\tau)|}{|x-y|}.$

###### Lemma 3.15

Let $s\in[0,T]$ and $\psi$ be defined as in (3.9). For any $t\in[s,T]$, let
$J_{s,t}$ be the Jacobian corresponding to the map
$\psi(t;s,\cdot):\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ . That is,

$\int_{\mathbb{R}^{d}}\zeta(y)dy:=\int_{\mathbb{R}^{d}}\zeta(\psi(t;s,x))J_{s,t}(x)\,dx,\qquad\forall~{}\zeta\in
C(\mathbb{R}^{d}).$ (3.11)

Then, the Jacobian $J_{s,t}$ is given as

$J_{s,t}(x)=e^{\int_{s}^{t}\nabla\cdot
V(\psi(\tau;s,x),\tau)\,d\tau},\qquad\forall~{}x\in{\mathbb{R}}^{d}.$ (3.12)

Proof. Let $x\in\mathbb{R}^{d}$ be given. Then, we have

$\displaystyle\frac{d}{dt}J_{s,t}(x)\big{|}_{t=\tau}$
$\displaystyle=\nabla\cdot V(\psi(\tau;s,x),\tau)J_{s,\tau}(x).$

Since $J_{s,s}\equiv 1$, we have

$\log J_{s,t}(x)=\int_{s}^{t}\nabla\cdot V(\psi(\tau;s,x),\tau)\,d\tau.$
(3.13)

This completes the proof.

###### Remark 3.16

We note that $\psi(t;s,\cdot)\circ\psi(s;t,\cdot)=Id$, that is,

$\psi(t;s,\psi(s;t,x))=x,\quad\forall~{}x\in{\mathbb{R}}^{d}.$

Exploiting this, we have

$J_{s,t}(\psi(s;t,x))=\frac{1}{J_{t,s}(x)}.$

###### Lemma 3.17

[KK-SIMA, Lemma 3.3] Let $s\in[0,T]$ and $\Psi$ be in (3.10). For any
$t\in[s,T]$ and $\mu,\nu\in\mathcal{P}_{p}(\mathbb{R}^{d})$, we have

$e^{-\int_{s}^{t}\mbox{Lip}(V(\tau))\,d\tau}W_{p}(\mu,\nu)\leq
W_{p}(\Psi(t;s,\mu),\Psi(t;s,\nu))\leq
e^{\int_{s}^{t}\mbox{Lip}(V(\tau))\,d\tau}W_{p}(\mu,\nu).$ (3.14)

###### Lemma 3.18

[KK-SIMA, Lemma 3.4] Let $\psi$ and $\Psi$ be defined as in (3.9) and (3.10),
respectively. If $\mu\in\mathcal{P}_{p}({\mathbb{R}}^{d})$ then
$\Psi(t;s,\mu)\in\mathcal{P}_{p}({\mathbb{R}}^{d})$. Furthermore, suppose
$\mu=\varrho\,dx$ and $\Psi(t;s,\mu)=\rho\,dx$ then

$\rho(\psi(t;s,x))J_{s,t}(x)=\varrho(x),\qquad a.e\quad x\in{\mathbb{R}}^{d},$
(3.15)

where $J_{s,t}$ is the Jacobian of the map $\psi(t;s,\cdot)$ as in (3.11). We
also have

$\int_{{\mathbb{R}}^{d}}\rho\log\rho\,dx=\int_{{\mathbb{R}}^{d}}\varrho\log\varrho\,dx-\int_{{\mathbb{R}}^{d}}\varrho\log
J_{s,t}\,dx.$ (3.16)

Moreover, if $\varrho\in L^{q}({\mathbb{R}}^{d})$ for $q\in[1,\infty]$, then
${\rho}\in L^{q}({\mathbb{R}}^{d})$ and we have

$\|{\rho}\|_{L^{q}({\mathbb{R}}^{d})}\leq\|\varrho\|_{L^{q}({\mathbb{R}}^{d})}e^{\frac{q-1}{q}\int_{s}^{t}\|\nabla\cdot
V\|_{L^{\infty}_{x}}\,d\tau},$ (3.17)

where $\frac{q-1}{q}=1$ if $q=\infty$.

###### Lemma 3.19

Let $\psi$ and $\Psi$ be defined as in (3.9) and (3.10), respectively. Suppose
that $V\in L^{1}(0,T;W^{1,\infty}({\mathbb{R}}^{d}))$.

  * (i)

If $V\in L^{1}(0,T;{\mathcal{C}}^{1,\alpha}({\mathbb{R}}^{d}))$ and
$\varrho\in\mathcal{P}_{2}({\mathbb{R}}^{d})\cap{\mathcal{C}}^{\alpha}({\mathbb{R}}^{d})$
for some $\alpha\in(0,1)$, then $\rho:=\Psi(t;s,\varrho)$ is also Hölder
continuous. More precisely, we have
$\|\rho\|_{{\mathcal{C}}^{\alpha}({\mathbb{R}}^{d})}<C$ where
$C=C(\|\varrho\|_{{\mathcal{C}}^{\alpha}({\mathbb{R}}^{d})},\,\int_{s}^{t}\|\nabla
V\|_{{\mathcal{C}}^{\alpha}({\mathbb{R}}^{d})}d\tau)$.

  * (ii)

If $V\in L^{1}(0,T;W^{2,\infty}({\mathbb{R}}^{d}))$, then, for any
$a>0,~{}q\geq 1$, we have

$\displaystyle\|\nabla\rho^{a}\|_{L^{q}({\mathbb{R}}^{d})}$ $\displaystyle\leq
e^{(a+2)\int_{s}^{t}\|\nabla
V\|_{L^{\infty}_{x}}\,d\tau}\left\\{\|\nabla\varrho^{a}\|_{L^{q}({\mathbb{R}}^{d})}+\|\varrho^{a}\|_{L^{p}_{x}}\times\left(a\int_{s}^{t}\|\nabla^{2}V\|_{L^{\infty}_{x}}\,d\tau\right)\right\\}.$
(3.18)

Proof. $(i)$ From (3.15), we have

$\rho(x)=\frac{{\varrho}(\psi(s;t,x))}{J_{s,t}(\psi(s;t,x))}.$ (3.19)

Let $f:=\varrho(\psi(s;t,x))$ and $g:=\frac{1}{J_{s,t}(\psi(s;t,x))}$, then
$\|\rho\|_{{\mathcal{C}}^{\alpha}({\mathbb{R}}^{d})}\leq\|f\|_{L^{\infty}_{x}}\|g\|_{\mathcal{C}^{\alpha}({\mathbb{R}}^{d})}+\|f\|_{{\mathcal{C}}^{\alpha}({\mathbb{R}}^{d})}\|g\|_{L^{\infty}_{x}}.$
We note that $\|f\|_{L^{\infty}_{x}}=\|\varrho\|_{L^{\infty}_{x}}$ and from
(3.12)

$\|g\|_{L^{\infty}_{x}}\leq e^{\int_{s}^{t}\|\nabla\cdot
V\|_{L^{\infty}_{x}}\,d\tau}.$

Next, we estimate $\|f\|_{{\mathcal{C}}^{\alpha}({\mathbb{R}}^{d})}$ that

$|\varrho(\psi(s;t,x))-\varrho(\psi(s;t,y))|\leq\|\varrho\|_{{\mathcal{C}}^{\alpha}}|\psi(s;t,x)-\psi(s;t,y)|^{\alpha}\leq\|\varrho\|_{{\mathcal{C}}^{\alpha}}e^{\alpha\int_{s}^{t}\|\nabla\cdot
V\|_{L^{\infty}_{x}}\,d\tau}|x-y|^{\alpha}.$

Hence, it gives

$\|f\|_{{\mathcal{C}}^{\alpha}}\leq\|\varrho\|_{{\mathcal{C}}^{\alpha}}e^{\alpha\int_{s}^{t}\|\nabla\cdot
V\|_{L^{\infty}_{x}}\,d\tau}.$

