# On the stabilizing effect of swimming in an active suspension

Dallas Albritton  School of Mathematics, Institute for Advanced Study, 1
Einstein Dr., Princeton, NJ 08540, USA<EMAIL_ADDRESS>and  Laurel
Ohm  Department of Mathematics, Princeton University, Princeton, NJ 08540, USA
<EMAIL_ADDRESS>
###### Abstract.

We consider a kinetic model of an active suspension of rod-like microswimmers.
In certain regimes, swimming has a stabilizing effect on the suspension. We
quantify this effect near homogeneous isotropic equilibria
$\overline{\psi}=\text{const}$. Notably, in the absence of particle
(translational and orientational) diffusion, swimming is the only stabilizing
mechanism. On the torus, in the non-diffusive regime, we demonstrate linear
Landau damping up to the stability threshold predicted in the applied
literature. With small diffusion, we demonstrate nonlinear stability of
arbitrary equilibrium values for pullers (rear-actuated swimmers) and enhanced
dissipation for both pullers and pushers (front-actuated swimmers) at small
concentrations. On the whole space, we prove nonlinear stability of the vacuum
equilibrium due to generalized Taylor dispersion.

###### Contents

  1. 1 Introduction
  2. 2 Landau damping
  3. 3 Taylor dispersion
  4. 4 Enhanced dissipation
  5. A Nondimensionalization
  6. B Strong solution theory
  7. C Proof of Lemma 4.3

##  1\. Introduction

We consider a kinetic model of a dilute suspension of rod-like microswimmers
proposed by Saintillan and Shelley [45, 44] and, independently, Subramanian
and Koch [51]. The number density $\psi(\bm{x},\bm{p},t)$ of swimmers with
center-of-mass position $\bm{x}\in\Omega^{d}$ and orientation $\bm{p}\in
S^{d-1}$, $d=2,3$, evolves according to a Smoluchowski equation:

$\partial_{t}\psi+\bm{p}\cdot\nabla_{x}\psi+\bm{u}\cdot\nabla_{x}\psi+\operatorname{div}_{p}\left[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla\bm{u}\bm{p})\psi\right]=\nu\Delta_{p}\psi+\kappa\Delta_{x}\psi\,,$
(1.1)

where $\operatorname{div}_{p}$ and $\Delta_{p}$ denote the divergence and
Laplacian, respectively, on $S^{d-1}$. We consider the domains
$\Omega^{d}=\mathbb{T}^{d}:=\mathbb{R}^{d}/(2\pi\mathbb{Z})^{d}$ and
$\Omega^{d}=\mathbb{R}^{d}$. The velocity field $\bm{u}(\bm{x},t)$ of the
surrounding fluid medium satisfies the Stokes equations forced by the
divergence of an active stress $\bm{\Sigma}$, which measures the local
alignment of swimmers:

$\displaystyle-\Delta\bm{u}+\nabla q$
$\displaystyle=\operatorname{div}\bm{\Sigma},\quad\operatorname{div}\bm{u}=0\,,$
(1.2) $\displaystyle\bm{\Sigma}(\bm{x},t)$
$\displaystyle=\iota\int_{S^{d-1}}\psi(\bm{x},\bm{p},t)\,\bm{p}\otimes\bm{p}\,d\bm{p}\,,\quad\iota\in\\{\pm\\}\,.$
(1.3)

The coefficient $\iota$ in the active stress $\bm{\Sigma}$ corresponds to the
sign of the force dipole $\iota(\bm{p}\cdot\nabla_{x})\delta\bm{p}$ exerted by
each swimmer on the fluid and depends on the swimming mechanism: $\iota=+$ for
_pullers_ (front-actuated swimmers), such as _E. coli_ , and $\iota=-$ for
_pushers_ (rear-actuated swimmers), such as _C. reinhardtii_.

It is instructive to rewrite (1.1) schematically as

$\partial_{t}\psi+\operatorname{div}_{x}(\dot{\bm{x}}\psi)+\operatorname{div}_{p}(\dot{\bm{p}}\psi)=\nu\Delta_{p}\psi+\kappa\Delta_{x}\psi\,,$
(1.4)

where the fluxes

$\dot{\bm{x}}=\bm{p}+\bm{u}\,,\quad\dot{\bm{p}}=(\mathbf{I}-\bm{p}\otimes\bm{p})\nabla\bm{u}\bm{p}$
(1.5)

signify that (i) particles swim in the direction of their orientation, (ii)
particles are advected by the surrounding fluid flow, and (iii) particles’
orientations evolve according to a Jeffery term [28] for the rotational
dynamics of an elongated particle in Stokes flow. The terms on the right-hand
side of (1.1) capture the orientational and center-of-mass diffusion,
respectively, with coefficients $0<\nu,\kappa\ll 1$. We refer to [48] for a
detailed derivation of the model, and we discuss our particular
nondimensionalization in Appendix A.

The model (1.1)-(1.3) may be considered as a minimal model in which to study
the large-scale flows generated by bacterial activity as seen in experiments.
(See [48] for comparisons between experimental, computational, and theoretical
results.) A key difference between this model and, for example, the
Doi–Edwards model for passive polymers [21, 22] is the presence of the
swimming term $\bm{p}\cdot\nabla_{x}\psi$ in (1.1). This term seems to be
necessary for the concentration fluctuations characteristic of certain
bacterial suspensions [45], but its role is quite complex. The simplest
setting in which to study the effects of swimming is near the uniform
isotropic equilibrium $\psi\equiv\text{constant }\overline{\psi}$. _What role
does swimming play in stabilizing the uniform isotropic equilibrium?_

In this paper, we identify and quantify three near-equilibrium effects of
swimming. On $\mathbb{T}^{d}$, the value of the constant $\overline{\psi}$ is
a free parameter (see Appendix A for our nondimensionalization of the model)
and plays an important role in our results. Furthermore, the expected behavior
of the system depends on whether we consider pullers or pushers in (1.3).
Defining the relative conformational entropy
$\mathcal{S}(t)=\int_{x,p}(\psi/\overline{\psi})\log\left(\psi/\overline{\psi}\right)\,d\bm{p}\,d\bm{x}$,
solutions of (1.1)-(1.3) on $\mathbb{T}^{d}$ formally satisfy the following
$H$-theorem:

$\displaystyle\overline{\psi}\frac{d\mathcal{S}}{dt}$ $\displaystyle=-\iota
d\int_{\mathbb{T}^{d}}\left\lvert\nabla\bm{u}\right\rvert^{2}\,d\bm{x}-4\int_{\mathbb{T}^{d}\times
S^{d-1}}\left(\nu|\nabla_{p}\sqrt{\psi}|^{2}+\kappa|\nabla_{x}\sqrt{\psi}|^{2}\right)\,d\bm{p}\,d\bm{x}\,,$
(1.6)

from which we can see that pullers ($\iota=+$) always have decreasing
conformational entropy, whereas pushers ($\iota=-$) may not.

Our main results may be summarized as follows ($d=2,3$ unless stated
otherwise):

  * •

_Landau damping_. Solutions of the linearized inviscid equations
($\nu=\kappa=0$) on $\mathbb{T}^{d}$ decay algebraically due to _phase mixing_
provided that $\iota=+$ or $\overline{\psi}<\overline{\psi}^{*}$, which is
given by a suitable Penrose condition.

  * •

_Taylor dispersion_. The vacuum state $\overline{\psi}=0$ in $\mathbb{R}^{3}$
is nonlinearly stable with respect to small perturbations due to the
dispersive effect of the operator $\bm{p}\cdot\nabla_{x}-\nu\Delta_{p}$. As a
byproduct of the analysis, arbitrarily large puller $(\iota=+)$ equilibria on
$\mathbb{T}^{d}$ are nonlinearly exponentially stable with respect to small
perturbations.

  * •

_Enhanced dissipation_. For small $\overline{\psi}\ll\nu^{-(\frac{1}{2}+)}$,
the equilibrium $\overline{\psi}$ is nonlinearly exponentially stable on
$\mathbb{T}^{d}$ with respect to small perturbations due to the hypocoercive
effect of the operator $\bm{p}\cdot\nabla_{x}-\nu\Delta_{p}$. Nearby solutions
converge to their $\bm{x}$-averages
$\langle\psi\rangle(\bm{p},t):=\int\psi(\bm{x},\bm{p},t)\,d\bm{x}$ in the
_enhancement time_ $O(\nu^{-(\frac{1}{2}+)})$. The $\bm{x}$-averages
$\langle\psi\rangle$ converge to $\overline{\psi}$ in the _diffusive time_
$O(\nu^{-1})$ and are _metastable_.

Our results on stability may be viewed as complementary to the wealth of
computational literature on dynamics in the unstable pusher region, where
perturbations to the uniform isotropic equilibrium can be seen to give rise to
the emergence of collective swimmer motion and large-scale flows [27, 31, 40,
43, 45, 44, 46, 47, 48, 51]. In particular, our results highlight the complex
role of swimming in these collective dynamics. Without swimming, the isotropic
state in pusher suspensions is always unstable for $0\leq\nu,\kappa\ll 1$
[40], and, as we see here, swimming has a clear stabilizing effect. However,
as noted in [45], swimming is also a necessary ingredient for the particle
density fluctuations observed in simulations. The interplay between the
destabilizing active stress (1.3) in the pusher case and stabilizing swimming
is perhaps worthy of further mathematical exploration.

###  1.1. Main results

In this section, we make the above informal assertions precise.

####  1.1.1. Landau damping

We consider the linearized inviscid equations:

$\displaystyle\partial_{t}f+\bm{p}\cdot\nabla_{x}f-d\overline{\psi}\nabla\bm{u}:\bm{p}\otimes\bm{p}$
$\displaystyle=0$ (1.7) $\displaystyle-\Delta\bm{u}+\nabla
q=\operatorname{div}\bm{\Sigma}\,,\quad\operatorname{div}\bm{u}$
$\displaystyle=0\,,$ (1.8)

with active stress

$\bm{\Sigma}[f](\bm{x},t)=\iota\int_{S^{d-1}}\bm{p}\otimes\bm{p}\,f(\bm{x},\bm{p},t)\,d\bm{p}\,,\qquad\iota\in\\{\pm\\}\,.$
(1.9)

Here $\overline{\psi}\geq 0$ is the constant background solution. Solutions
satisfy

$\frac{1}{2}\frac{d}{dt}\lVert f\rVert_{L^{2}_{x,p}}^{2}+\iota
d\overline{\psi}\lVert\nabla\bm{u}\rVert_{L^{2}_{x}}=0\,,$ (1.10)

which is a linearized version of the $H$-theorem (1.6).

To better understand the underlying decay mechanism, we consider
$\overline{\psi}=0$ and perform a mode-by-mode analysis for
$f(\bm{x},\bm{p},t)=h(\bm{p},t)e^{i\bm{k}\cdot\bm{x}}$ and $f^{\rm
in}=f(\cdot,0)$. Without loss of generality, we may take $\bm{k}=k\bm{e}_{1}$
and solve for $h$ as

$h=e^{-ikp_{1}t}h^{\rm in}\,.$ (1.11)

Over time, the solution develops large oscillations and is transferred to
higher and higher frequencies in $\bm{p}$. This increasingly oscillatory
behavior for an isolated spatial mode can be observed in numerical simulations
by Hohenegger and Shelley [27, Figure 3]. Notably, $h$ converges to zero
_weakly_ in $L^{2}$ but not strongly, and the convergence can be quantified in
negative Sobolev spaces via the method of stationary phase, see Section 2.
This is known as _phase mixing_ in kinetic theory.

It is observed in [27, 45, 51] that the linearized operator about large
$\overline{\psi}$ admits unstable eigenvalues in the case of pushers but not
pullers. Away from this pusher instability, we show that the phase mixing
described above persists, causing the velocity field $\bm{u}$ to quickly
decay. We call this phenomenon _Landau damping_ , terminology again borrowed
from kinetic theory.

Letting $f$ denote the solution to the linearized system, we show the
following:

######  Theorem 1.1 (Linear Landau damping).

Let $f^{\rm in}\in L^{2}_{x}H^{d+1}_{p}(\mathbb{T}^{d}\times S^{d-1})$.
Suppose that $\iota=+$ or $\overline{\psi}<\overline{\psi}^{*}$. Then the
velocity field $\bm{u}$ generated by the solution $f\in
C_{t}L^{2}_{x,p}(\mathbb{T}^{d}\times S^{d-1}\times\mathbb{R}_{+})$ to the
linearized PDE satisfies

$\int\|\nabla\bm{u}(\cdot,t)\|_{L^{2}_{x}}^{2}\langle
t\rangle^{d-\varepsilon}\,dt\lesssim_{\overline{\psi},\varepsilon}\|f^{\rm
in}\|_{L^{2}_{x}H^{d+1}_{p}}^{2}\,$ (1.12)

for all $\varepsilon>0$, where $\langle t\rangle=\sqrt{1+t^{2}}$. The solution
$f$ decomposes as

$f=f_{\rm lin}+g\,,$ (1.13)

where $f_{\rm lin}=e^{-\bm{p}\cdot\nabla_{x}t}f^{\rm in}$ satisfies

$\|f_{\rm lin}(\cdot,t)\|_{L^{2}_{x}H^{-(d-1)}_{p}}\lesssim\langle
t\rangle^{-\frac{d-1}{2}}\|f^{\rm in}\|_{L^{2}_{x}H^{d-1}_{p}}\,,$ (1.14)

and
$g=d\overline{\psi}\int_{0}^{t}e^{-\bm{p}\cdot\nabla_{x}(t-s)}(\nabla\bm{u}:\bm{p}\otimes\bm{p})(\cdot,s)\,ds$
satisfies, for all $\varepsilon>0$,

$\int\|g(\cdot,t)\|_{L^{2}_{x}H^{-(d+1)}_{p}}^{2}\langle
t\rangle^{d-\varepsilon}\,dt\lesssim_{\overline{\psi},\varepsilon}\|f^{\rm
in}\|_{L^{2}_{x}H^{d+1}_{p}}^{2}\,.$ (1.15)

In the pusher case ($\iota=-$), the stability threshold $\overline{\psi}^{*}$,
defined in Lemma 2.2, corresponds precisely to the eigenvalue crossing studied
in the computational literature [27, 40, 44, 45, 47, 51]. It arises in the
_Penrose condition_ (terminology borrowed from kinetic theory) on the Volterra
equation for $\nabla\bm{u}$ in the proof of Theorem 1.1.

The standard setting in which to investigate Landau damping is the _Vlasov-
Poisson equation_

$\partial_{t}f+\bm{v}\cdot\nabla_{x}f+\bm{E}\cdot\nabla_{v}f=0\,,$ (1.16)
$\bm{E}(\bm{x},t)=\iota\nabla\Delta^{-1}\varrho\,,\quad\varrho=\int_{v}f\,d\bm{v}\,,\quad\iota\in\\{\pm\\}\,,$
(1.17)

near homogeneous equilibria $\bar{f}(\bm{v})$ (typically radial or
Maxwellian), where $\iota=+$ is the repulsive (ionic) case and $\iota=-$ is
the attractive (gravitational) case. Landau damping in this context has been
thoroughly studied in the PDE community [39, 9, 25, 13]. We will discuss below
certain important differences between our model (1.1)-(1.3) and the Vlasov-
Poisson equations (1.16)-(1.17).

####  1.1.2. Taylor dispersion

For particles swimming with speed $U_{0}$, various sources (see, e.g., [34, p.
282] and [48, p. 326]) predict _generalized Taylor dispersion_ , namely,
effective $\bm{x}$-diffusion
$\left(\kappa+\frac{U_{0}^{2}}{2d\nu}\right)\Delta_{x}$ in the regime
$0<\nu\ll 1$ of weak orientational diffusion. The hallmark of Taylor
dispersion [23, 2, 36, 54] is the inverse dependence of the effective
viscosity coefficient on $\nu$.

We begin by extracting the Taylor dispersion effect in the linearized PDE with
$\overline{\psi}=0$:

$\partial_{t}f+\bm{p}\cdot\nabla_{x}f=\nu\Delta_{p}f\,.$ (1.18)

If $f$ solves the above PDE, then $g=e^{\kappa\Delta_{x}t}f$ solves the
analogous PDE with additional diffusion $\kappa\Delta_{x}$, so there is no
loss of generality in setting $\kappa=0$ at the linear level. (This is a
common feature of the linearized operators in this paper.)

Let $f=h(\bm{p},t)e^{i\bm{k}\cdot\bm{x}}$ and $k=|\bm{k}|$. In Theorem 3.1
($\nu\leq k$) and Theorem 4.1 ($\nu\geq k$), we prove

$\|h\|_{L^{2}}\leq e^{-c_{0}\lambda_{\nu,k}t}\|h^{\rm in}\|_{L^{2}}\,,$ (1.19)

where

$\lambda_{\nu,k}=\begin{cases}\frac{k^{2}}{\nu}&k\leq\nu\\\
\frac{\nu^{1/2}k^{1/2}}{1+\left\lvert\log(\nu/k)\right\rvert}&k\geq\nu\,.\end{cases}$
(1.20)

The inverse dependence of $\lambda_{\nu,k}$ on $\nu$ is seen in low modes
$k\leq\nu$. However, as is characteristic of Taylor dispersion, this inverse
dependence is not “seen” until the diffusive time $O(\nu^{-1})$, that is, once
the $k\geq\nu$ modes have been damped. The particular $\nu^{1/2}$ dependence
of $\lambda_{\nu,k}$ is sometimes known as _enhanced dissipation_ , and we
explain it in the next section.

At the nonlinear level, the dispersive effect111Although this is called Taylor
dispersion, mathematically the operator $\bm{p}\cdot\nabla_{x}-\nu\Delta_{p}$
would not typically be called dispersive. In low modes, it creates a diffusive
(parabolic) operator. allows us to prove

######  Theorem 1.2 (Nonlinear stability of vacuum in $\mathbb{R}^{3}$).

Let $d=3$, $\nu\in(0,1]$, and $\kappa>0$. Let $0\leq f^{\rm in}\in
L^{1}_{x}L^{2}_{p}\cap H^{2}_{x}L^{2}_{p}$. If
$0<\varepsilon_{0}\ll\min(\nu,\kappa)$ and222Throughout the paper, $a\ll b$ is
used in hypotheses to mean that there exists an absolute constant $m_{0}$ such
that $a\leq m_{0}b$.

$\varepsilon:=\|f^{\rm in}\|_{(L^{1}_{x}\cap
H^{2}_{x})L^{2}_{p}}\leq\varepsilon_{0}\,,$ (1.21)

then the strong solution $f$ exists globally in time and satisfies the decay
estimate

$\|f(\cdot,t)\|_{L^{\infty}_{x}L^{2}_{p}}+\|f(\cdot,t)\|_{(\dot{H}^{d/2}\cap\dot{H}^{2})_{x}L^{2}_{p}}\lesssim
r(\nu t)\varepsilon\,,$ (1.22)

where

$r(s)=\langle s\rangle^{-\frac{d}{2}}(\log(2+s))^{2}\,.$ (1.23)

For completeness, we develop a basic theory of strong solutions in Appendix B,
see specifically Definition B.1 and Theorem B.2.

Note that Theorem 1.2 holds for both pullers and pushers near equilibrium,
which is consistent with the observation of Saintillan–Shelley [43, 46] in
many-particle simulations that the diffusivity of individual swimmers is not
dependent on the emergence of large-scale flows in the suspension. This may be
contrasted with enhanced diffusion of tracer particles in swimmer suspensions,
a well-studied phenomenon experimentally, which does rely on collective motion
of the swimmers and is different from the effect studied here.

Our proof of the linearized estimates (see Theorem 3.1) is based on a well-
chosen energy estimate for the solution and two moments. It is inspired by
[12, Section 3], which is itself inspired by work of Guo on collisional
kinetic equations, e.g., [26]. These energy estimates capture the Taylor
dispersion effect (though not enhanced dissipation for modes $k\geq\nu$) in
the puller setting for _arbitrary_ $\overline{\psi}\geq 0$ (that is, the non-
local term may be non-perturbative). As a consequence, we show

######  Theorem 1.3 (Nonlinear stability of puller equilibrium on
$\mathbb{T}^{d}$).

Let $\nu\leq 1$, $\kappa>0$, $\iota=+$, and $\overline{\psi}\geq 0$. Let
$f^{\rm in}\in H^{2}_{x}L^{2}_{p}$ with $\int f^{\rm in}\,d\bm{x}\,d\bm{p}=0$.
If $0<\varepsilon_{0}\ll\min(\nu,\kappa)$ and

$\varepsilon:=\|f^{\rm in}\|_{H^{2}_{x}L^{2}_{p}}\leq\varepsilon_{0}\,,$
(1.24)

then the strong solution $\psi=\overline{\psi}+f$ exists globally in time and
the perturbation $f$ satisfies the decay estimate

$\|f(\cdot,t)\|_{H^{2}_{x}L^{2}_{p}}\lesssim e^{-\delta_{0}\nu t}\varepsilon$
(1.25)

for some $\delta_{0}>0$.

Notably, there is no size restriction on $\overline{\psi}$.

####  1.1.3. Enhanced dissipation

We now address the $k\geq\nu$ regime in (1.19)-(1.20). Define
$\lambda_{\nu}=\lambda_{\nu,1}$ in (1.20). If we consider (1.18) on
$\mathbb{T}^{d}$, then the minimal non-zero wavenumber is $k=1$, and we see
that the modes with $k\geq 1$ are damped in the enhanced dissipation time
$O(\lambda_{\nu}^{-1})$, logarithmically slower than $O(\nu^{-1/2})$, when
$\nu\leq 1$. The decay of the $k=0$ mode
$f_{0}(\bm{p},t):=\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{x}f(\bm{x},\bm{p},t)\,d\bm{x}$ is not enhanced, as
it solves the heat equation:

$\partial_{t}f_{0}-\nu\Delta_{p}f_{0}=0\,.$ (1.26)

To extract the enhanced dissipation timescale, we define $f_{\neq}:=f-f_{0}$.

For sufficiently small $\overline{\psi}$, we demonstrate that perturbations to
the equilibrium $\overline{\psi}$ exhibit nonlinear enhanced dissipation in
the case of both pullers and pushers.

######  Theorem 1.4 (Nonlinear enhanced dissipation for small concentrations).

Let
$\nu,\kappa>0$, $0\leq f^{\rm in}\in H^{2}_{x}L^{2}_{p}$, and
$\overline{\psi}\ll\lambda_{\nu}^{-1}$. If
$0<\varepsilon_{0}\ll\min(\kappa^{1/2},\nu^{1/2})\lambda_{\nu}^{1/2}$ and

$\varepsilon:=\lVert f_{\neq}^{\rm
in}\rVert_{H^{2}_{x}L^{2}_{p}}\leq\varepsilon_{0}\quad\text{and}\quad\lVert
f^{\rm in}_{0}\rVert_{L^{2}_{p}}\leq\varepsilon_{0}\,,$ (1.27)

then the nonzero modes of the strong solution $\psi=\overline{\psi}+f$ to
(4.47) satisfy the enhanced decay rate

$\lVert f_{\neq}(\cdot,t)\rVert_{H^{2}_{x}L^{2}_{p}}\lesssim
e^{-\delta_{\neq}\lambda_{\nu}t}\varepsilon$ (1.28)

for some $\delta_{\neq}>0$. Furthermore, the zero mode satisfies the bound

$\lVert f_{0}\rVert_{L^{2}_{p}}\lesssim e^{-\delta_{0}\nu t}\bigg{(}\lVert
f^{\rm in}_{0}\rVert_{L^{2}_{p}}+\nu^{-1}\varepsilon^{2}\bigg{)}$ (1.29)

for some $\delta_{0}>0$.

Note that the notion of “small” $\overline{\psi}$ here is larger than might be
immediately expected for a perturbative result: $\lambda_{\nu}$ rather than
$\nu$.

We highlight a useful heuristic regarding the enhancement timescale
$\nu^{-1/2}$. Consider the PDE (1.18) in dimension $d=2$ and, writing
$\bm{p}=\cos\theta\bm{e}_{1}+\sin\theta\bm{e}_{2}$, consider a plane wave
solution (note the time rescaling)

$f(\bm{x},\bm{p},t)=h(\theta,s)e^{i\bm{k}\cdot\bm{x}},\quad\bm{k}=k\bm{e}_{1},\quad
s=kt\,.$ (1.30)

Then $h$ satisfies

$\partial_{s}h+i\cos\theta h-\frac{\nu}{k}\partial_{\theta}^{2}h=0\,.$ (1.31)

Heuristically, the evolution $\partial_{s}h+i\cos\theta h=0$, which dominates
for short times, creates large gradients in $\theta$. This can be seen from
applying $\partial_{\theta}$ to the explicit formula $h=e^{-i\cos\theta
s}h^{\rm in}$. Once these gradients are sufficiently large, the evolution
$\partial_{s}h-\nu\partial_{\theta}^{2}h/k=0$ smooths them; this is the source
of the “enhanced dissipation.”

The enhancement is slowest near $\partial_{\theta}\cos\theta=0$, where the
“shearing” caused by $i\cos\theta$ is slowest. We examine the critical point
$\theta=0$ more closely. The characteristic scales at $\theta=0$ are
$\widehat{\theta}=(\nu/k)^{1/4}$ and $\widehat{s}=(\nu/k)^{-1/2}$, in the
sense that if we “zoom in” at $\theta=0$ by writing
$H(\Theta,S)=e^{is}h(\theta,s)$, where $\Theta=\theta/\widehat{\theta}$ and
$S=s/\widehat{s}$, we see the following leading order behavior in $\nu/k$:

$\partial_{S}H+\frac{i}{2}\Theta^{2}H-\partial_{\Theta}^{2}H\approx 0\,.$
(1.32)

We note a strong analogy between (1.32) and the evolution of a passive scalar
in a two-dimensional shear flow $\bm{u}=(b(y),0)$:

$\partial_{t}f+b(y)\partial_{x}f-\nu\Delta_{x,y}f=0\text{ on
}\mathbb{T}^{2}\times\mathbb{R}_{+}\,.$ (1.33)

Around a critical point $y_{0}$ at which
$b^{\prime}(y_{0})=\cdots=b^{(N)}(y_{0})=0$ but $b^{(N+1)}(y_{0})\neq 0$, a
_shear layer_ where the scalar $f$ is dissipated more slowly becomes readily
visible in simulations; its characteristic length scale in $y$ is
$\nu^{1/(N+3)}$, and its characteristic “enhancement” timescale is
$\nu^{-(N+1)/(N+3)}$, which is known to be sharp [19]. Our setting corresponds
to $N=1$, which is akin to Poiseuille flow, and the PDE (1.32) is
approximately satisfied inside our “shear layer” at $\theta=0$.