Next, we note that

$\displaystyle g(x)=e^{-\int_{s}^{t}\nabla\cdot
V(\psi(\tau;s,\psi(s;t,x)),\tau)\,d\tau}=e^{-\int_{s}^{t}\nabla\cdot
V(\psi(\tau;t,x),\tau)\,d\tau},$

which gives us

$\displaystyle|g(x)-g(y)|$ $\displaystyle=\left|e^{-\int_{s}^{t}\nabla\cdot
V(\psi(\tau;t,x),\tau)\,d\tau}-e^{-\int_{s}^{t}\nabla\cdot
V(\psi(\tau;t,y),\tau)\,d\tau}\right|$ (3.20) $\displaystyle\leq
e^{\int_{s}^{t}\|\nabla\cdot
V\|_{L^{\infty}_{x}}\,d\tau}\int_{s}^{t}|\nabla\cdot
V(\psi(\tau;t,x),\tau)-\nabla\cdot V(\psi(\tau;t,y),\tau)|\,d\tau$
$\displaystyle\leq e^{\int_{s}^{t}\|\nabla\cdot
V\|_{L^{\infty}_{x}}\,d\tau}\int_{s}^{t}\|\nabla\cdot
V(\tau)\|_{{\mathcal{C}}^{\alpha}}|\psi(\tau;t,x)-\psi(\tau;t,y)|^{\alpha}\,d\tau$
$\displaystyle\leq e^{\int_{s}^{t}\|\nabla\cdot
V\|_{L^{\infty}_{x}}\,d\tau}\int_{s}^{t}\|\nabla\cdot
V(\tau)\|_{{\mathcal{C}}^{\alpha}}\,d\tau~{}e^{\alpha\int_{s}^{t}Lip(V(\theta))d\theta}|x-y|^{\alpha}.$

We combine (3.19)-(3.20) to get, for any $x,~{}y\in{\mathbb{R}}^{d}$,

$\displaystyle|\rho(x)-\rho(y)|$ $\displaystyle\leq
e^{(\alpha+1)\int_{s}^{t}\|\nabla
V\|_{L^{\infty}_{x}}\,d\tau}\left(\|\varrho\|_{L^{\infty}_{x}}\int_{s}^{t}\|\nabla\cdot
V(\tau)\|_{{\mathcal{C}}^{\alpha}}\,d\tau+\|\varrho\|_{{\mathcal{C}}^{\alpha}}\right)|x-y|^{\alpha}.$

$(ii)$ First of all, we exploit $\psi(\tau;s,\psi(s;t,x))=\psi(\tau;t,x)$ in
(3.19) and get

$\displaystyle\rho(x)=\varrho(\psi(s;t,x))~{}e^{-\int_{s}^{t}\nabla\cdot
V(\psi(\tau;t,x),\tau)\,d\tau}.$

Hence, for $a>1$, we have

$\displaystyle\rho^{a}(x)=\varrho^{a}(\psi(s;t,x))~{}e^{-a\int_{s}^{t}\nabla\cdot
V(\psi(\tau;t,x),\tau)\,d\tau},$

and

$\displaystyle\nabla\rho^{a}(x)$
$\displaystyle=\nabla\varrho^{a}(\psi(s;t,x))~{}\nabla_{x}\psi(s;t,x)~{}e^{-a\int_{s}^{t}\nabla\cdot
V(\psi(\tau;t,x),\tau)\,d\tau}$ (3.21)
$\displaystyle\quad+\varrho^{a}(\psi(s;t,x))~{}e^{-a\int_{s}^{t}\nabla\cdot
V(\psi(\tau;t,x)\,d\tau)}\left(-a\int_{s}^{t}\nabla^{2}V(\psi(\tau;t,x))\nabla_{x}\psi(\tau;t,x)\,d\tau\right).$

From Lemma 3.14, we note

$\left|\nabla_{x}\psi(s;t,x)\right|\leq e^{\int_{s}^{t}\|\nabla
V\|_{L^{\infty}_{x}}\,d\tau}.$ (3.22)

We combine (3.21) and (3.22) to get

$\displaystyle\left|\nabla\rho^{a}(x)\right|$
$\displaystyle\leq\left\\{\left|\nabla\varrho^{a}(\psi(s;t,x))\right|+\varrho^{a}(\psi(s;t,x))\times\left(a\int_{s}^{t}\|\nabla^{2}V\|_{L^{\infty}_{x}}\,d\tau\right)\right\\}e^{(a+1)\int_{s}^{t}\|\nabla
V\|_{L^{\infty}_{x}}\,d\tau}.$ (3.23)

Next, we note that for any nonnegative function $f$

$\int_{{\mathbb{R}}^{d}}f(\psi(s;t,x))dx=\int_{{\mathbb{R}}^{d}}f(y)J_{t,s}(y)\,dy\leq
e^{\int_{s}^{t}\|\nabla
V\|_{L^{\infty}_{x}}\,d\tau}\int_{{\mathbb{R}}^{d}}f(y)\,dy.$ (3.24)

Combining (3.23) and (3.24), we have (3.18).

##  4 A priori estimates

Here we provide a priori estimates of a regular solution of (1.1) given as
following definition.

###### Definition 4.1

Let $q\in[1,\infty)$ and $V$ be a measurable vector field. We say that a
nonnegative measurable function $\rho$ is a regular solution of (1.1) with
$\rho_{0}\in L^{1}(\mathbb{R}^{d})\cap{\mathcal{C}}^{\alpha}(\mathbb{R}^{d})$
for some $\alpha\in(0,1)$ if the followings are satisfied:

  * (i)

It holds that

$\rho\in L^{1}(Q_{T})\cap
L^{\infty}(0,T;\mathcal{C}^{\alpha}(\mathbb{R}^{d})),\quad\nabla\rho^{\frac{m+q-1}{2}}\in
L^{2}(Q_{T}),\quad\text{and}\quad V\in L^{1}_{\operatorname{loc}}(Q_{T}).$

  * (ii)

For any $\varphi\in{\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d}\times[0,T))$,
it holds that

$\iint_{Q_{T}}\left\\{\rho\varphi_{t}-\nabla\rho^{m}\cdot\nabla\varphi+\rho
V\cdot\nabla\varphi\right\\}\,dx\,dt=\int_{\mathbb{R}^{d}}\rho_{0}\varphi(\cdot,0)\,dx.$

By following similar arguments as in [Vaz07, Proposition 9.15], for
$\rho_{0}\in\mathcal{P}(\mathbb{R}^{d})$, we have that a regular solution
$\rho$ of (1.1) carries the mass conservation property in time; that is,
$\|\rho(\cdot,t)\|_{L^{1}(\mathbb{R}^{d})}=1$ a.e. $t\in[0,T]$. Throughout
this section, positive constants $m$, $d$, $p$, $q$, and $T$ are given, hence
we omit the dependence of $c$ and $C$, unless we need to specify the
dependencies clearly.

###  4.1  Supplementary estimates of speed and $p$-th moment

Note that (1.1) can be written as

$\partial_{t}\rho+\nabla\cdot\left(w\rho\right)=0,\quad\text{where}\quad
w:=-\frac{\nabla\rho^{m}}{\rho}+V.$ (4.1)

Due to Lemma 3.11, (4.1) can be seen as a curve in
${\mathcal{P}}_{p}({\mathbb{R}}^{d})$ whose speed at time $t$ is limited by
$\|w(\cdot,t)\|_{L^{p}(\rho(t))}$. In the following lemma, we deliver the
estimate of $\|w\|_{L^{p}(0,T;L^{p}(\rho(t)))}$.