Finally, we mention briefly the techniques involved in Theorem 1.4. First, in
Theorem 4.1, we prove linear enhanced dissipation for $\overline{\psi}=0$ via
the hypocoercivity method (see, for example, [6, 7, 24] in the context of
shear flows (1.33)). The hypocoercivity method has the advantage of being
elementary, although without adjustment it produces a logarithmic loss in the
exponent $\lambda_{\nu}$ compared to $\nu^{1/2}$; this is technical in nature,
and there are methods to remove it [57, 56, 1]. To prove the nonlinear
enhanced dissipation in Theorem 1.4, we rely crucially on the structure of the
nonlinear terms. On the whole, our approach is partially inspired by [20].

###  1.2. Comparison with existing literature

####  1.2.1. Classical kinetic theory

While the kinetic model has similarities with the Vlasov-Poisson equation
(1.16) and its collisional cousins, a key difference is that the phase mixing
effect of the swimming term $\bm{p}\cdot\nabla_{x}$ is not as strong as that
of the free-streaming term $\bm{v}\cdot\nabla_{x}$. For example, solutions of
$e^{-it\bm{v}\cdot\nabla_{x}}$ on the torus decay exponentially to their mean-
in-$\bm{x}$, provided the initial data is analytic-in-$\bm{v}$. Meanwhile,
solutions of $e^{-it\bm{p}\cdot\nabla_{x}}$ decay only polynomially, as
$t^{-\frac{d-1}{2}}$, which is not even time-integrable. This is the
difference between non-stationary versus stationary phase, see Lemma 2.1.
Luckily, the particular structure of the nonlocal term allows us to improve
the decay rate of $\nabla\bm{u}$ by a factor of $t^{-1}$. The enhanced
dissipation rates are different as well: $e^{-c\nu^{1/2}t}$ rather than
$e^{-c\nu^{1/3}t}$ observed in, e.g., [12]. Finally, while the nonlocal term
$\nabla_{x}\bm{u}:\bm{p}\otimes\bm{p}$ in the linearization might be compared
to $\bm{E}\cdot\nabla_{v}\bar{f}$ in (1.16), there is no general theory to
handle these nonlocal terms, and each is treated separately.

####  1.2.2. Complex fluids

Although there is a vast PDE literature on related models without swimming
(see the survey [37]), to the authors’ knowledge, there is a single PDE work
on the model (1.1)-(1.3). In [15], Chen and Liu establish the existence of
global weak entropy solutions to (1.1) coupled with the Navier-Stokes
equations or the Stokes equations (1.2)-(1.3) for the velocity. Notably, in
dimension two, the weak solutions they construct in the Stokes setting are
_unique_. The regularity theory of (1.1)-(1.3) is complicated by the absence
of a maximum principle for the density $\varrho$.

The system (1.1)-(1.3) belongs to a broader class of micro-macro models
commonly used to describe passive immersed polymers. Such models couple a
microscopic description of the immersed particles (in the form of a Fokker-
Planck equation which depends on a configuration variable $\bm{p}$) to a
macroscopic description of the suspension in the form of a forced Stokes or
Navier-Stokes equation depending only on $\bm{x}$.

Different microscopic descriptions lead to two main types of models. The first
treats the polymers as elastic dumbbells (two beads attached by a spring). We
refer to [33, 37] for an overview of well-posedness results here. In the
second class of models (sometimes called Doi-type), to which the system
(1.1)-(1.3) belongs, the immersed polymers are treated as rigid rods. The many
existence and uniqueness results for this class of models include [3, 14, 16,
17, 18, 32, 35, 38, 41, 49, 53, 58]. Note that no swimming is included in
these previous results.

Finally, we mention two works which do incorporate swimming. Jiang et al. [29]
provide a proof of local well-posedness for a microscopic “self-organized
kinetic” model coupled with Navier-Stokes and rigorously justify the
hydrodynamic limit to a macroscopic closure model. Further related work on
swimmers includes Kanzler and Schmeiser [30], who consider a kinetic transport
model for myxobacteria in which the particles interact via collisions rather
than through a surrounding fluid medium. They show existence, uniqueness, and
decay to equilibrium for sufficiently large particle diffusivity using
hypocoercivity.

###  1.3. Future directions

The model (1.1)-(1.3) is in its early stages of development from a rigorous
PDE perspective, and many questions remain. Below, we focus only on questions
concerning the near-equilibrium behavior:

####  1.3.1. Nonlinear Landau damping

A natural but challenging question is whether solutions exhibit _nonlinear_
Landau damping in the absence of dissipation. A comparison with previous works
on nonlinear Landau damping in the Vlasov-Poisson equation [39, 10, 25] and
inviscid damping in two-dimensional Euler near the Couette flow [8] indicate
the potential difficulty of this question, especially since arbitrarily fast
polynomial decay to equilibrium is not anticipated.

####  1.3.2. Precise description of the Taylor dispersion

It would be desirable to capture more precise asymptotics of the Taylor
dispersion rather than the upper bound we prove in Theorem 3.1. We expect this
to be possible by homogenization or, in a different vein, the center manifold
approach in [5]. There may be additional difficulties in capturing this at the
nonlinear level. Furthermore, it would be interesting to prove the correct
analogue of Theorem 1.2 on $\mathbb{R}^{2}$. In this context, the term
$\bm{u}\cdot\nabla_{x}\psi$ is “critical” and could modify the leading order
asymptotics.

####  1.3.3. Stable-in-$\nu$ Landau damping and the viscous Penrose condition

Our expectation is that the Landau damping and enhanced dissipation phenomena
persist for $0<\nu\ll 1$ provided that the Penrose condition is satisfied.
More specifically, we expect that whenever $\iota=+$ or
$\overline{\psi}<\overline{\psi}^{*}$, then for sufficiently small
$\nu\ll_{\overline{\psi}}1$, the linearized equations exhibit Landau damping
and enhanced dissipation. See [13] and [11] for analogous theorems in the
Vlasov–Poisson–Landau and two-dimensional Navier-Stokes settings. In our
setting, the key difficulty seems to be to perturb the Penrose condition to
positive viscosity. For this purpose, one requires Landau damping estimates
for the viscous operator $\partial_{t}f+\bm{p}\cdot\nabla_{x}f-\nu\Delta_{p}f$
which appear to be unknown. This corresponds to stable-in-$\nu$ mixing
estimates for shear flows with critical points.

####  1.3.4. Zero translational diffusion

We anticipate that much of our analysis can be generalized to $\kappa=0$, in
which case the model is quasilinear-in-$\bm{x}$. One approach would be to
incorporate a term $\overline{\bm{u}}\cdot\nabla_{x}$ into the Guo and
hypocoercivity schemes, where $\overline{\bm{u}}$ is a known function.

##  2\. Landau damping

In this section, we prove the linear Landau damping result of Theorem 1.1. We
consider the linearized inviscid equations (1.7)-(1.9) and perform a mode-by-
mode analysis in $\bm{x}$. Writing
$f(\bm{x},\bm{p},t)=h(\bm{p},t)e^{i\bm{k}\cdot\bm{x}}$,
$\bm{k}\in\mathbb{R}^{d}\setminus\\{0\\}$, we have that
$\bm{u}=\widehat{\bm{u}}_{k}e^{i\bm{k}\cdot\bm{x}}$, where

$\widehat{\bm{u}}_{k}=i\left\lvert\bm{k}\right\rvert^{-1}({\bf
I}-\overline{\bm{k}}\otimes\overline{\bm{k}})\widehat{\bm{\Sigma}}_{k}\overline{\bm{k}}\,.$
(2.1)

Here $\overline{\bm{k}}=\bm{k}/|\bm{k}|$,
$\bm{\Sigma}=\widehat{\bm{\Sigma}}_{k}e^{i\bm{k}\cdot\bm{x}}$, and

$\widehat{\nabla\bm{u}}_{k}=-({\bf
I}-\overline{\bm{k}}\otimes\overline{\bm{k}})\widehat{\bm{\Sigma}}_{k}\overline{\bm{k}}\otimes\overline{\bm{k}}\,.$
(2.2)

Upon rotating, we may take $\bm{k}=k\bm{e}_{1}$, and upon rescaling time and
replacing $\overline{\psi}$ by $\overline{\psi}/k$, we may consider
$\bm{k}=\bm{e}_{1}$ without loss of generality. With these simplifications,
observe that $\widehat{\nabla\bm{u}}$ (we omit the subscript $k$) has the
following structure:

$\widehat{\nabla\bm{u}}=-({\bf
I}-\bm{e}_{1}\otimes\bm{e}_{1})\widehat{\bm{\Sigma}}\,\bm{e}_{1}\otimes\bm{e}_{1}=-\begin{pmatrix}0&0&\cdots&0\\\
\Sigma_{21}&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\Sigma_{d1}&0&\cdots&0\\\ \end{pmatrix}\,.$ (2.3)

In particular, the matrix $\widehat{\nabla\bm{u}}$ consists of nonzero entries
only in the first column, with zero first row. The structure of this matrix,
particularly the zero first entry, will be very important for achieving the
decay rate in Theorem 1.1.

The proof of Theorem 1.1 thus reduces to studying decay properties of the
following equation for $h(\bm{p},t)$:

$\partial_{t}h+ip_{1}h-d\overline{\psi}\widehat{\nabla\bm{u}}:\bm{p}\otimes\bm{p}=0\,.$
(2.4)

We begin by quantifying the decay when $\overline{\psi}=0$. We study the free
transport equation

$\partial_{t}h+ip_{1}h=0\,,\quad h(\cdot,0)=h^{\rm in}\,,$ (2.5)

whose solution

$h=e^{-ip_{1}t}h^{\rm in}$ (2.6)

decays due to phase mixing, which is measured in a negative Sobolev norm.

######  Lemma 2.1 (Stationary phase estimates).

(i) If $h^{\rm in}\in H^{d-1}(S^{d-1})$ and $h$ is defined by (2.6), then

$\|h(\cdot,t)\|_{H^{-(d-1)}}\lesssim\langle t\rangle^{-\frac{d-1}{2}}\|h^{\rm
in}\|_{H^{d-1}}\,.$ (2.7)

(ii) If moreover $h^{\rm in}\in H^{d+1}(S^{d-1})$ and $h^{\rm
in}(\bm{e}_{1})=h^{\rm in}(-\bm{e}_{1})=0$, then

$\|h(\cdot,t)\|_{H^{-(d+1)}}\lesssim\langle
t\rangle^{-\frac{d-1}{2}-1}\|h^{\rm in}\|_{H^{d+1}}\,.$ (2.8)

Here $\langle t\rangle=\sqrt{1+t^{2}}$ is the Japanese bracket.

Recall that

$\lVert h\rVert_{H^{k}}^{2}:=\lVert
h\rVert_{L^{2}}^{2}+\lVert\nabla^{k}h\rVert_{L^{2}}^{2}$ (2.9)

and

$\lVert h\rVert_{H^{-k}}:=\sup_{\lVert g\rVert_{H^{k}}=1}\left\lvert\int
hg\right\rvert\,,\quad k\geq 0\,.$ (2.10)

###### Proof.

Our goal is to estimate the oscillatory integral

$\int_{S^{d-1}}e^{-ip_{1}t}h^{\rm in}\phi\,d\bm{p}\,,$ (2.11)

where $\phi\in H^{m}(S^{d-1})$ is arbitrary and $m\in\\{d-1,d+1\\}$ depending
on whether we are proving (i) or (ii). Then $h^{\rm in}\phi\in
W^{m,1}(S^{d-1})$ and

$\|h^{\rm in}\phi\|_{W^{m,1}(S^{d-1})}\lesssim\|h^{\rm
in}\|_{H^{m}(S^{d-1})}\|\phi\|_{H^{m}(S^{d-1})}\,.$ (2.12)

If $m=d+1$ and $h^{\rm in}$ vanishes at $\bm{e}_{1}$ and $-\bm{e}_{1}$, then
the function $h^{\rm in}\phi$ also vanishes at $\bm{e}_{1}$ and $-\bm{e}_{1}$.
The desired estimates follow from the method of stationary phase, which
describes the $t$-asymptotics of oscillatory integrals of the type (2.11);
see, for example, [50, Chapter 8]. ∎

We now consider $\overline{\psi}\geq 0$ and seek a closed equation for
$\widehat{\nabla\bm{u}}$.333It would also be natural to study a closed
equation for the active stress $\bm{\Sigma}$. However, to see the improved
decay, e.g., in (2.12), we will need to take advantage of the structure of
$\nabla\bm{u}$.  This type of argument is familiar from kinetic theory; see
[55, Chapter 3] regarding the Vlasov equation. By Duhamel’s formula, we have

$h(\bm{p},t)=e^{-ip_{1}t}h^{\rm
in}(\bm{p})+d\overline{\psi}\int_{0}^{t}e^{-ip_{1}(t-s)}\widehat{\nabla\bm{u}}(s):\bm{p}\otimes\bm{p}\,ds\,.$
(2.13)

Multiplying (2.13) by $\iota\bm{p}\otimes\bm{p}$ and integrating in $\bm{p}$,
by definition of the active stress $\bm{\Sigma}$ (1.9), we obtain

$\widehat{\bm{\Sigma}}[h]=\widehat{\bm{\Sigma}}[e^{-ip_{1}t}h^{\rm in}]+\iota
d\overline{\psi}\int_{0}^{t}\bm{p}\otimes\bm{p}\,e^{-ip_{1}(t-s)}\widehat{\nabla\bm{u}}(s):\bm{p}\otimes\bm{p}\,ds\,.$
(2.14)

Using the definition of $\widehat{\nabla\bm{u}}$ (2.3), we may then multiply
(2.14) on the left by $({\bf I}-\bm{e}_{1}\otimes\bm{e}_{1})$ and on the right
by $\bm{e}_{1}\otimes\bm{e}_{1}$ to obtain a Volterra equation for
$\widehat{\nabla\bm{u}}$:

$\widehat{\nabla\bm{u}}[h]=\widehat{\nabla\bm{u}}[e^{-ip_{1}t}h^{\rm
in}]-\iota
d\overline{\psi}\int_{0}^{t}K(t-s)\,\widehat{\nabla\bm{u}}[h](s)\,ds\,.$
(2.15)

Here, $K(t)$ is an operator-valued kernel which acts on the tensor
$\widehat{\nabla\bm{u}}$ by

$K(t)\widehat{\nabla\bm{u}}=({\bf
I}-\bm{e}_{1}\otimes\bm{e}_{1})\bigg{(}\int_{S^{d-1}}\bm{p}\otimes\bm{p}\,e^{-ip_{1}t}\,\widehat{\nabla\bm{u}}:\bm{p}\otimes\bm{p}\,d\bm{p}\bigg{)}\,\bm{e}_{1}\otimes\bm{e}_{1}\,,$
(2.16)

that is, for $j\geq 2$,

$(K(t)\widehat{\nabla\bm{u}})_{j1}=-\int_{S^{d-1}}p_{1}p_{j}e^{-ip_{1}t}\,\widehat{\nabla\bm{u}}:\bm{p}\otimes\bm{p}\,d\bm{p}\,,$
(2.17)

and vanishes in all other components.

Notice that, due to the special structure (2.3) of $\widehat{\nabla\bm{u}}$,
we may apply Lemma 2.1, estimate (ii), to obtain the following decay
estimates:

$\displaystyle|\widehat{\nabla\bm{u}}[e^{-ip_{1}t}h^{\rm in}]|$
$\displaystyle\lesssim\langle t\rangle^{-\frac{d-1}{2}-1}\|h^{\rm
in}\|_{H^{d+1}(S^{d-1})}\,,$ (2.18)
$\displaystyle|K(t)\widehat{\nabla\bm{u}}|$ $\displaystyle\lesssim\langle
t\rangle^{-\frac{d-1}{2}-1}|\widehat{\nabla\bm{u}}|\,.$ (2.19)

To solve the Volterra equation (2.15) for $\widehat{\nabla\bm{u}}$, as is
standard, we begin by taking the Fourier-Laplace transform in time, which we
denote by

$\mathcal{L}[g](\lambda)=\int_{0}^{+\infty}e^{-\lambda
t}g(t)\,dt\,,\qquad\lambda\in\mathbb{C}\,.$ (2.20)

_A priori_ , $\mathcal{L}\widehat{\nabla\bm{u}}$ is only well defined for
$\operatorname{Re}(\lambda)\gg\overline{\psi}$ when $\iota=-$, although it is
automatically well defined for $\operatorname{Re}\lambda>0$ when $\iota=+$.
This can be seen from the energy estimate (1.10). By the decay estimates
(2.18) and (2.19), $\widehat{\nabla\bm{u}}[e^{-ip_{1}t}h^{\rm in}]$ and $K$
are time-integrable, and their Fourier-Laplace transforms are well defined and
continuous for $\operatorname{Re}\lambda\geq 0$. We have

$\mathcal{L}\widehat{\nabla\bm{u}}[h]=\mathcal{L}\widehat{\nabla\bm{u}}[e^{-ip_{1}t}h^{\rm
in}]-\iota
d\overline{\psi}(\mathcal{L}K)\mathcal{L}\widehat{\nabla\bm{u}}[h]\,,$ (2.21)

which formally can be solved for $\mathcal{L}\widehat{\nabla\bm{u}}$:

$\mathcal{L}\widehat{\nabla\bm{u}}[h]=(I+\iota
d\overline{\psi}\mathcal{L}K)^{-1}\mathcal{L}\widehat{\nabla\bm{u}}[e^{-ip_{1}t}h^{\rm
in}]\,,$ (2.22)

provided that $I+\iota d\overline{\psi}\mathcal{L}K$ is invertible. Note that
the inverse in (2.22) is not a matrix inverse, since in this case $I$ and
$\mathcal{L}K$ are really linear operators acting on tensors of the form (2.3)
via (2.16). As long as $(I+\iota d\overline{\psi}\mathcal{L}K)^{-1}$ is
finite, we can solve for $\mathcal{L}\widehat{\nabla\bm{u}}$. We characterize
the invertibility of $I+\iota d\overline{\psi}\mathcal{L}K$ in the following
lemma.

######  Lemma 2.2 (Penrose condition).

The _Penrose condition_

$\sup_{\operatorname{Re}\lambda\geq 0}\|(I+\iota
d\overline{\psi}\mathcal{L}K)^{-1}\|\leq C<+\infty\,$ (2.23)

is satisfied unconditionally in the case of pullers ($\iota=+$) with constant
$C$ independent of $\overline{\psi}$. For pushers ($\iota=-$), the condition
(2.23) is equivalent to

$\iota
d\overline{\psi}\int_{S^{d-1}}\frac{p_{1}^{2}p_{j}^{2}}{\lambda+ip_{1}}\,d\bm{p}=1$
(2.24)

having no solution for $\operatorname{Re}(\lambda)\geq 0$.

Let $\overline{\psi}^{*}$ denote the supremum of $\psi$ such that the Penrose
condition is satisfied for all $\overline{\psi}<\psi$. We will see in the
course of the proof that indeed $\overline{\psi}^{*}>0$ is well defined.

The Fourier-Laplace transforms of $\mathcal{L}K$ and
$\mathcal{L}\widehat{\nabla\bm{u}}[e^{-ip_{1}\cdot}h^{\rm in}]$ are analytic
in $\operatorname{Re}\lambda>0$ and continuous in
$\operatorname{Re}\lambda\geq 0$. When the Penrose condition is satisfied, the
right-hand side of (2.22) is the unique analytic continuation of
$\mathcal{L}\widehat{\nabla\bm{u}}$ into $\operatorname{Re}\lambda>0$. Using
the notation $\lambda=\sigma+i\tau$, this analytic continuation belongs to
$L^{\infty}_{\sigma}L^{2}_{\tau}(\\{\sigma>0\\})$ and is known to attain its
boundary data continuously. Hence, the Paley-Wiener theorem (see [42, Theorem
19.2, p. 372]) guarantees that $\mathcal{L}\widehat{\nabla\bm{u}}$ is the
Fourier-Laplace transform of an $L^{2}(\mathbb{R}_{+})$ function, which must
be $\widehat{\nabla\bm{u}}$ by uniqueness of the Fourier-Laplace transform.
See also the argument in [55, p. 41].

We emphasize that if the integral equation (2.24) has no solution (that is,
the operator $I+\iota d\overline{\psi}\mathcal{L}K)$ is invertible pointwise),
then the Penrose condition (2.23) is satisfied for some constant $C$ (that is,
the operator is invertible uniformly). This is because $\mathcal{L}K$ is
continuous and decays as $|\lambda|\to+\infty$ with
$\operatorname{Re}\lambda\geq 0$.

######  Remark 2.3.

Note that the integral equation (2.24) is precisely the dispersion relation
arising in the eigenvalue problem for the linearized operator (1.7), which has
been studied in detail by various authors [27, 40, 44, 45, 47, 51, 52]. In
particular, any $\lambda$ satisfying (2.24) is an eigenvalue of the linearized
operator (1.7). In the pusher case ($\iota=-$), the Penrose condition reduces
to the dispersion relation having no solution for
$\operatorname{Re}(\lambda)\geq 0$, i.e. the linearized operator has no
unstable or marginally stable eigenvalue.

The implicit dispersion relation (2.24) can be solved numerically for
$\lambda$ as a function of the parameter $\overline{\psi}$ (see [27, 45, 44,
47]; note that a different nondimensionalization from (A.4) is commonly used)
from which we observe that (2.24) has a solution with
$\operatorname{Re}(\lambda)\geq 0$ for
$\overline{\psi}\geq\overline{\psi}^{*}$ for some $\overline{\psi}^{*}$. In
this situation the linearized equations (1.7)-(1.8) have growing modes which
give rise to pattern formation and “bacterial turbulence” observed in
numerical simulations [40, 45, 46, 47, 48].

To satisfy the Penrose condition (2.23) in the case of pushers,
$\overline{\psi}$ must therefore be taken to be sufficiently small. The
threshold value of $\overline{\psi}^{*}$ exactly corresponds to the eigenvalue
crossing observed in numerical studies [27, 40, 44, 45, 47, 51].

###### Proof of Lemma 2.2 (Penrose condition).

To see how $\mathcal{L}K$ acts on tensors $\bm{\Xi}$ of the form (2.3), we
first note that since $\bm{\Xi}$ is only nonzero in rows 2 through $d$ of the
first column, we have

$\bm{\Xi}:\bm{p}\otimes\bm{p}=\sum_{\ell=2}^{d}\Xi_{\ell 1}p_{\ell}p_{1}\,.$
(2.25)

Therefore, we may write $\mathcal{L}K\bm{\Xi}$ as

$\displaystyle\mathcal{L}K\bm{\Xi}(\lambda)$
$\displaystyle=\int_{0}^{\infty}e^{-\lambda
t}\int_{S^{d-1}}e^{-ip_{1}t}\,\bm{M}(\bm{\Xi},\bm{p})\,d\bm{p}\,dt$ (2.26)
$\displaystyle=\int_{S^{d-1}}\frac{1}{\lambda+ip_{1}}\,\bm{M}(\bm{\Xi},\bm{p})\,d\bm{p},$

where $\bm{M}(\bm{\Xi},\bm{p})$ is again a matrix of the form (2.3),

$\bm{M}(\bm{\Xi},\bm{p})=\begin{pmatrix}0&0&\cdots&0\\\ M_{21}&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ M_{d1}&0&\cdots&0\\\ \end{pmatrix},$ (2.27)

with

$M_{j1}=p_{1}^{2}p_{j}\sum_{\ell=2}^{d}\Xi_{\ell 1}p_{\ell}.$ (2.28)

Since odd functions in $p_{j}$ integrate to zero on $S^{d-1}$ for each
$j=1,\dots,d$, the integral of each $M_{j1}$ in (2.26) is only nonzero when
$\ell=j$. Thus $\mathcal{L}K$ acts on $\bm{\Xi}$ via entrywise multiplication:

$(\mathcal{L}K\Xi)_{j1}=\gamma_{j}(\lambda)\Xi_{j1},\qquad j\neq 1,$ (2.29)

where

$\gamma_{j}(\lambda)=\int_{S^{d-1}}\frac{p_{1}^{2}p_{j}^{2}}{\lambda+ip_{1}}\,d\bm{p}.$
(2.30)

In particular, we have that the operator $I+\iota
d\overline{\psi}\mathcal{L}K$ sends

$I+\iota d\overline{\psi}\mathcal{L}K:\;\;\Xi_{jm}\mapsto\begin{cases}(1+\iota
d\overline{\psi}\gamma_{j})\Xi_{j1}&\text{if }m=1\text{ and }j\neq 1\\\
\Xi_{jm}&\text{else}.\end{cases}$ (2.31)

The invertibility of $I+\iota d\overline{\psi}\mathcal{L}K$ (2.23) thus comes
down to verifying that

$\iota d\overline{\psi}\gamma_{j}(\lambda)\neq-1$ (2.32)

for all $\operatorname{Re}(\lambda)\geq 0$. We must therefore analyze the
integral $\gamma_{j}(\lambda)$ in greater detail. Writing
$\lambda=\sigma+i\tau$, we consider the behavior of
$\operatorname{Im}(\gamma_{j})$ and $\operatorname{Re}(\gamma_{j})$ for
$\sigma\geq 0$.

For the imaginary part, we need only to verify that
$\operatorname{Im}(\gamma_{j})$ is finite and uniformly bounded. This follows
from the integrability of $K$. For $\sigma>0$, the real part of $\gamma_{j}$
is given by

$\operatorname{Re}(\gamma_{j})=\int_{S^{d-1}}\frac{\sigma}{\sigma^{2}+(\tau+p_{1})^{2}}p_{1}^{2}p_{j}^{2}\,d\bm{p},$
(2.33)

which is clearly finite and nonnegative.