###### Lemma 4.2

Let $m>1$, $1\leq q\leq m$, and
$1<p\leq{\lambda_{q}}:=1+\frac{d(q-1)+q}{d(m-1)+q}$. Suppose that
$\rho:Q_{T}\mapsto{\mathbb{R}}$ is a nonnegative measurable function
satisfying

$\rho\in
L^{\infty}(0,T;L^{q}(\mathbb{R}^{d}))\quad\text{and}\quad\rho^{\frac{q+m-1}{2}}\in
L^{2}(0,T;W^{1,2}(\mathbb{R}^{d})).$

Furthermore, assume that

$V\in\mathcal{S}_{m,q}^{(q_{1},q_{2})}\ \text{ for }\
\begin{cases}{\lambda_{q}}\leq
q_{2}\leq\frac{{\lambda_{q}}(q+m-1)}{q+m-2},&\text{ if }\ d>2,\vspace{1 mm}\\\
{\lambda_{q}}\leq q_{2}<\frac{{\lambda_{q}}(q+m-1)}{q+m-2},&\text{ if }\
d=2.\end{cases}$ (4.2)

Then, for any $\varepsilon>0$, there exists constant $c=c(\varepsilon)$ such
that

$\iint_{Q_{T}}\left|\frac{\nabla\rho^{m}}{\rho}\right|^{p}\rho\,dx\,dt\leq\varepsilon\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt+cT\left(\sup_{0\leq
t\leq
T}\int_{\mathbb{R}^{d}}\rho^{q}\,dx\right)^{\frac{2\theta}{d(1-\theta)}},$
(4.3)
$\iint_{Q_{T}}|V|^{p}\rho\,dx\,dt\leq\varepsilon\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt+cT^{\beta_{1}\frac{{\lambda_{q}}-p}{{\lambda_{q}}}}\|\rho\|_{L^{q,\infty}_{x,t}}^{\beta\beta_{1}\frac{qp(q_{1}-{\lambda_{q}})}{q_{1}{\lambda_{q}}}}\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}^{p\beta_{1}},$
(4.4)

where $\theta=\frac{dp(m-q)}{(2-p)[(d+2)q+(m-2)d]}$,
$\beta=1-\frac{(d-2)(q_{1}(1-q)+q{\lambda_{q}})}{(q_{1}-{\lambda_{q}})(d(m-1)+2q)}$,
and
$\beta_{1}=\frac{q_{2}{\lambda_{q}}}{q_{2}({\lambda_{q}}-p)+p{\lambda_{q}}}$.

Proof. For simplicity, let $\lambda:=\lambda_{q}$. First we prove (4.3). If
$1\leq q<m$, then $1<p\leq{\lambda}<2$ and we observe that

$\left|\frac{\nabla\rho^{m}}{\rho}\right|^{p}\rho=\left(\frac{2m}{q+m-1}\right)^{p}\rho^{\frac{2-p+p(m-q)}{2}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{p}.$

By applying Young’s inequality with $\frac{p}{2}+\frac{2-p}{2}=1$, we have the
following, for any $\varepsilon>0$,

$\iint_{Q_{T}}\left|\frac{\nabla\rho^{m}}{\rho}\right|^{p}\rho\,dx\,dt\leq\frac{\varepsilon}{2}\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt+c(\varepsilon)\iint_{Q_{T}}\rho^{\frac{2-p+p(m-q)}{2-p}}\,dx\,dt.$
(4.5)

By Lemma 3.2 for $d>2$ (or Lemma 3.3, for $d=2$), it holds that $\rho\in
L_{x,t}^{(q+m-1)+\frac{2q}{d}}$. Therefore, we set

$1\leq\frac{2-p+p(m-q)}{2-p}\leq q+m-1+\frac{2q}{d}\ \iff\ 1<p\leq
1+\frac{d(q-1)+q}{d(m-1)+q}.$

To handle the second term on the RHS of (4.5), let us set

$\frac{2-p+p(m-q)}{2-p}=(1-\theta)+\left(q+m-1+\frac{2q}{d}\right)\theta,\quad\text{for}\quad\theta\in(0,1).$

Then the Hölder inequality and Lemma 3.2 yield

$\displaystyle\iint_{Q_{T}}\rho^{\frac{2-p+p(m-q)}{2-p}}\,dx\,dt$
$\displaystyle\leq\|\rho\|_{L_{x,t}^{1}}^{1-\theta}\|\rho\|_{L^{q,\infty}_{x,t}}^{\frac{2\theta
q}{d}}\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x,t}}^{2\theta}$
(4.6)
$\displaystyle\leq\frac{\varepsilon}{2}\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x,t}}^{2}+cT\|\rho_{0}\|_{L^{1}(\mathbb{R}^{d})}\|\rho\|_{L^{q,\infty}_{x,t}}^{\frac{2\theta
q}{d(1-\theta)}},$

by taking the Young’s inequality with $(1-\theta)+\theta=1$ and applying the
mass conservation property. Hence the combination of (4.5) and (4.6) yield
(4.3) for $1<q<m$.

If $q=m$, then we observe that ${\lambda}_{q}=2$ and

$\left|\frac{\nabla\rho^{m}}{\rho}\right|^{2}\rho=\left(\frac{2m}{2m-1}\right)^{2}\left|\nabla\rho^{\frac{2m-1}{2}}\right|^{2}.$

For any $1<p<2$, Young’s inequality with $\frac{p}{2}+\frac{2-p}{2}=1$ gives

$\left|\frac{\nabla\rho^{m}}{\rho}\right|^{p}\rho\leq\varepsilon\left|\frac{\nabla\rho^{m}}{\rho}\right|^{2}\rho+c(\varepsilon)\rho,$

that completes the proof of (4.3) by taking integration on $Q_{T}$.

Now we prove (4.4). First, by taking the Hölder inequality with
$\frac{{\lambda}}{q_{1}}+\frac{q_{1}-{\lambda}}{q_{1}}=1$ for
$q_{1}>{\lambda}$, we have

$\int_{\mathbb{R}^{d}}|V|^{{\lambda}}\rho\,dx\leq\|V\|_{L^{q_{1}}_{x}}^{{\lambda}}\|\rho\|_{L^{r_{1}}_{x}}\quad\text{where}\quad
r_{1}:=\frac{q_{1}}{q_{1}-{\lambda}}.$ (4.7)

Let us set

$q\leq r_{1}\leq\frac{d(q+m-1)}{d-2}\quad\text{
and}\quad\theta:=\frac{q}{r_{1}}\left(1-\frac{(r_{1}-q)(d-2)}{2q+d(m-1)}\right).$
(4.8)

Then by (3.4) and (3.5) in Lemma 3.4, we yield, from (4.7), that

$\displaystyle\int_{\mathbb{R}^{d}}|V|^{{\lambda}}\rho\,dx$
$\displaystyle\leq\|V\|_{L^{q_{1}}_{x}}^{{\lambda}}\|\rho\|_{L^{q}_{x}}^{\theta}\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x}}^{\frac{2(1-\theta)}{q+m-1}}$
(4.9)
$\displaystyle\leq\varepsilon\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x}}^{2}+c(\varepsilon)\|\rho\|_{L^{q}_{x}}^{\frac{\theta(q+m-1)}{q+m-1-(1-\theta)}}\|V\|_{L^{q_{1}}_{x}}^{\frac{{\lambda}(q+m-1)}{q+m-1-(1-\theta)}},$

by taking Young’s inequality with
$\frac{1-\theta}{q+m-1}+\frac{q+m-1-(1-\theta)}{q+m-1}=1$. Then now let us set

$q_{2}:=\frac{{\lambda}(q+m-1)}{q+m-1-(1-\theta)}=\frac{q_{1}{\lambda}(2q+d(m-1))}{q_{1}(2q+d(m+q-2))-dq{\lambda}}.$
(4.10)

Then the direct computation gives that the pairs $(q_{1},q_{2})$ satisfy
$\mathcal{S}_{m,q}^{(q_{1},q_{2})}$. Moreover, from (4.8) and (4.10), we
deduce that

$\frac{d{\lambda}(q+m-1)}{2+d(q+m-2)}\leq
q_{1}\leq\frac{q{\lambda}}{q-1}\quad\Longrightarrow\quad{\lambda}\leq
q_{2}\leq\frac{{\lambda}(q+m-1)}{q+m-2}.$

Moreover, by taking integration in terms of $t$ to (4.9), it gives

$\iint_{Q_{T}}|V|^{{\lambda}}\rho\,dx\,dt\leq\varepsilon\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x,t}}^{2}+c(\varepsilon)\|\rho\|_{L^{q,\infty}_{x,t}}^{\frac{\theta
q_{2}}{{\lambda}}}\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}.$ (4.11)

For any $1<p<{\lambda}$, we take the Hölder inequality with
$\frac{p}{{\lambda}}+\frac{{\lambda}-p}{{\lambda}}=1$ and apply the mass
conservation property

$\iint_{Q_{T}}|V|^{p}\rho\,dx\,dt\leq\left(\iint_{Q_{T}}|V|^{{\lambda}}\rho\,dx\,dt\right)^{\frac{p}{{\lambda}}}\left(\iint_{Q_{T}}\rho\,dx\,dt\right)^{\frac{{\lambda}-p}{{\lambda}}}\leq
T^{\frac{{\lambda}-p}{{\lambda}}}\left(\iint_{Q_{T}}|V|^{{\lambda}}\rho\,dx\,dt\right)^{\frac{p}{{\lambda}}}.$
(4.12)

Then by taking Young’s inequality to the combined inequality of (4.12) and
(4.11), we obtain (4.4).