In particular, $\gamma_{j}(\lambda)$ is well-defined with
$\operatorname{Re}(\gamma_{j})\geq 0$ for all $\lambda$ with
$\operatorname{Re}(\lambda)\geq 0$. In the case of pullers ($\iota=+$), we
thus have that (2.32) is satisfied for any value of $\overline{\psi}$. For
pushers ($\iota=-$), on the other hand, in order to ensure that (2.32) is
satisfied, $\overline{\psi}$ must be chosen such that

$d\overline{\psi}\gamma_{j}(\lambda)\neq 1$ (2.34)

for any $\lambda$ with $\operatorname{Re}(\lambda)\geq 0$. This is the Penrose
condition. ∎

###### Proof of Theorem 1.1 (Linear Landau damping).

We return to our mode-by-mode analysis. To begin, we observe that due to the
time decay estimates (2.18) and (2.19) of
$\widehat{\nabla\bm{u}}[e^{-ip_{1}t}]$ and $K$, respectively, we have the
following regularity estimates on the Fourier-Laplace transforms:

$\|\mathcal{L}\widehat{\nabla\bm{u}}[e^{-ip_{1}\cdot}h^{\rm
in}](\sigma+i\tau)\|_{H^{s}_{\tau}}\lesssim_{s}\|h^{\rm
in}\|_{H^{d+1}_{p}}\,,$ (2.35)
$\|\mathcal{L}K(\sigma+i\tau)\|_{H^{s}_{\tau}}\lesssim_{s}1\,,$ (2.36)

for all $\sigma\geq 0$ and $s\in[0,d/2)$. Consider the function

$F(m)=(1+\iota d\overline{\psi}m)^{-1}-1\,.$ (2.37)

Since $\iota=+$ or $\overline{\psi}<\overline{\psi}^{*}$, the Penrose
condition implies that $F\circ\gamma_{j}$ is bounded. Since $F$ is smooth on
an open neighborhood of the range of $\gamma_{j}$ and $F(0)=0$, and in light
of the regularity estimate (2.36), the composition
$F\circ\gamma_{j}(\sigma+i\tau)$ belongs to $H^{s}_{\tau}$ for all
$s\in[0,d/2)$. See, for example, Theorem 2.87 in [4]. Since $H^{s}_{\tau}$ is
an algebra for $s>1/2$, we have

$\|(I+\iota
d\overline{\psi}\mathcal{L}K)^{-1}\mathcal{L}\widehat{\nabla\bm{u}}[e^{-ip_{1}\cdot}h^{\rm
in}]\|_{H^{s}_{\tau}}\lesssim_{s,\overline{\psi}}\|h^{\rm
in}\|_{H^{d+1}_{p}}\,,$ (2.38)

for all $s\in(1/2,d/2)$. Finally, the equality (2.22) and Fourier inversion on
$\sigma=0$ yield

$\int|\widehat{\nabla\bm{u}}|^{2}\langle
t\rangle^{2s}\,dt\lesssim_{s,\overline{\psi}}\|h^{\rm
in}\|_{H^{d+1}_{p}}^{2}\,,$ (2.39)

for all $s\in(1/2,d/2)$. This validates (1.12). To complete the theorem, we
must justify (1.15), namely, that

$g(\bm{p},t):=d\overline{\psi}\int_{0}^{t}e^{-ip_{1}(t-s)}\widehat{\nabla\bm{u}}:\bm{p}\otimes\bm{p}\,ds$
(2.40)

satisfies

$\int\|g(\cdot,t)\|_{H^{-(d+1)}_{p}}^{2}\langle
t\rangle^{d-\varepsilon}\,dt\lesssim_{s,\overline{\psi}}\|h^{\rm
in}\|_{H^{d+1}_{p}}^{2}\,.$ (2.41)

Again, we may take the Fourier-Laplace transform, this time of (2.40), and
observe that $e^{-ip_{1}t}p_{1}p_{j}$, $j\geq 2$, has the same time decay as
the kernel $K$. Since $H^{s}_{\tau}$ is a multiplicative algebra, the proof
follows. ∎

######  Remark 2.4 (Pointwise-in-time decay).

The representation formula (2.22) can be rewritten as

$\mathcal{L}\widehat{\nabla\bm{u}}=\mathcal{L}\widehat{\nabla\bm{u}}[e^{-ip_{1}\cdot}h^{\rm
in}]+(I+\iota d\overline{\psi}\mathcal{L}K)^{-1}(-\iota
d\overline{\psi}\mathcal{L}K)\mathcal{L}\widehat{\nabla\bm{u}}[e^{-ip_{1}\cdot}h^{\rm
in}]\,.$ (2.42)

In principle, one can obtain pointwise-in-time decay estimates on
$\widehat{\nabla\bm{u}}$ by studying the Green’s function
$\mathcal{L}^{-1}[(I+\iota d\overline{\psi}\mathcal{L}K)^{-1}(-\iota
d\overline{\psi}\mathcal{L}K)]$ pointwise-in-time. A natural way to proceed is
to study the optimal regularity of $\mathcal{L}K(i\tau)$ in
$L^{1}_{\tau}$-based spaces rather than $H^{s}_{\tau}$. One can obtain an
explicit formula for $\mathcal{L}K(i\tau)$ by sending $\sigma\to 0^{+}$. We
leave its analysis to future work.

##  3\. Taylor dispersion

In this section, we consider stability due to Taylor dispersion near vacuum
$\overline{\psi}=0$ in the whole space $\mathbb{R}^{3}$. Our linearized
estimates will also allow us to prove stability of the puller equilibrium
$\overline{\psi}\equiv\text{const.}$ on the torus $\mathbb{T}^{d}$, $d=2,3$.
Notably, when $\iota=+$, stability holds regardless of the size of
$\overline{\psi}$.

To begin, we study the linearized equation

$\partial_{t}f+\bm{p}\cdot\nabla_{x}f-\nu\Delta_{p}f-d\overline{\psi}\nabla\bm{u}:\bm{p}\otimes\bm{p}=0\,,$
(3.1)

whose basic energy estimate is

$\frac{1}{2}\frac{d}{dt}\|f\|_{L^{2}}^{2}+d\iota\overline{\psi}\|\nabla\bm{u}\|_{L^{2}}^{2}=-\nu\|\nabla_{p}f\|_{L^{2}}^{2}\,.$
(3.2)

Notice that the right-hand side of (3.2) only controls
$f-\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}f\,d\bm{p}$, and this by itself is not
enough to prove exponential decay. In view of this, we consider the density
(or concentration) $\varrho$ and momentum (or the nematic order parameter
times $\varrho$) $\bm{m}$,

$\varrho(\bm{x},t)=\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}f(\bm{x},\bm{p},t)\,dp\,,\quad\bm{m}(\bm{x},t)=\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}f(\bm{x},\bm{p},t)\,d\bm{p}\,.$
(3.3)

Here and in what follows, integration is with respect to the normalized
measure on the sphere ($\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\,d\bm{p}=1$). The density and momentum
satisfy the PDEs

$\partial_{t}\varrho+\operatorname{div}_{x}\bm{m}=0\,,$ (3.4)
$\partial_{t}\bm{m}+\operatorname{div}_{x}\left(\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}f\,d\bm{p}\right)+\nu\lambda\bm{m}=0\,,$
(3.5)

where $\lambda$ is the first nonzero eigenvalue of the Laplacian on the unit
sphere. To derive (3.4), we integrate (3.1) in $\bm{p}$ and observe that
$\nabla\bm{u}:\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}\,d\bm{p}=d^{-1}\,{\rm
tr}\;\nabla\bm{u}=d^{-1}\,\operatorname{div}\bm{u}=0$ in the non-local term.
To derive (3.5), we integrate $\bm{p}$ times (3.1) in $\bm{p}$ and observe
that $\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\Delta_{p}f\,d\bm{p}=\lambda\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}f\,d\bm{p}=\lambda\bm{m}$ in the
dissipation term and $(\nabla\bm{u}:\bm{p}\otimes\bm{p})\bm{p}$ is odd-
in-$\bm{p}$ in the nonlocal term. Notice that (3.4) does not contain a damping
term for the density, whereas (3.5) already contains a damping term for the
momentum.

As mentioned in the introduction, our method is inspired by [12, Section 3],
which itself is inspired by work of Guo. We introduce the macro-micro
decomposition

$f=\varrho+g\,.$ (3.6)

Assume that $\int_{x}\varrho=\int\kern-3.39996pt\int_{x,p}f=0$. Notice that

$\bm{m}=\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}g\,d\bm{p}\,.$ (3.7)

With the above decomposition (3.6), we have the following refinement of the
flux in (3.5):

$\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}f\,d\bm{p}=\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}(\varrho+g)\,d\bm{p}=\frac{1}{d}\varrho{\bf
I}+\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}g\,d\bm{p}\,.$ (3.8)

Thus, (3.5) can be rewritten as

$\partial_{t}\bm{m}+\frac{1}{d}\nabla_{x}\varrho+\operatorname{div}_{x}\left(\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}g\,d\bm{p}\right)+\nu\lambda\bm{m}=0\,.$
(3.9)

Crucially, (3.9) will produce the desired damping in $\varrho$ when integrated
against $|\nabla_{x}|^{-2}\nabla_{x}\varrho$. Below, we refine the above
reasoning to demonstrate

######  Theorem 3.1 (Linear Taylor dispersion).

Let $\nu>0$, $\bm{k}\in\mathbb{R}^{d}\setminus\\{0\\}$, and $k=|\bm{k}|$.
Suppose that $f=h(\bm{p},t)e^{i\bm{k}\cdot\bm{x}}$ solves the linearized PDE
(3.1) and $f^{\rm in}=h^{\rm in}(\bm{p})e^{i\bm{k}\cdot\bm{x}}$. There exists
an absolute constant $c_{0}>0$ satisfying the following property: If $\iota=+$
or $\overline{\psi}\ll\mu_{\nu,k}$, then

$\|h(\cdot,t)\|_{L^{2}_{p}}\lesssim e^{-c_{0}\mu_{\nu,k}t}\|h^{\rm
in}\|_{L^{2}_{p}}\,,\quad\forall t>0\,,$ (3.10)

where

$\mu_{\nu,k}=\begin{cases}\nu&k\geq\nu\\\
\frac{k^{2}}{\nu}&k\leq\nu\end{cases}\,.$ (3.11)

###### Proof.

Upon rotating, we may assume that $\bm{k}=k\bm{e}_{1}$ without loss of
generality. Furthermore, upon rescaling time and replacing $\nu$ and
$\overline{\psi}$ by $\nu/k$ and $\overline{\psi}/k$, respectively, we may
assume $k=1$. Thus, we consider (3.1) on $\mathbb{T}^{d}\times
S^{d-1}\times\mathbb{R}_{+}$ below.

Define

$G=\langle|\nabla_{x}|^{-2}\nabla_{x}\varrho,\bm{m}\rangle\,,$ (3.12)

where the brackets denote the $L^{2}$ inner product. Using (3.4) and (3.9), we
compute

$\displaystyle\partial_{t}G$
$\displaystyle=-\langle|\nabla_{x}|^{-2}\nabla_{x}\operatorname{div}_{x}\bm{m},\bm{m}\rangle$
(3.13)
$\displaystyle\qquad-\langle|\nabla_{x}|^{-2}\nabla_{x}\varrho,\frac{1}{d}\nabla_{x}\varrho+\operatorname{div}_{x}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}g\,d\bm{p}+\nu\lambda\bm{m}\rangle$
$\displaystyle=\|\,|\nabla_{x}|^{-1}\operatorname{div}_{x}\bm{m}\|_{L^{2}}^{2}-\frac{1}{d}\|\varrho\|_{L^{2}}^{2}$
$\displaystyle\qquad-\langle|\nabla_{x}|^{-1}\nabla_{x}\varrho,|\nabla_{x}|^{-1}\operatorname{div}_{x}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{S^{d-1}}\bm{p}\otimes\bm{p}g\,d\bm{p}\rangle-\nu\lambda\langle|\nabla_{x}|^{-2}\nabla_{x}\varrho,\bm{m}\rangle$
$\displaystyle\leq
C_{d}\|g\|_{L^{2}}^{2}-\frac{1}{2d}\|\varrho\|_{L^{2}}^{2}-\nu\lambda G\,,$

where we recall that $\bm{m}$ is estimated by $g$, see (3.7).