Recalling $p$-th power $\langle
x\rangle^{p}=\left(1+|x|^{2}\right)^{\frac{p}{2}}$ for $1<p\leq 2$, we note
the following properties:

$\nabla\langle
x\rangle^{p}=p\,x\left(1+|x|^{2}\right)^{\frac{p-2}{2}}\quad\text{so}\quad\left|\nabla\langle
x\rangle^{p}\right|<p\langle x\rangle^{p-1},$ (4.13)

and

$\Delta\langle
x\rangle^{p}=p\frac{1+(p-1)|x|^{2}}{\left(1+|x|^{2}\right)^{2-p/2}}<p\frac{1+|x|^{2}}{\left(1+|x|^{2}\right)^{2-p/2}}=p\left(1+|x|^{2}\right)^{\frac{p-2}{2}}\leq
p.$ (4.14)

In the following lemma, we compute $p$-th moment estimate assuming that $V$
satisfies (4.2).

###### Lemma 4.3

($p$-th moment estimate) Let $m>1$, $1\leq q\leq m$, and
$1<p\leq{\lambda_{q}}:=1+\frac{d(q-1)+q}{d(m-1)+q}$. Suppose that
$\rho:[0,T]\mapsto\mathcal{P}({\mathbb{R}}^{d})$ is narrowly continuous, and
is a regular solution with sufficiently fast decay at infinity in spatial
variables of (1.1). If $V\in{\mathcal{C}}^{\infty}(\bar{Q}_{T})$ satisfies
(4.2), then it holds

$\displaystyle\sup_{0\leq t\leq T}$
$\displaystyle\int_{\mathbb{R}^{d}}\,\rho(\cdot,t)\langle
x\rangle^{p}\,dx\leq\int_{\mathbb{R}^{d}}\,\rho_{0}\langle
x\rangle^{p}\,dx+\iint_{Q_{T}}\rho\langle x\rangle^{p}\,dx\,dt$ (4.15)
$\displaystyle+\varepsilon\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt+c_{1}T+c_{2}T^{\beta_{1}\frac{{\lambda_{q}}-p}{{\lambda_{q}}}}\|\rho\|_{L^{q,\infty}_{x,t}}^{\beta\beta_{1}\frac{qp(q_{1}-{\lambda_{q}})}{q_{1}{\lambda_{q}}}}\|V\|_{\mathcal{S}_{m,q}}^{p\beta_{1}},$

where $c_{1}=c_{1}(\varepsilon,p)$, $c_{2}=c_{2}(\varepsilon,m,d,p,q)$,
$\beta=1-\frac{(d-2)(q_{1}(1-q)+q{\lambda_{q}})}{(q_{1}-{\lambda_{q}})(d(m-1)+2q)}$,
and
$\beta_{1}=\frac{q_{2}{\lambda_{q}}}{q_{2}({\lambda_{q}}-p)+p{\lambda_{q}}}$.

Proof. For simplicity, let $\lambda:=\lambda_{q}$. By testing $\langle
x\rangle^{p}$ for $p>1$ to (1.1) and taking the integration by parts, we
obtain

$\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho\langle
x\rangle^{p}\,dx=\int_{\mathbb{R}^{d}}\left[\Delta\rho^{m}-\nabla\left(\rho
V\right)\right]\langle
x\rangle^{p}\,dx=\int_{\mathbb{R}^{d}}\left[\rho^{m}\Delta\langle
x\rangle^{p}+\rho V\cdot\nabla\langle x\rangle^{p}\right]\,dx.$

By taking the integration in $t\in[0,T]$, we have

$\sup_{0\leq t\leq T}\int_{\mathbb{R}^{d}}\rho\langle
x\rangle^{p}\,dx\leq\int_{\mathbb{R}^{d}}\rho_{0}\langle
x\rangle^{p}\,dx+\underbrace{\iint_{Q_{T}}\left|\rho^{m}\Delta\langle
x\rangle^{p}\right|\,dx\,dt}_{:=J_{1}}+\underbrace{\iint_{Q_{T}}\left|\rho
V\cdot\nabla\langle x\rangle^{p}\right|\,dx\,dt}_{:=J_{2}}.$ (4.16)

Let us set

$m=(1-\theta)+\frac{d(q+m-1)}{d-2}\theta\quad\iff\quad\theta=\frac{(m-1)(d-2)}{d(q+m-1)-(d-2)}\in(0,1).$

Then by (4.14), Hölder inequality, and Lemma 3.1, we have

$\displaystyle J_{1}$ $\displaystyle\leq p\iint_{Q_{T}}\rho^{m}\,dx\,dt\leq
p\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}\rho\,dx\right)^{1-\theta}\left(\int_{\mathbb{R}^{d}}\rho^{\frac{d(q+m-1)}{d-2}}\,dx\right)^{\theta}\,dt$
(4.17) $\displaystyle\leq
p\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}\rho\,dx\right)^{1-\theta}\left(\int_{\mathbb{R}^{d}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\right)^{\frac{\theta
d}{d-2}}\,dt$
$\displaystyle\leq\frac{\varepsilon}{2}\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt+c(p,\varepsilon)T\|\rho_{0}\|_{L^{1}(\mathbb{R}^{d})}^{\frac{d(q-1)+2m}{d(q-1)+2}},$

by mass conservation property and Young’s inequality for $\frac{\theta
d}{d-2}+\frac{d(1-\theta)-2}{d-2}=1$.

Now let us handle $J_{2}$. First, consider the case $p={\lambda}$. Then by
(4.13), we take Hölder inequality to get

$\iint_{Q_{T}}\rho|V|\langle
x\rangle^{{\lambda}-1}\,dx\,dt\leq\iint_{Q_{T}}\rho\langle
x\rangle^{{\lambda}}\,dx\,dt+\iint_{Q_{T}}\rho|V|^{{\lambda}}\,dx\,dt.$

Then we apply (4.11) to handle the last term on RHS which gives, for
$\beta=1-\frac{(r_{1}-q)(d-2)}{d(m-1)+2q}$,

$\iint_{Q_{T}}\rho|V|\langle
x\rangle^{{\lambda}-1}\,dx\,dt\leq\iint_{Q_{T}}\rho\langle
x\rangle^{{\lambda}}\,dx\,dt+\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}^{{\lambda}}\|\rho\|_{L^{q,\infty}_{x,t}}^{\frac{q\beta}{r_{1}}}\,\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x,t}}^{\frac{2}{r_{2}}}.$

On the other hand, we consider the case $1<p<{\lambda}$ in $J_{2}$ from
(4.16). Then we apply (4.13) and Young’s inequality with
$\frac{p-1}{p}+\frac{1}{p}=1$ to obtain

$\displaystyle J_{2}$ $\displaystyle\leq p\iint_{Q_{T}}\rho|V|\langle
x\rangle^{p-1}\,dx\,dt\leq\iint_{Q_{T}}\rho\langle
x\rangle^{p}\,dx\,dt+c(p)\iint_{Q_{T}}\rho|V|^{p}\,dx\,dt$
$\displaystyle\leq\iint_{Q_{T}}\rho\langle
x\rangle^{p}\,dx\,dt+c(p)\left(\iint_{Q_{T}}\rho|V|^{{\lambda}}\,dx\,dt\right)^{\frac{p}{{\lambda}}}\left(\iint_{Q_{T}}\rho\,dx\,dt\right)^{\frac{{\lambda}-p}{{\lambda}}}$
$\displaystyle\leq\iint_{Q_{T}}\rho\langle
x\rangle^{p}\,dx\,dt+c(p)T^{\frac{{\lambda}-p}{{\lambda}}}\|\rho_{0}\|_{L^{1}(\mathbb{R}^{d})}^{\frac{{\lambda}-p}{{\lambda}}}\left(\iint_{Q_{T}}\rho|V|^{{\lambda}}\,dx\,dt\right)^{\frac{p}{{\lambda}}},$

by taking another Hölder inequality with
$\frac{p}{{\lambda}}+\frac{{\lambda}-p}{{\lambda}}=1$, and applying the mass
conservation property. By (4.11) and Young’s inequality with
$\frac{p}{{\lambda}r_{2}}+\frac{{\lambda}r_{2}-p}{{\lambda}r_{2}}=1$ , we have
the following, for any $\varepsilon>0$,

$J_{2}\leq\iint_{Q_{T}}\rho\langle
x\rangle^{p}\,dx\,dt+\frac{\varepsilon}{2}\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x,t}}^{2}+c(\varepsilon,p)T^{\frac{r_{2}({\lambda}-p)}{{\lambda}r_{2}-p}}\|\rho\|_{L^{q,\infty}_{x,t}}^{\frac{qp\beta
r_{2}}{r_{1}({\lambda}r_{2}-p)}}\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}^{\frac{p{\lambda}r_{2}}{{\lambda}r_{2}-p}}.$
(4.18)

The combination of the above with (4.16) and (4.17) completes the proof.

###### Remark 4.4

A similar estimate as in above lemma holds for $L^{q}$-weak solutions that are
narrowly continuous in time variable. For the sketch of proof, let
$\eta:{\mathbb{R}}^{d}\rightarrow[0,1]$ be a smooth cut-off function
satisfying

$\eta(x)=\ \begin{cases}1,&\text{ if }\ |x|\leq 1,\\\ 0,&\text{ if }\ |x|\geq
2.\end{cases}$

Let $\rho$ be a $L^{q}$-weak solution of (1.1) such that
$\rho:[0,T]\mapsto{\mathcal{P}}({\mathbb{R}}^{d})$ is narrowly continuous. For
$\delta>0$, we define $\eta_{\delta}(x):=\eta(\delta x)$. As in Lemma 4.3, by
testing $\langle x\rangle^{p}\eta_{\delta}$ to (1.1), we obtain

$\displaystyle\sup_{0\leq t\leq T}\int_{\mathbb{R}^{d}}\rho\left(\langle
x\rangle^{p}\eta_{\delta}(x)\right)\,dx$
$\displaystyle\leq\int_{\mathbb{R}^{d}}\rho_{0}\left(\langle
x\rangle^{p}\eta_{\delta}(x)\right)\,dx$ (4.19)
$\displaystyle\quad+\underbrace{\iint_{Q_{T}}\left|\rho^{m}\Delta\left(\langle
x\rangle^{p}\eta_{\delta}(x)\right)\right|\,dx\,dt}_{:=J_{3}}+\underbrace{\iint_{Q_{T}}\left|\rho
V\cdot\nabla\left(\langle
x\rangle^{p}\eta_{\delta}(x)\right)\right|\,dx\,dt.}_{:=J_{4}}$

We note that $\|\nabla\langle
x\rangle^{p}\nabla\eta_{\delta}\|_{L_{x}^{\infty}},~{}\|\langle
x\rangle^{p}\Delta\eta_{\delta}\|_{L_{x}^{\infty}}\lesssim\delta^{2-p}$. Hence

$\displaystyle J_{3}$
$\displaystyle\lesssim\iint_{Q_{T}}\left|\rho^{m}\Delta\langle
x\rangle^{p}\right|\eta_{\delta}(x)\,dx\,dt+\delta^{2-p}\iint_{\\{\frac{1}{\delta}\leq|x|\leq\frac{2}{\delta}\\}\times[0,T]}\rho^{m}\,dx\,dt$
(4.20) $\displaystyle\rightarrow\iint_{Q_{T}}\left|\rho^{m}\Delta\langle
x\rangle^{p}\right|\,dx\,dt\quad\mbox{as}~{}~{}\delta\rightarrow 0.$

Similarly, we note $|\langle x\rangle^{p}\nabla\eta_{\delta}|\lesssim|\langle
x\rangle^{p-1}|$ and have

$\displaystyle J_{4}$ $\displaystyle\lesssim\iint_{Q_{T}}\left|\rho
V\cdot\nabla\langle
x\rangle^{p}\right|\eta_{\delta}(x)\,dx\,dt+\iint_{Q_{T}}\rho|V|\langle
x\rangle^{p-1}\,dx\,dt$ (4.21)
$\displaystyle\rightarrow\iint_{Q_{T}}\left|\rho V\cdot\nabla\langle
x\rangle^{p}\right|\,dx\,dt+\iint_{Q_{T}}\rho|V|\langle
x\rangle^{p-1}\,dx\,dt,\quad\mbox{as}~{}~{}\delta\rightarrow 0.$

Finally, we take $\delta\rightarrow 0$ in (4.19). Then, we combine
(4.17)–(4.18), (4.20) and (4.21) to have

$\displaystyle\sup_{0\leq t\leq T}$
$\displaystyle\int_{\mathbb{R}^{d}}\,\rho(\cdot,t)\langle
x\rangle^{p}\,dx\leq\int_{\mathbb{R}^{d}}\,\rho_{0}\langle
x\rangle^{p}\,dx+2\iint_{Q_{T}}\rho\langle x\rangle^{p}\,dx\,dt$ (4.22)
$\displaystyle+\varepsilon\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt+c_{1}T+c_{2}T^{\beta_{1}\frac{{\lambda_{q}}-p}{{\lambda_{q}}}}\|\rho\|_{L^{q,\infty}_{x,t}}^{\beta\beta_{1}\frac{qp(q_{1}-{\lambda_{q}})}{q_{1}{\lambda_{q}}}}\|V\|_{\mathcal{S}_{m,q}}^{p\beta_{1}},$

where $c_{1},\,c_{2},\,\beta$ and $\beta_{1}$ are same as in Lemma 4.3.

###  4.2 Energy estimates

This subsection is composed with energy estimates for two different cases:
first when $\int_{\mathbb{R}^{d}}\rho_{0}\log\rho_{0}\,dx<\infty$ and second
when $\|\rho_{0}\|_{L^{q}({\mathbb{R}}^{d})}<\infty$ for $q>1$.

####  4.2.1 Energy estimates for case:
$\int_{\mathbb{R}^{d}}\rho_{0}\log\rho_{0}\,dx<\infty$.

Under additional assumption $\rho_{0}\log\rho_{0}\in L^{1}(\mathbb{R}^{d})$,
not merely $\rho_{0}\in L^{1}(\mathbb{R}^{d})$, we obtain a priori estimate
categorized three cases depending on the range of $m$ and the conditions on
$V$.

###### Proposition 4.5

Let $m>1$, $1<p\leq{\lambda_{1}}:=1+\frac{1}{d(m-1)+1}$. Assume that
$\rho_{0}\in\mathcal{P}_{p}(\mathbb{R}^{d})\cap{\mathcal{C}}^{\alpha}(\mathbb{R}^{d})$
holds

$\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx<\infty.$ (4.23)

Suppose that $\rho$ is a regular solution of (1.1) with sufficiently fast
decay at infinity in spatial variables.

  * (i)

Let $1<m\leq 2$. Assume that

$V\in\mathcal{S}_{m,1}^{(q_{1},q_{2})}\cap{\mathcal{C}}^{\infty}(\bar{Q}_{T})\quad\text{for}\quad\begin{cases}2\leq
q_{2}\leq\frac{m}{m-1},&\text{ if }d>2,\vspace{1 mm}\\\ 2\leq
q_{2}<\frac{m}{m-1},&\text{ if }d=2.\end{cases}$ (4.24)

Then the following estimate holds

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}\times\\{t\\}}\rho\left(\left|\log\rho\right|+\langle
x\rangle^{p}\right)\,dx+\frac{2}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt\leq
C,$ (4.25)

where
$C=C(\|V\|_{\mathcal{S}_{m,1}^{(q_{1},q_{2})}},\,\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx)$.