Define

$\Phi=\frac{1}{2}\|f\|_{L^{2}}^{2}+\varepsilon G\,.$ (3.14)

Since $G$ is not inherently sign-definite, we require $\varepsilon\ll 1$ to
ensure

$C^{-1}\|f\|_{L^{2}}^{2}\leq\Phi\leq C\|f\|_{L^{2}}^{2}\,.$ (3.15)

We seek a differential inequality for $\Phi$ by summing (3.2) and
$\varepsilon$ times (3.13):

$\frac{d\Phi}{dt}\leq-d\iota\overline{\psi}\|\nabla\bm{u}\|_{L^{2}}^{2}-\nu
c_{d}\|g\|_{L^{2}}^{2}-\frac{\varepsilon}{2d}\|\varrho\|_{L^{2}}^{2}-\varepsilon\nu\lambda
G+C_{d}\varepsilon\|g\|_{L^{2}}^{2}\,,$ (3.16)

where we employ the Poincaré inequality
$c_{d}\|g\|_{L^{2}}^{2}\leq\|\nabla_{p}f\|_{L^{2}}^{2}$ in (3.2). We require
$\varepsilon\ll\nu$ to ensure that $C_{d}\varepsilon\|g\|_{L^{2}}^{2}$ may be
absorbed:

$\frac{d\Phi}{dt}\leq-d\iota\overline{\psi}\|\nabla\bm{u}\|_{L^{2}}^{2}-\frac{\nu
c_{d}}{2}\|g\|_{L^{2}}^{2}-\frac{\varepsilon}{2d}\|\varrho\|_{L^{2}}^{2}-\varepsilon\nu\lambda
G\,.$ (3.17)

The above inequality has $O(\nu)$ damping in $g$ and $O(\varepsilon)$ damping
in $\varrho$.

Due to the rescaling of $\nu$ by $k$, we have two cases to consider,
corresponding to long and short wavelengths, respectively:

_Case 1. $\nu\leq 1$._ In this setting, the requirement $\varepsilon\ll\nu$ is
enough to close an estimate for $\Phi$, since
$\varepsilon\nu\lambda|G|\ll\nu^{2}\|f\|_{L^{2}}^{2}$, and hence this term may
be absorbed into the $O(\nu)$ damping in $f$ and $\varrho$.

_Case 2. $\nu\geq 1$._ In this setting, we further require $\varepsilon\ll
1/\nu$, say, $\varepsilon=\delta/\nu$. Then, using Young’s inequality,

$\varepsilon\nu\lambda|G|=\delta\lambda|G|\leq\eta\nu\|g\|_{L^{2}}^{2}+C\eta^{-1}\delta^{2}\nu^{-1}\|\varrho\|_{L^{2}}^{2}\,.$
(3.18)

We choose $\eta\ll 1$ and $\delta\ll C^{-1}\eta$ to ensure that the above term
may be absorbed into the $O(\nu)$ damping in $f$ and $O(\delta/\nu)$ damping
in $\varrho$.

In either case, we have

$\frac{d\Phi}{dt}\leq-d\iota\overline{\psi}\|\nabla\bm{u}\|_{L^{2}}^{2}-3c_{0}\mu_{\nu,1}\Phi$
(3.19)

for some $c_{0}>0$. When $\iota=+$, the first term on the right-hand side is
non-positive. When $\iota=-$, we require $\overline{\psi}\ll\mu_{\nu,1}$ to
absorb this term. The proof is complete. ∎

We now define the linear operator

$-L_{\nu,\kappa}^{\overline{\psi},\iota}f:=\bm{p}\cdot\nabla_{x}f-d\overline{\psi}\nabla\bm{u}[f]:\bm{p}\otimes\bm{p}-\nu\Delta_{p}f-\kappa\Delta_{x}f$
(3.20)

and its associated semigroup

$S(t)=e^{tL_{\nu,\kappa}^{\overline{\psi},\iota}}\,.$ (3.21)

Note that since Fourier multipliers in $\bm{x}$ commute, we have

$e^{t\kappa\Delta_{x}}e^{tL_{\nu,0}^{\overline{\psi},\iota}}=e^{tL_{\nu,0}^{\overline{\psi},\iota}}e^{t\kappa\Delta_{x}}=S(t)\,.$
(3.22)

We show the following:

######  Corollary 3.2 (Smoothing estimates).

Let $T>0$. Suppose that $f=h(\bm{p},t)e^{i\bm{k}\cdot\bm{x}}$ solves

$\partial_{t}f-L_{\nu,\kappa}^{\overline{\psi},\iota}f=\operatorname{div}_{p}\bm{g}$
(3.23)

with $f^{\rm in}=0$ for some
$\bm{g}=\widehat{\bm{g}}_{k}(\bm{p},t)e^{i\bm{k}\cdot\bm{x}}$ for $t\in(0,T)$.
Suppose that $\iota=+$. Then

$\|h(\cdot,T)\|_{L^{2}_{p}}^{2}\lesssim\nu^{-1}\int_{0}^{T}e^{-2c_{0}\mu_{\nu,k}(T-s)}\|\widehat{\bm{g}}_{k}\|_{L^{2}_{p}}^{2}\,ds\,.$
(3.24)

###### Proof.

Again, without loss of generality, $\bm{k}=k\bm{e}_{1}$ with $k=|\bm{k}|$, and
we consider the PDEs on the torus. Observe that for mode-$\bm{k}$ solutions of
the initial-value problem,

$\nu\int_{0}^{t}e^{2c_{0}\mu_{\nu,k}s}\int_{x,p}|\nabla_{p}S(s)f^{\rm
in}|^{2}\,d\bm{x}\,d\bm{p}\,ds\lesssim\|f^{\rm in}\|_{L^{2}}^{2}\,,$ (3.25)

which is a simple corollary of the energy estimates in Theorem 3.1. Consider
the solution operator $f^{\rm in}\mapsto\nabla_{p}S(\cdot)f^{\rm in}:L^{2}\to
L^{2}_{w}(\mathbb{T}^{d}\times S^{d-1}\times(0,T))$, where $w$ refers to the
weight $e^{c_{0}\mu_{\nu,k}t}$ in the above estimate, and we restrict to
mode-$\bm{k}$ functions. The adjoint operator
$L^{2}_{w^{-1}}(\mathbb{T}^{d}\times S^{d-1}\times(0,T))\to L^{2}$ produces
the $t^{\prime}=0$ trace of the solution of the PDE

$-\partial_{t^{\prime}}\widetilde{f}-\bm{p}\cdot\nabla_{x}\widetilde{f}-d\overline{\psi}\nabla\bm{u}[\widetilde{f}]:\bm{p}\otimes\bm{p}-\nu\Delta_{p}\widetilde{f}-\kappa\Delta_{x}\widetilde{f}=\operatorname{div}_{p}\bm{g}$
(3.26)

backward-in-time with $\widetilde{f}(\cdot,T)=0$ and a given $\bm{g}$ in mode
$\bm{k}$. Its solution satisfies

$\|\widetilde{f}(\cdot,0)\|_{L^{2}}^{2}\lesssim\nu^{-1}\int_{0}^{T}e^{-2c_{0}\mu_{\nu,k}s}\int_{x,p}|\bm{g}|^{2}\,d\bm{x}\,d\bm{p}\,ds\,.$
(3.27)

The desired estimate is obtained by rewriting $t=T-t^{\prime}$ and reflecting
$\bm{p}\to-\bm{p}$. ∎

###### Proof of Theorem 1.3 (Puller stability).

We consider the PDE for the perturbation $f$, namely,

$\partial_{t}f-L_{\nu,\kappa}^{\bar{\psi},\iota}f+\bm{u}\cdot\nabla_{x}f+\operatorname{div}_{p}[(\mathbf{I}-\bm{p}\otimes\bm{p})\nabla\bm{u}\bm{p}f]=0\,.$
(3.28)

Since $\iota=+$, we omit this from the notation. By Duhamel’s formula, we have

$f(\cdot,t)=S(t)f^{\rm in}+B(f,f)(\cdot,t)\,,$ (3.29)

where

$B(f,g)=B_{1}(f,g)+B_{2}(f,g)\,,$ (3.30)
$B_{1}(f,g)(\cdot,t)=-\int_{0}^{t}S(t-s)\operatorname{div}_{x}(\bm{u}[g]f)(\cdot,s)\,ds\,,$
(3.31)
$B_{2}(f,g)(\cdot,t)=-\int_{0}^{t}S(t-s)\operatorname{div}_{p}[(\mathbf{I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}[g]\bm{p})f](\cdot,s)\,ds\,.$
(3.32)

Our estimates will be in the function space

$X_{T}=\\{f\in C_{t}H^{2}_{x}L^{2}_{p}:\|f\|_{X_{T}}<+\infty\\}\,,$ (3.33)

where

$\|f\|_{X_{T}}:=\sup_{t\in[0,T]}e^{\delta_{0}\nu
t}\|f(\cdot,t)\|_{H^{2}_{x}L^{2}_{p}}\,.$ (3.34)

Here, $\delta_{0}<c_{0}$, where $c_{0}$ is the decay rate from the linear
theory (Theorem 3.1 and Corollary 3.2). Note that since we are on
$\mathbb{T}^{d}$ and $\nu\leq 1$, only the case $\mu_{\nu,k}=\nu$ is relevant.

Suppose the _bootstrap assumption_ : $\|f\|_{X_{T}}\leq C_{0}\varepsilon$,
where $\varepsilon_{0}>0$ and $C_{0}>2$ will be determined in the course of
the proof. We will demonstrate

$\|f\|_{X_{T}}\leq\frac{C_{0}}{2}\varepsilon\,.$ (3.35)

Hence, the bootstrap assumption can be propagated forward-in-time to complete
the proof. This Gronwall-type strategy is common in the long-time behavior of
nonlinear PDEs.

To begin, we estimate $B_{1}$:

$\displaystyle\|B_{1}(f,f)(\cdot,t)\|_{H^{2}_{x}L^{2}_{p}}$
$\displaystyle\leq\int_{0}^{t}\|S(t-s)\operatorname{div}_{x}(\bm{u}f)(\cdot,s)\|_{H^{2}_{x}L^{2}_{p}}\,ds$
(3.36)
$\displaystyle\lesssim\int_{0}^{t}(\kappa(t-s))^{-1/2}e^{-c_{0}\nu(t-s)}\|\bm{u}f(\cdot,s)\|_{H^{2}_{x}L^{2}_{p}}\,ds\,.$

Since $H^{2}_{x}$ is an algebra, we have

$\displaystyle\|B_{1}(f,f)(\cdot,t)\|_{H^{2}_{x}L^{2}_{p}}$
$\displaystyle\lesssim\kappa^{-1/2}C_{0}^{2}\varepsilon^{2}\int_{0}^{t}(t-s)^{-1/2}e^{-c_{0}\nu(t-s)}e^{-\delta_{0}\nu
s}\,ds\,$ (3.37)
$\displaystyle\lesssim\kappa^{-1/2}\nu^{-1/2}C_{0}^{2}\varepsilon^{2}e^{-\delta_{0}\nu
t}\,.$

Next, we estimate $B_{2}$. First, we revisit the smoothing estimate in
Corollary 3.2. If $\bm{g}\in X_{t}$, then we may conclude

$\displaystyle\|h(\cdot,t)\|_{H^{2}_{x}L^{2}_{p}}^{2}$
$\displaystyle\lesssim\nu^{-1}\int_{0}^{t}e^{-2c_{0}\nu(t-s)}\|\bm{g}\|_{H^{2}_{x}L^{2}_{p}}^{2}\,ds$
(3.38)
$\displaystyle\lesssim\nu^{-1}\|\bm{g}\|_{X_{t}}^{2}\int_{0}^{t}e^{-2c_{0}\nu(t-s)}e^{-2\delta_{0}\nu
s}\,ds$ $\displaystyle\lesssim\nu^{-2}\|\bm{g}\|_{X_{t}}^{2}e^{-2\delta_{0}\nu
t}\,.$

Plugging in $\bm{g}=(\mathbf{I}-\bm{p}\otimes\bm{p})(\nabla\bm{u}\bm{p})f$ and
employing that $H^{2}_{x}$ is an algebra, we have

$\|B_{2}(f,f)\|_{X_{T}}\lesssim\nu^{-1}C_{0}^{2}\varepsilon^{2}\,.$ (3.39)

Finally, the contribution from $f^{\rm in}$ is estimated as follows:

$\|Sf^{\rm in}\|_{X_{T}}\lesssim\|f^{\rm
in}\|_{H^{2}_{x}L^{2}_{p}}\lesssim\varepsilon\,.$ (3.40)

To conclude the bootstrap argument, we estimate the sum of (3.37), (3.39), and
(3.40) from above. Specifically, we require that

$\kappa^{-1/2}\nu^{-1/2}C_{0}^{2}\varepsilon^{2}+\nu^{-1}C_{0}^{2}\varepsilon^{2}+\varepsilon\ll
C_{0}\varepsilon\,.$ (3.41)

To ensure this, it is enough to require

$\kappa^{-1/2}\nu^{-1/2}C_{0}\varepsilon_{0}+\nu^{-1}C_{0}\varepsilon_{0}+C_{0}^{-1}\ll
1\,.$ (3.42)

To conclude, we choose $C_{0}\gg 1$ and $\varepsilon_{0}\ll\min(\nu,\kappa)$.
This completes the proof. ∎

Finally, we use the linear Taylor dispersion estimates of Theorem 3.1 to prove
Theorem 1.2 on $\mathbb{R}^{3}$. Our proof will utilize Besov spaces
$(\dot{B}^{s}_{2,q})_{x}L^{2}_{p}$. We review them below, although we cannot
review the whole theory here and therefore assume a certain familiarity, for
example, see the presentation in [4, Chapter 2]. Let $f\in
L^{2}_{x}L^{2}_{p}$. There exists a smooth function $\varphi$, compactly
supported on the annulus $\\{3/4<|\xi|<8/3\\}$, and satisfying that
$\sum_{j}\varphi(2^{-j}\xi)=1$ when $\xi\neq 0$. Define the Littlewood-Paley
projections

$\widehat{P_{j}f}=\varphi(2^{-j}\cdot)\widehat{f}\,,$ (3.43) $P_{\leq
j}:=\sum_{k\leq j}P_{k}\,,$ (3.44)

and $P_{<j}$, $P_{\geq j}$, $P_{>j}$ similarly. Then the Besov norms
$\|\cdot\|_{(\dot{B}^{s}_{2,q})_{x}L^{2}_{p}}$,
$(s,q)\in\mathbb{R}\times[1,\infty]$, are defined according to

$\|f\|_{(\dot{B}^{s}_{2,q})_{x}L^{2}_{p}}:=\left\lVert\left(2^{js}\|P_{j}f\|_{L^{2}_{x}L^{2}_{p}(\mathbb{R}^{d}\times
S^{d-1})}\right)\right\rVert_{\ell^{q}_{j}(\mathbb{Z})}\,.$ (3.45)

Membership in the Besov space is determined by finiteness of the above
norm.444When $s>d/2$ and $(s,q)\neq(d/2,1)$, the above Besov norms are merely
seminorms, and corresponding Besov spaces are only well defined up to
polynomials. We will not require this level of subtlety here. When $q=2$, the
Besov spaces are equal to the homogeneous Sobolev spaces.