  * (ii)

Assume that

$V\in\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}\cap{\mathcal{C}}^{\infty}(\bar{Q}_{T})\quad\text{for}\quad\begin{cases}\frac{2+d(m-1)}{1+d(m-1)}<\tilde{q}_{2}\leq\frac{m}{m-1},&\text{
if }d>2,\vspace{1 mm}\\\
\frac{2+d(m-1)}{1+d(m-1)}<\tilde{q}_{2}<\frac{m}{m-1},&\text{ if
}d=2.\end{cases}$ (4.26)

Then the estimate (4.25) holds where
$C=C(\|V\|_{\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}},\,\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx).$

  * (iii)

Let $\nabla\cdot V=0$. Assume that

$V\in\mathcal{S}_{m,1}^{(q_{1},q_{2})}\cap{\mathcal{C}}^{\infty}(\bar{Q}_{T})\quad\text{for}\quad\begin{cases}{\lambda_{1}}\leq
q_{2}\leq\frac{{\lambda_{1}}m}{m-1},&\text{ if }d>2,\vspace{1 mm}\\\
{\lambda_{1}}\leq q_{2}<\frac{{\lambda_{1}}m}{m-1},&\text{ if
}d=2.\end{cases}$ (4.27)

Then the estimate (4.25) holds where
$C=C(\|V\|_{\mathcal{S}_{m,1}^{(q_{1},q_{2})}},\,\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx).$

Proof. First, we recall the $p$-entropy inequality (for example, see [DL89,
CKL14])

$\int_{\mathbb{R}^{d}}\rho\left|\log\rho\right|\,dx\leq\int_{\mathbb{R}^{d}}\rho\log\rho\,dx+\int_{\mathbb{R}^{d}}\rho\langle
x\rangle^{p}\,dx+c(d),$ (4.28)

which gives

$0\leq\int_{\mathbb{R}^{d}}\rho\log\rho\,dx-\int_{\mathbb{R}^{d}}\rho\left(\log\rho\right)_{-}\,dx\leq\int_{\mathbb{R}^{d}}\rho\log\rho\,dx+d\int_{\mathbb{R}^{d}}\rho\langle
x\rangle^{p}\,dx+c(d).$ (4.29)

By testing $\log\rho$ to (1.1), we obtain

$\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho\log\rho\,dx+\frac{4}{m}\int_{\mathbb{R}^{d}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx=\int_{\mathbb{R}^{d}}V\cdot\nabla\rho\,dx=:\mathcal{J}.$
(4.30)

$(i)$ For $1<m<2$, we write RHS of (4.30) as

$\mathcal{J}=\int_{\mathbb{R}^{d}}V\cdot\nabla\rho\,dx\leq\frac{2}{m}\int_{\mathbb{R}^{d}}\left|V\right|\rho^{1-\frac{m}{2}}\left|\nabla\rho^{\frac{m}{2}}\right|\,dx.$

By Hölder inequality with
$\frac{1}{q_{1}}+\frac{1}{2}+\frac{q_{1}-2}{2q_{1}}=1$ for $q_{1}>2$, we have

$\mathcal{J}\leq
c\|V\|_{L^{q_{1}}_{x}}\left\|\nabla\rho^{\frac{m}{2}}\right\|_{L^{2}_{x}}\left\|\rho^{\frac{2-m}{2}}\right\|_{L^{\frac{2q_{1}}{q_{1}-2}}_{x}}\leq
c\|V\|_{L^{q_{1}}_{x}}\left\|\nabla\rho^{\frac{m}{2}}\right\|_{L^{2}_{x}}\left\|\rho\right\|^{\frac{2-m}{2}}_{L^{\frac{(2-m)q_{1}}{q_{1}-2}}_{x}}.$
(4.31)

When $m=2$, the above inequality corresponds to $\mathcal{J}\leq
c\|V\|_{L_{x}^{2}}\left\|\nabla\rho^{\frac{m}{2}}\right\|_{L^{2}_{x}}$.

For convinence, we denote

$r_{1}:=\frac{(2-m)q_{1}}{q_{1}-2},\quad\text{and}\quad\theta:=\frac{1}{r_{1}}\left(1-\frac{(r_{1}-1)(d-2)}{2+d(m-1)}\right).$

Then by (3.2) in Lemma 3.4, we find that

$\begin{cases}2<\frac{md}{1+(m-1)(d-1)}\leq q_{1}\leq\frac{2}{m-1},&\text{ if
}d>2,\\\ 2<q_{1}\leq\frac{2}{m-1},&\text{ if }d=2.\end{cases}$ (4.32)

Moreover, the combination of (3.4) and (3.5) in Lemma 3.4 and (4.31) implies
that

$\displaystyle\mathcal{J}$ $\displaystyle\leq
c\|V\|_{L^{q_{1}}_{x}}\left\|\nabla\rho^{\frac{m}{2}}\right\|_{L^{2}_{x}}\left\|\rho\right\|^{\frac{2-m}{2}}_{L^{\frac{(2-m)q_{1}}{q_{1}-2}}_{x}}\leq
c\|V\|_{L^{q_{1}}_{x}}\left\|\rho\right\|^{\frac{\theta(2-m)}{2}}_{L^{1}_{x}}\left\|\nabla\rho^{\frac{m}{2}}\right\|^{1+\frac{(1-\theta)(2-m)}{m}}_{L^{2}_{x}}$
(4.33)
$\displaystyle\leq\varepsilon\left\|\nabla\rho^{\frac{m}{2}}\right\|^{2}_{L^{2}_{x}}+c(\varepsilon)\|V\|^{\frac{2m}{m-(1-\theta)(2-m)}}_{L^{q_{1}}_{x}},$

by taking Young’s inequality. Since $V$ is in
$\mathcal{S}_{m,1}^{(q_{1},q_{2})}$, it follows that

$\frac{2m}{m-(1-\theta)(2-m)}=\frac{q_{1}(2+d(m-1))}{q_{1}(1+d(m-1))-d}=q_{2}.$

Moreover, (4.32) corresponds to (4.24).

We combine (4.30) and (4.33), and then take integration in terms of $t$ with
$\varepsilon=\frac{2}{m}$ to have

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\rho\log\rho\,dx+\frac{2}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt\leq\int_{\mathbb{R}^{d}}\rho_{0}\log\rho_{0}\,dx+c\|V\|^{q_{2}}_{\mathcal{S}_{m,1}^{(q_{1},q_{2})}}.$
(4.34)

To (4.34), we apply (4.29) and add $p$-th moment estimate (4.15) with
$\varepsilon=\frac{1}{m}$ to obtain

$\displaystyle\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\left(\rho\left|\log\rho\right|+\rho\langle
x\rangle^{p}\right)\,dx+\frac{1}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt$
(4.35)
$\displaystyle\leq\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx+\iint_{Q_{T}}\rho\langle
x\rangle^{p}\,dx\,dt+C(\|V\|_{\mathcal{S}_{m,1}^{(q_{1},q_{2})}}),$

where $(q_{1},q_{2})$ given in (4.24) that satisfies (4.2) too. To handle the
integral of $p$-th moment on RHS of (4.35), we apply the Grönwall’s inequality
that leads to

$\displaystyle\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\left(\rho\left|\log\rho\right|+\rho\langle
x\rangle^{p}\right)\,dx+\frac{1}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt$
$\displaystyle\leq
c\left[\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx+C(\|V\|_{\mathcal{S}_{m,1}^{(q_{1},q_{2})}})\right].$

Therefore, we complete to obtain (4.25) under (4.24).

$(ii)$ First we take the integration by parts and apply Hölder inequality for
$\frac{1}{\tilde{q}_{1}}+\frac{\tilde{q}_{1}-1}{\tilde{q}_{1}}=1$ to have

$\mathcal{J}=\int_{\mathbb{R}^{d}}V\cdot\nabla\rho\,dx=\int_{\mathbb{R}^{d}}\rho\nabla\cdot
V\,dx\leq\|\nabla
V\|_{L_{x}^{\tilde{q}_{1}}}\|\rho\|_{L_{x}^{\frac{\tilde{q}_{1}}{\tilde{q}_{1}-1}}}.$
(4.36)

For convenience, we denote

$\tilde{r}_{1}:=\frac{\tilde{q}_{1}}{\tilde{q}_{1}-1}\quad\text{and}\quad\theta:=\frac{1}{\tilde{r}_{1}}\left(1-\frac{(\tilde{r}_{1}-1)(d-2)}{2+d(m-1)}\right).$

Then Lemma 3.4 with $\tilde{r}_{1}$ in the place for $r_{1}$ implies that

$\begin{cases}1<\frac{md}{2+d(m-1)}\leq\tilde{q}_{1}\leq\infty,&\text{ if
}d>2,\\\ 1<\tilde{q}_{1}<\infty,&\text{ if }d=2.\end{cases}$ (4.37)

Then the combination of (3.4) and (3.5) in Lemma 3.4 and (4.36) gives

$\mathcal{J}\leq c\|\nabla
V\|_{L_{x}^{\tilde{q}_{1}}}\left\|\nabla\rho^{\frac{m}{2}}\right\|_{L_{x}^{2}}^{\frac{m}{m-(1-\theta)}}\leq\varepsilon\left\|\nabla\rho^{\frac{m}{2}}\right\|_{L_{x}^{2}}^{2}+c(\varepsilon)\|\nabla
V\|_{L_{x}^{\tilde{q}_{1}}}^{\frac{m}{m-(1-\theta)}},$ (4.38)

by taking Young’s inequality. Because of
$V\in\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}$, it gives

$\frac{m}{m-(1-\theta)}=\frac{\tilde{q}_{1}(2+d(m-1))}{\tilde{q}_{1}(2+d(m-1))-2d}=\tilde{q}_{2}.$

Moreover, (4.37) corresponds to (4.26).

Now we combine (4.30) and (4.38), and then take integration in terms of $t$
with $\varepsilon=\frac{1}{m}$ to have

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\rho\log\rho\,dx+\frac{3}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt\leq\int_{\mathbb{R}^{d}}\rho_{0}\log\rho_{0}\,dx+c\|V\|^{\tilde{q}_{2}}_{\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}}.$
(4.39)

To (4.39), we apply (4.29) and add $p$-th moment estimate (4.15) with
$\varepsilon=\frac{1}{m}$ to obtain

$\displaystyle\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\left(\rho\left|\log\rho\right|+\rho\langle
x\rangle^{p}\right)\,dx+\frac{2}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt$
$\displaystyle\leq\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx+\iint_{Q_{T}}\rho\langle
x\rangle^{p}\,dx\,dt+C(\|V\|_{\mathcal{S}_{m,1}^{(q_{1},q_{2})}},\|V\|_{\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}}),$

under (4.26) and $V\in\mathcal{S}_{m,1}^{(q_{1},q_{2})}$ holding (4.2) for
$q=1$. By Remark 1.2 $(ii)$, we see that (4.26) yields

$\displaystyle\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\left(\rho\left|\log\rho\right|+\rho\langle
x\rangle^{p}\right)\,dx+\frac{2}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt$
$\displaystyle\leq\int_{\mathbb{R}^{d}}\left(\rho_{0}\log\rho_{0}+\rho_{0}\langle
x\rangle^{p}\right)\,dx+\iint_{Q_{T}}\rho\langle
x\rangle^{p}\,dx\,dt+C(\|V\|_{\tilde{\mathcal{S}}_{m,1}^{(\tilde{q}_{1},\tilde{q}_{2})}}).$

Then we complete the proof by taking Grönwall’s inequality to handle the
integral of $p$-th moment as in (4.35).

$(iii)$ Because of (4.36) and $\nabla\cdot V=0$, we observe that RHS of (4.30)
vanishes. Therefore, we obtain the following by taking integral to (4.30) with
repect to $t$:

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\rho\log\rho\,dx+\frac{4}{m}\iint_{Q_{T}}\left|\nabla\rho^{\frac{m}{2}}\right|^{2}\,dx\,dt\leq\int_{\mathbb{R}^{d}}\rho_{0}\log\rho_{0}\,dx.$
(4.40)

For $V$ satisfying (4.2) which is (4.27) when $q=1$, we add (4.15) to (4.40).
Then as above, we apply (4.29) and Grönwall’s inequality to complete the
proof. We note that (4.27) comes directly from (4.2).

####  4.2.2 Energy estimates for case: $\rho_{0}\in L^{q}(\mathbb{R}^{d}),\
q>1$.

In the following proposition, there are three cases for $m,q>1$. First case is
when $q-m+1\geq 0$ holds. The second case cares all $m,q>1$ by prescribing
conditions on $\nabla V$, and the last case is when $\nabla\cdot V=0$.

###### Proposition 4.6

Let $m,q>1$ and $\rho_{0}\in\mathcal{P}(\mathbb{R}^{d})\cap
L^{q}(\mathbb{R}^{d})\cap{\mathcal{C}}^{\alpha}(\mathbb{R}^{d})$. Suppose that
$\rho$ is a regular solution of (1.1) with sufficiently fast decay at infinity
in spatial variables.

  * (i)

Let $q\geq m-1$. Assume that

$V\in\mathcal{S}_{m,q}^{(q_{1},q_{2})}\cap{\mathcal{C}}^{\infty}(\bar{Q}_{T})\quad\text{for}\quad\begin{cases}2\leq
q_{2}\leq\frac{q+m-1}{m-1},&\text{ if }d>2,\vspace{1 mm}\\\ 2\leq
q_{2}<\frac{q+m-1}{m-1},&\text{ if }d=2.\end{cases}$ (4.41)

Then the following estimate holds:

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\rho^{q}(\cdot,t)\,dx+\frac{2qm(q-1)}{(q+m-1)^{2}}\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt\leq
C,$ (4.42)

where
$C=C(\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}},\,\|\rho_{0}\|_{L^{q}(\mathbb{R}^{d})})$.

  * (ii)

Assume that

$V\in\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}\cap{\mathcal{C}}^{\infty}(\bar{Q}_{T})\quad\text{for}\quad\begin{cases}1\leq\tilde{q}_{2}\leq\frac{q+m-1}{m-1},&\text{
if }d>2,\vspace{1 mm}\\\ 1\leq\tilde{q}_{2}<\frac{q+m-1}{m-1},&\text{ if
}d=2.\end{cases}$ (4.43)

Then (4.42) holds with
$C=C(\|V\|_{\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}},\,\|\rho_{0}\|_{L^{q}(\mathbb{R}^{d})})$.

  * (iii)

Assume that $V\in{\mathcal{C}}^{\infty}(\bar{Q}_{T})$ and $\nabla\cdot V=0$.
Then the following estimate holds

$\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{d}}\rho^{q}(\cdot,t)\,dx+\frac{qm(q-1)}{(q+m-1)^{2}}\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt\leq\int_{\mathbb{R}^{d}}\rho_{0}^{q}\,dx.$
(4.44)

Proof. Let $q>1$. By testing $\varphi=q\rho^{q-1}$ to (1.1), we have

$\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{q}\,dx+\frac{4mq(q-1)}{(q+m-1)^{2}}\int_{\mathbb{R}^{d}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx=q\int_{\mathbb{R}^{d}}V\rho\nabla\rho^{q-1}\,dx=:\mathcal{J}.$
(4.45)

$(i)$ In case $q\geq m-1$, we observe that

$\mathcal{J}=q\int_{\mathbb{R}^{d}}V\rho\nabla\rho^{q-1}\,dx=\frac{2(q-1)}{q+m-1}\int_{\mathbb{R}^{d}}V\rho^{\frac{q-m+1}{2}}\nabla\rho^{\frac{q+m-1}{2}}\,dx.$
(4.46)

First, let $q>m-1$. Then by Hölder inequality with
$\frac{1}{q_{1}}+\frac{1}{2}+\frac{q_{1}-2}{2q_{1}}=1$ for $q_{1}>2$, we have