We introduce the spaces $(\dot{B}^{s}_{2,q})_{x}L^{2}_{p}$ because they are
$L^{2}$-based (hence, amenable to the $L^{2}$-based linear estimates we
derived previously) yet live at different scalings and satisfy desirable
embeddings, namely,

$\displaystyle\|f\|_{L^{\infty}_{x}L^{2}_{p}}$
$\displaystyle\lesssim\|f\|_{(\dot{B}^{d/2}_{2,1})_{x}L^{2}_{p}}\,,$ (3.46)
$\displaystyle\|f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}$
$\displaystyle\lesssim\|f\|_{L^{1}_{x}L^{2}_{p}}\,.$ (3.47)

We mention also the real interpolation inequality

$\|f\|_{(\dot{B}^{s}_{2,1})_{x}L^{2}_{p}}\lesssim\|f\|_{(\dot{B}^{s_{1}}_{2,\infty})_{x}L^{2}_{p}}^{\theta}\|f\|_{(\dot{B}^{s_{2}}_{2,\infty})_{x}L^{2}_{p}}^{1-\theta}\,,$
(3.48)

where $s=\theta s_{1}+(1-\theta)s_{2}$. Notably, elliptic regularity is well
behaved in the above Besov spaces regardless of $q\in[1,+\infty]$.

Define

$Y=(\dot{B}^{d/2}_{2,1}\cap\dot{H}^{2})_{x}L^{2}_{p}\,,\quad
Z=(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}\cap Y\,.$ (3.49)

Note that $Y$ is a multiplicative algebra. For $T\in(0,+\infty]$, consider the
function space

$\|f\|_{X_{T}}:=\sup_{t\in(0,T)}r^{-1}(\nu
t)\|f(\cdot,t)\|_{(\dot{B}^{d/2}_{2,1}\cap\dot{H}^{2})_{x}L^{2}_{p}}\,,\quad
r(s):=\langle s\rangle^{-\frac{d}{2}}(\log(2+s))^{2}\,.$ (3.50)

Let $j_{0}\in\mathbb{Z}$ be the greatest integer such that $2^{j_{0}}\leq\nu$;
note that $j_{0}\leq 0$.

To prove Theorem 1.2, we require two further linear estimates, stated in
Lemmas 3.3 and 3.4. For the remainder of the section,
$S(t)=e^{tL^{0,\iota}_{\nu,k}}$, that is, $\overline{\psi}=0$.

######  Lemma 3.3.

For $\nu\in(0,1]$ and $f^{\rm in}\in Z$, we have

$\|S(\cdot)f^{\rm in}\|_{X_{\infty}}+\sup_{t\in(0,+\infty)}\|S(t)f^{\rm
in}\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\lesssim\|f^{\rm in}\|_{Z}$
(3.51)

###### Proof.

We have the propagation estimates

$\sup_{t\in(0,+\infty)}\|S(t)f^{\rm in}\|_{Z}\leq\|f^{\rm in}\|_{Z}\,,$ (3.52)

which imply the second half of the estimate in (3.51). This estimate will also
be used to control the solution for $t\leq\nu^{-1}$, so we focus on
$t>\nu^{-1}$. We use the estimates in Theorem 3.1. For $j\geq j_{0}$, we have
exponential decay:

$\|S(t)P_{j}f^{\rm in}\|_{Y}\lesssim e^{-\delta_{0}\nu t}\|P_{j}f^{\rm
in}\|_{Y}\,.$ (3.53)

For $j<j_{0}$, we have diffusive decay: for any real $s_{2}>s_{1}$,

$2^{js_{2}}\|S(t)P_{j}f^{\rm in}\|_{L^{2}_{x}L^{2}_{p}}\lesssim
2^{j(s_{2}-s_{1})}e^{-\delta_{0}2^{2j}\nu^{-1}t}2^{js_{1}}\|P_{j}f^{\rm
in}\|_{L^{2}_{x}L^{2}_{p}}\,.$ (3.54)

In particular, summing in $j<j_{0}$ and using that
$\int_{0}^{+\infty}y^{\alpha}e^{-y}\,dy\lesssim_{\alpha}1$ for all $\alpha>0$,
we have

$\|S(t)P_{<j_{0}}f^{\rm
in}\|_{(\dot{B}^{s_{2}}_{2,1})_{x}L^{2}_{p}}\lesssim_{s_{2}-s_{1}}(t/\nu)^{(s_{2}-s_{1})/2}\|P_{<j_{0}}f^{\rm
in}\|_{(\dot{B}^{s_{1}}_{2,\infty})_{x}L^{2}_{p}}\,,\quad\forall
s_{2}>s_{1}\,,\;\forall t>0\,.$ (3.55)

Choosing $s_{1}=-d/2$ and $s_{2}\in\\{d/2,2\\}$, we obtain for $t>\nu^{-1}$,

$\displaystyle\|S(t)P_{<j_{0}}f^{\rm in}\|_{(\dot{B}^{d/2}_{2,1}\cap
H^{2})_{x}L^{2}_{p}}$ $\displaystyle\lesssim(t/\nu)^{-d/2}\|P_{<j_{0}}f^{\rm
in}\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}$ (3.56)
$\displaystyle\lesssim\nu^{d}(\nu t)^{-\frac{d}{2}}\|f^{\rm
in}\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\,.$

Together, (3.53), (3.56), and standard embeddings complete the proof of
(3.51). ∎

######  Lemma 3.4.

Let $d=3$, $T>0$, and $\nu\in(0,1]$. Consider $\bm{g}=\bm{g}(\bm{x},\bm{p},t)$
satisfying

$\|\bm{g}(\cdot,t)\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\lesssim r(\nu
t)N\,,\quad\forall t\in(0,T)\,,$ (3.57) $\|\bm{g}\|_{X_{T}}\lesssim N\,,$
(3.58)

for $r$ as in (3.50) and upper bound $N\geq 0$. Then
$q:=\int_{0}^{t}S(t-s)\operatorname{div}_{p}\bm{g}\,ds$ satisfies

$\|q\|_{X_{T}}+\sup_{t\in(0,T)}\|q(\cdot,t)\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\lesssim\nu^{-1}N\,.$
(3.59)

###### Proof.

We require the estimates in Theorem 3.1 and Corollary 3.2. For
$t\leq\nu^{-1}$, we use the propagation estimate
$e^{-c_{0}\mu_{\nu,j}(t-s)}\lesssim 1$ in Corollary 3.2. Hence,

$\|\int_{0}^{t}S(t-s)\operatorname{div}_{p}\bm{g}\,ds\|_{\dot{H}^{\sigma}_{x}L^{2}_{p}}\lesssim\nu^{-1}N\,,$
(3.60)

for all $\sigma\in[0,2]$. Interpolating in $\sigma$ (3.48), we also have

$\|\int_{0}^{t}S(t-s)\operatorname{div}_{p}\bm{g}\,ds\|_{(\dot{B}^{d/2}_{2,1})_{x}L^{2}_{p}}\lesssim\nu^{-1}N\,.$
(3.61)

The propagation estimate also implies the second half of (3.59).

For $t\geq\nu^{-1}$ and $j\geq j_{0}-1$, we use the exponential decay
estimates

$\displaystyle\|P_{j\geq
j_{0}}q(\cdot,t)\|_{\dot{H}^{\sigma}_{x}L^{2}_{p}}^{2}$
$\displaystyle\lesssim\sum_{j\geq
j_{0}}2^{2j\sigma}\|\int_{0}^{t}S(t-s)\operatorname{div}_{p}P_{j}\bm{g}\,ds\|_{L^{2}_{x,p}}^{2}$
(3.62)
$\displaystyle\lesssim\nu^{-1}N^{2}\int_{0}^{t}e^{-2c_{0}\nu(t-s)}r^{2}(\nu
s)\,ds\lesssim N^{2}\nu^{-2}r^{2}(\nu t)\,,$

valid for all $\sigma\in[0,2]$. Interpolating in $\sigma$, we similarly
control $\|P_{j\geq j_{0}}q(\cdot,t)\|_{\dot{B}^{d/2}_{2,1}}$.

For $t\geq\nu^{-1}$ and $j<j_{0}$, we have, for any $s_{2}>s_{1}$,

$\displaystyle
2^{js_{2}}\|\int_{0}^{t}S(t-s)\operatorname{div}_{p}P_{j}\bm{g}\,ds\|_{L^{2}_{x,p}}$
(3.63)
$\displaystyle\qquad\lesssim\nu^{-1/2}\left(\int_{0}^{t}2^{2j(s_{2}-s_{1})}e^{-2\delta_{0}2^{2j}(t-s)/\nu}2^{2js_{1}}\|P_{j}\bm{g}(\cdot,s)\|_{L^{2}_{x,p}}^{2}\,ds\right)^{1/2}\,.$

Here, we choose $s_{1}=-d/2$ and $s_{2}=\sigma\geq d/2$. Employing $\nu\leq
1$, we have

$\displaystyle
2^{j\sigma}\|\int_{0}^{t}S(t-s)\operatorname{div}_{p}P_{j}\bm{g}\,ds\|_{L^{2}_{x,p}}$
(3.64)
$\displaystyle\qquad\lesssim\nu^{-1/2}Nj^{-3/2}\left(\int_{0}^{t}j^{3}2^{2j(\sigma+d/2)}e^{-2\delta_{0}2^{2j}\langle
t-s\rangle}r^{2}(\nu s)\,ds\right)^{1/2}$
$\displaystyle\qquad\lesssim\nu^{-1/2}Nj^{-3/2}\left(\int_{0}^{t}\langle
t-s\rangle^{-(\sigma+\frac{d}{2})}(\log(1+\langle t-s\rangle))^{3}r^{2}(\nu
s)\,ds\right)^{1/2}$ $\displaystyle\qquad\lesssim\nu^{-1}Nj^{-3/2}r(\nu t)\,.$

Summing in $j$ with $\sigma=d/2$, we obtain the desired estimate in
$(\dot{B}^{d/2}_{2,1})_{x}L^{2}_{p}$. Notice that the logarithmic factor is
lost in the $j^{-3/2}$ coefficient, which is needed to sum the right-hand
side. There is an analogous (simpler) estimate in $\dot{H}^{\sigma}$,
$\sigma\in(d/2,2]$. ∎

###### Proof of Theorem 1.2.

We suppose the _bootstrap assumption_ :

$\|f\|_{X_{T}}+\sup_{t\in(0,T)}\|f(\cdot,t)\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\leq
C_{0}\varepsilon\,,$ (3.65)

where $\varepsilon_{0}>0$ and $C_{0}\geq 10$ will be determined in the course
of the proof. We seek to demonstrate that (3.65) holds with $C_{0}/2$ on the
right-hand side instead of $C_{0}.$ Again, we estimate Duhamel’s formula
(3.29) with $B$, $B_{1}$, and $B_{2}$ defined by (3.30)-(3.32), where now
$S(t)=e^{tL^{0,\iota}_{\nu,\kappa}}$.