$\mathcal{J}\leq
c\|V\|_{L^{q_{1}}_{x}}\left\|\nabla\rho^{\frac{m+q-1}{2}}\right\|_{L^{2}_{x}}\|\rho\|_{L_{x}^{\frac{q_{1}(q-m+1)}{q_{1}-2}}}^{\frac{q-m+1}{2}}.$

Furthermore, when $q=m-1$, we have

$\mathcal{J}=c\int_{\mathbb{R}^{d}}V\nabla\rho^{\frac{q+m-1}{2}}\,dx\leq
c\|V\|_{L^{2}_{x}}\left\|\nabla\rho^{\frac{m+q-1}{2}}\right\|_{L^{2}_{x}}\leq\varepsilon\left\|\nabla\rho^{\frac{m+q-1}{2}}\right\|_{L^{2}_{x}}^{2}+c(\varepsilon)\|V\|_{L^{2}_{x}}^{2}.$

For simplicity, let

$r_{1}:=\frac{q_{1}(q-m+1)}{q_{1}-2},\quad\text{and}\quad\theta:=\frac{q}{r_{1}}\left(1-\frac{(r_{1}-q)(d-2)}{2q+d(m-1)}\right).$

Then by (3.2) in Lemma 3.4, we see that

$\begin{cases}2<\frac{d(q+m-1)}{q+(d-1)(m-1)}\leq
q_{1}\leq\frac{2q}{m-1},&\text{ if }d>2,\vspace{1 mm}\\\
2<q_{1}\leq\frac{2q}{m-1},&\text{ if }d=2.\end{cases}$ (4.47)

Then by (3.4) and (3.5) in Lemma 3.4, we deduce

$\mathcal{J}\leq
c\|V\|_{L^{q_{1}}_{x}}\left\|\nabla\rho^{\frac{m+q-1}{2}}\right\|_{L^{2}_{x}}^{1+\frac{(1-\theta)(q-m+1)}{q+m-1}}\|\rho\|_{L_{x}^{q}}^{\frac{\theta(q-m+1)}{2}}\leq\varepsilon\left\|\nabla\rho^{\frac{m+q-1}{2}}\right\|_{L^{2}_{x}}^{2}+c(\varepsilon)\|V\|_{L^{q_{1}}_{x}}^{q_{2}}\|\rho\|_{L_{x}^{q}}^{q_{2}\frac{\theta(q-m+1)}{2}},$
(4.48)

by taking Young’s inequality. Due to $V\in\mathcal{S}_{m,q}^{(q_{1},q_{2})}$,
it gives

$\frac{2(q+m-1)}{(q+m-1)-(1-\theta)(q-m+1)}=\frac{q_{1}(2+q_{m,d})}{q_{1}(1+q_{m,d})-d}=q_{2}.$

The conditions (4.47) corrresponds to (4.41).

Now the combination of (4.45) and (4.48) with
$\varepsilon=\frac{4mq(q-1)}{(q+m-1)^{2}}$ yields that

$\frac{d}{dt}\int_{\mathbb{R}^{d}}\rho^{q}\,dx\leq
c\|V\|_{L^{q_{1}}_{x}}^{q_{2}}\|\rho\|_{L_{x}^{q}}^{q_{2}\frac{\theta(q-m+1)}{2}}\quad\Longrightarrow\quad\frac{d}{dt}\left(\int_{\mathbb{R}^{d}}\rho^{q}\,dx\right)^{1-\sigma}\leq
c\|V\|^{q_{2}}_{L^{q_{1}}_{x}},$

because $\sigma:=\frac{\theta q_{2}(q-m+1)}{2q}=\frac{\theta
q_{2}(q_{1}-2)}{2qq_{1}}<1$. By taking the integration in terms of $t$, and
taking power by $\frac{1}{1-\sigma}$, we obtain the following :

$\sup_{0\leq t\leq T}\int_{\mathbb{R}^{d}}\rho^{q}(t)\,dx\leq
c\int_{\mathbb{R}^{d}}\rho_{0}^{q}\,dx+c\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}^{c}.$
(4.49)

We combine (4.45) and (4.48) with $\varepsilon=\frac{2mq(q-1)}{(q+m-q)^{2}}$
and take the integration in terms of $t$, which yields

$\displaystyle\sup_{0\leq t\leq T}\int_{\mathbb{R}^{d}}\rho^{q}\,dx$
$\displaystyle+\frac{2mq(q-1)}{(q+m-1)^{2}}\iint_{Q_{T}}\left|\nabla\rho^{\frac{q+m-1}{2}}\right|^{2}\,dx\,dt$
(4.50)
$\displaystyle\leq\int_{\mathbb{R}^{d}}\rho_{0}^{q}\,dx+c\|V\|_{\mathcal{S}_{m,q}^{(q_{1},q_{2})}}\sup_{0\leq
t\leq T}\|\rho(\cdot,t)\|_{L_{x}^{q}}^{q_{2}\frac{\theta(q-m+1)}{2}}.$

Therefore, we obtain (4.42) by substituting (4.49) into (4.50).

$(ii)$ Here, we take the integration by parts that gives:

$\mathcal{J}=q\int_{\mathbb{R}^{d}}V\rho\nabla\rho^{q-1}\,dx=(q-1)\int_{\mathbb{R}^{d}}V\nabla\rho^{q}\,dx=-(q-1)\int_{\mathbb{R}^{d}}\nabla
V\rho^{q}\,dx.$ (4.51)

By taking Hölder inequality with
$\frac{1}{\tilde{q}_{1}}+\frac{\tilde{q}_{1}-1}{\tilde{q}_{1}}=1$ for
$\tilde{q}_{1}>1$, we have

$\mathcal{J}\leq c\|\nabla
V\|_{L_{x}^{\tilde{q}_{1}}}\|\rho\|_{L^{\frac{q\tilde{q}_{1}}{\tilde{q}_{1}-1}}_{x}}^{q}.$

For simplicity, let

$\tilde{r}_{1}:=\frac{q\tilde{q}_{1}}{\tilde{q}_{1}-1}\quad\text{ and
}\quad\theta:=\frac{q}{\tilde{r}_{1}}\left(1-\frac{(\tilde{r}_{1}-q)(d-2)}{2q+d(m-1)}\right).$

Then by (3.2) in Lemma 3.4 for $\tilde{r}_{1}$ instead of $r_{1}$, it gives

$\begin{cases}1<\frac{d+q_{m,d}}{2+q_{m,d}}\leq\tilde{q}_{1}\leq\infty,&\text{
if }d>2,\vspace{1 mm}\\\ 1<\tilde{q}_{1}\leq\infty,&\text{ if
}d=2.\end{cases}$ (4.52)

Then by (3.4) and (3.5) in Lemma 3.4, we deduce

$\mathcal{J}\leq c\|\nabla
V\|_{L_{x}^{\tilde{q}_{1}}}\|\rho\|_{L^{q}_{x}}^{q\theta}\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x}}^{\frac{2q(1-\theta)}{q+m-1}}\leq\varepsilon\left\|\nabla\rho^{\frac{q+m-1}{2}}\right\|_{L^{2}_{x}}^{2}+c(\varepsilon)\|\nabla
V\|_{L_{x}^{\tilde{q}_{1}}}^{\tilde{q}_{2}}\|\rho\|_{L^{q}_{x}}^{q\theta\tilde{q}_{2}},$
(4.53)

by taking Young’s inequality. Because of
$V\in\tilde{\mathcal{S}}_{m,q}^{(\tilde{q}_{1},\tilde{q}_{2})}$, it gives

$\frac{q+m-1}{(q+m-1)-(1-\theta)q}=\frac{\tilde{q}_{1}(2+q_{m,d})}{\tilde{q}_{1}(2+q_{m,d})-d}=\tilde{q}_{2}.$

Also, we observe that (4.52) corresponds to (4.43).

Now we combine (4.45) and (4.53) with
$\varepsilon=\frac{4mq(q-1)}{(q-m+1)^{2}}$ which yields

$\sup_{0\leq t\leq T}\int_{\mathbb{R}^{d}}\rho^{q}(\cdot,t)\,dx\leq c\|\nabla