_Step 0. Initial data_. By Lemma 3.3, we have

$\|S(\cdot)f^{\rm in}\|_{X_{\infty}}+\sup_{t\in(0,+\infty)}\|S(t)f^{\rm
in}\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\lesssim\|f^{\rm
in}\|_{Z}\lesssim\varepsilon\,.$ (3.66)

_Step 1. $B_{1}$ term_. Again, by Lemma 3.3, we have

$\displaystyle\|B_{1}(f,f)(\cdot,t)\|_{Y}$
$\displaystyle\leq\int_{0}^{t}\|S(t-s)\operatorname{div}_{x}(\bm{u}f)(\cdot,s)\|_{Y}\,ds$
(3.67)
$\displaystyle\lesssim\int_{0}^{t}r^{-1}(\nu(t-s))\|e^{\kappa(t-s)\Delta_{x}}\operatorname{div}_{x}(\bm{u}f)(\cdot,s)\|_{Z}\,ds\,.$

We estimate the products $\bm{u}f$ and $\bm{u}\cdot\nabla_{x}f$. We begin by
recording preliminary estimates. Since $\|f(\cdot,t)\|_{L^{1}_{x,p}}=M$ is
conserved, we have

$\|\bm{u}(\cdot,t)\|_{\dot{B}^{1-d/2}_{2,\infty}}\lesssim M\,.$ (3.68)

Additionally, we have

$\|\bm{u}(\cdot,t)\|_{\dot{B}^{1+d/2}_{2,1}\cap\dot{H}^{3}}\lesssim\|f(\cdot,t)\|_{(\dot{B}^{d/2}_{2,1}\cap\dot{H}^{2})_{x}L^{2}_{p}}\,.$
(3.69)

By real interpolation, we have

$\|\bm{u}\|_{\dot{B}^{s}_{2,1}}\lesssim
M^{\theta}\|f\|_{\dot{B}^{d/2}_{2,1}}^{1-\theta}\,,$ (3.70)

where $s=\theta(1-d/2)+(1-\theta)(1+d/2)$ and $\theta\in(0,1)$. Similarly,

$\|f\|_{(\dot{B}^{s^{\prime}}_{2,1})_{x}L^{2}_{p}}\lesssim\|f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}^{\theta}\|f\|_{(\dot{B}^{d/2}_{2,1})_{x}L^{2}_{p}}^{1-\theta}\,,$
(3.71)

where $s^{\prime}=\theta(-d/2)+(1-\theta)(d/2)$ and $\theta\in(0,1)$.

To estimate $\bm{u}f$, we use (3.70) with $s=d/2$ and $\theta=1/3$. Then

$\|\bm{u}f(\cdot,t)\|_{(\dot{B}^{d/2}_{2,1})_{x}L^{2}_{p}}\lesssim
M^{1/3}\|f(\cdot,t)\|_{(\dot{B}^{d/2}_{2,1})_{x}L^{2}_{p}}^{5/3}\,,$ (3.72)

since $\dot{B}^{d/2}_{2,1}$ is a multiplicative algebra, and

$\displaystyle\|\bm{u}f(\cdot,t)\|_{\dot{H}^{2}_{x}L^{2}_{p}}$
$\displaystyle\lesssim\|\bm{u}\nabla^{2}_{x}f\|_{L^{2}_{x,p}}+\|\nabla\bm{u}\otimes\nabla_{x}f\|_{L^{2}_{x,p}}+\|\nabla^{2}\bm{u}f\|_{L^{2}_{x,p}}$
(3.73) $\displaystyle\lesssim
M^{1/3}\|f(\cdot,t)\|_{Y}^{5/3}+\|f(\cdot,t)\|_{Y}^{2}\,.$

In summary,

$\|\bm{u}f(\cdot,s)\|_{Y}\lesssim M^{1/3}\|f\|_{Y}^{5/3}+\|f\|_{Y}^{2}\,,$
(3.74)

and we estimate

$\|e^{\kappa(t-s)\Delta_{x}}\operatorname{div}_{x}(\bm{u}f)\|_{Y}\lesssim(\kappa(t-s))^{-1/2}(M^{1/3}\|f\|_{Y}^{5/3}+\|f\|_{Y}^{2})\,.$
(3.75)

To estimate $\bm{u}\cdot\nabla_{x}f$, we use $\theta=5/6$ in (3.70) and
$\theta=1/6$ in (3.71) to obtain

$\|\bm{u}\|_{L^{2}_{x}}\lesssim M^{5/6}\|f\|_{Y}^{1/6}\,,$ (3.76)
$\|\nabla_{x}f\|_{L^{2}_{x,p}}\lesssim\|f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}^{1/6}\|f\|_{Y}^{5/6}\,.$
(3.77)

Upon multiplying the two, we have

$\|\bm{u}\cdot\nabla_{x}f\|_{L^{1}_{x}L^{2}_{p}}\lesssim
M^{5/6}\|f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}^{1/6}\|f\|_{Y}\,$
(3.78)

and the estimate

$\|e^{\kappa(t-s)\Delta_{x}}\bm{u}\cdot\nabla_{x}f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\lesssim
M^{5/6}\|f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}^{1/6}\|f\|_{Y}\,.$
(3.79)

We combine (3.75) and (3.79) to estimate (3.67):

$\displaystyle\|B_{1}\|_{X_{T}}$
$\displaystyle\lesssim(\kappa^{-1/2}\nu^{-1/2}+\nu^{-1})(M+\|f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}+\|f\|_{X_{T}})\|f\|_{X_{T}}$
(3.80)
$\displaystyle\lesssim(\kappa^{-1/2}\nu^{-1/2}+\nu^{-1})C_{0}^{2}\varepsilon^{2}\,.$

Here, we use crucially that $d=3$ to ensure that the kernel in (3.67) decays
faster than $t^{-1}$. The case $d=2$ seems to be critical for treating
$\bm{u}\cdot\nabla_{x}f$ perturbatively. Additionally, for all $t\in(0,T)$, we
have

$\displaystyle\|B_{1}(\cdot,t)\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}$
$\displaystyle\leq\|\int_{0}^{t}S(t-s)(\bm{u}\cdot\nabla_{x}f)(\cdot,s)\,ds\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}$
(3.81) $\displaystyle\lesssim\int_{0}^{t}r(\nu
s)\,ds\,M^{5/6}\|f\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}^{1/6}\|f\|_{X_{T}}\lesssim\nu^{-1}C_{0}^{2}\varepsilon^{2}\,.$

_Step 2. $B_{2}$ term_. For this, we must estimate the product
$\nabla\bm{u}f$. Since $f\mapsto\nabla\bm{u}$ is a zeroth-order operator and
$Y$ is an algebra, we have

$\|\nabla\bm{u}f\|_{Y}\lesssim\|f\|_{Y}^{2}\,.$ (3.82)

For the $\dot{B}^{-d/2}_{2,\infty}$ part of $Z$, we use that
$\|\nabla\bm{u}\|_{\dot{B}^{-d/2}_{2,\infty}}\lesssim M$ and continuity555This
follows from the characterization
$\dot{B}^{-d/2}_{2,\infty}=(\dot{B}^{d/2}_{2,1})^{*}$ and the algebra property
of $\dot{B}^{d/2}_{2,1}$. of the product $(f,g)\mapsto
fg:\dot{B}^{-d/2}_{2,\infty}\times\dot{B}^{d/2}_{2,1}\to\dot{B}^{-d/2}_{2,\infty}$.
Using these bounds in Lemma 3.4, we obtain

$\|B_{2}\|_{X_{T}}+\sup_{t\in(0,T)}\|B_{2}(\cdot,t)\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\lesssim\nu^{-1}C_{0}^{2}\varepsilon^{2}\,.$
(3.83)

_3\. Conclusion_. Combining the above estimates (3.66), (3.80), and (3.83)
into Duhamel’s formula (3.29), we have

$\|f\|_{X_{T}}+\sup_{t\in(0,T)}\|f(\cdot,t)\|_{(\dot{B}^{-d/2}_{2,\infty})_{x}L^{2}_{p}}\leq
C\varepsilon+C(\kappa^{-1/2}\nu^{-1/2}+\nu^{-1})C_{0}^{2}\varepsilon^{2}\,.$
(3.84)

To ensure that the right-hand side is bounded above by $C_{0}\varepsilon/2$,
we choose $C_{0}\gg 1$ and $\varepsilon_{0}\ll\min(\nu,\kappa)$. This
completes the proof. ∎

##  4\. Enhanced dissipation

###  4.1. Linear enhancement

In this section, we consider the linearized system of equations

$\displaystyle\partial_{t}f+\bm{p}\cdot\nabla
f-\nu\Delta_{p}f-d\overline{\psi}\nabla\bm{u}:\bm{p}\otimes\bm{p}$
$\displaystyle=0$ (4.1) $\displaystyle-\Delta\bm{u}+\nabla
q=\iota\int_{S^{d-1}}\bm{p}\otimes\bm{p}\,\nabla_{x}f\,d\bm{p},\quad\operatorname{div}\bm{u}$
$\displaystyle=0$ (4.2)

and show that solutions decay at an enhanced rate as long as $\overline{\psi}$
is sufficiently small.

As in previous sections, it will be convenient to work mode-by-mode in
$\bm{x}$. Our results will be stated for each mode $\bm{k}$, but we will often
suppress the $\bm{k}$-dependence in our notation.

We begin by considering the case $\overline{\psi}=0$. In this setting, we
obtain the following enhancement result.

######  Theorem 4.1 (Linear enhancement for $\overline{\psi}=0$).

Suppose $f(\bm{x},\bm{p},t)=f_{k}(\bm{p},t)e^{i\bm{k}\cdot\bm{x}}$,
$\bm{k}\in\mathbb{R}^{d}\setminus\\{0\\}$, and let
$k=\left\lvert\bm{k}\right\rvert$. For $0<\nu<1$, there exist constants
$\overline{a}_{1}$, $\overline{a}_{2}$, $\overline{a}_{3}>0$ such that if
$\nabla_{p}f^{\rm in}$ and $({\bf I}-\bm{p}\otimes\bm{p})\nabla_{x}f^{\rm in}$
are both in $L^{2}$, then the functional

$\displaystyle\Phi_{k}(t)$ $\displaystyle=\frac{1}{2}\lVert
f\rVert_{L^{2}}^{2}+\frac{\overline{a}_{1}}{2}\nu^{1/2}k^{-1/2}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}+\overline{a}_{2}k^{-1}\langle({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f,\nabla_{p}f\rangle$ (4.3)
$\displaystyle\hskip
170.71652pt+\frac{\overline{a}_{3}}{2}\nu^{-1/2}k^{-3/2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}$

satisfies the fast decay estimate

$\Phi_{k}(t)\leq e^{-\alpha_{0}\nu^{1/2}k^{1/2}t}\Phi_{k}(0),\qquad t\geq 0,$
(4.4)

where $\alpha_{0}=\overline{a}_{3}$ is explicit.

For $f^{\rm in}$ in $L^{2}$ only, the decay rate of (4.4) is modified by a
logarithmic factor:

$\lVert f_{k}(\cdot,t)\rVert_{L^{2}}\leq\beta_{0}^{1/2}\lVert f^{\rm
in}_{k}\rVert_{L^{2}}e^{-\frac{\alpha_{0}}{2}\lambda_{k}t},\qquad\lambda_{\nu,k}=\frac{\nu^{1/2}k^{1/2}}{1+\left\lvert\log\nu\right\rvert+\log
k}$ (4.5)

where
$\beta_{0}=e(1+\frac{3}{4}\alpha_{0}\overline{a}_{1}+\frac{3}{2}\overline{a}_{3})$.

In particular, defining the linear operator

$\mathcal{L}_{\nu}f:=\bm{p}\cdot\nabla f-\nu\Delta_{p}f\,,$ (4.6)

we make note of the operator estimate

$\lVert e^{\mathcal{L}_{\nu}t}\rVert_{L^{2}\to
L^{2}}\leq\beta_{0}^{1/2}e^{-\frac{\alpha_{0}}{2}\lambda_{\nu,k}t}.$ (4.7)

As a consequence of Theorem 4.1, we obtain the following enhancement result
for small swimmer concentrations $\overline{\psi}$, regardless of the sign of
the active stress in (4.2).

######  Corollary 4.2 (Linear enhancement for small $\overline{\psi}$).

For $\overline{\psi}$ satisfying

$\overline{\psi}\leq\lambda_{\nu}\frac{\alpha_{0}}{8d\beta_{0}^{1/2}},$ (4.8)

the solution $f$ to the full linearized system (4.1)–(4.2) satisfies

$\lVert f(\cdot,t)\rVert_{L^{2}}\leq
2e^{-\frac{\alpha_{0}}{4}\lambda_{\nu}t}\lVert f^{\rm in}\rVert_{L^{2}}\,,$
(4.9)

provided that $\int_{x}f^{\rm in}\,d\bm{x}=0$. Here,
$\lambda_{\nu}=\lambda_{\nu,1}=\frac{\nu^{1/2}}{1+\left\lvert\log\nu\right\rvert}$.

####  4.1.1. Enhancement for $\overline{\psi}=0$

For Theorem 4.1, we consider the equation

$\partial_{t}f+\bm{p}\cdot\nabla_{x}f-\nu\Delta_{p}f=0$ (4.10)

and proceed via a similar hypocoercivity argument to [6, 20]. Recall that we
will work mode-by-mode in $\bm{x}$ but suppress the $k$-dependence in our
notation of functions. We consider the functional $\Phi(t)$ defined by (4.3)
with coefficients $a_{1}$, $a_{2}$, and $a_{3}$ yet to be determined:

$\Phi(t)=\frac{1}{2}\lVert
f\rVert_{L^{2}}^{2}+\frac{a_{1}}{2}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}+a_{2}\langle({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f,\nabla_{p}f\rangle+\frac{a_{3}}{2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}\,.$ (4.11)

###### Proof of Theorem 4.1.

Our goal is to choose $a_{1}$, $a_{2}$, and $a_{3}$ such that the bound (4.4)
holds.

We first notice that

$\left\lvert a_{2}\langle({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f,\nabla_{p}f\rangle\right\rvert\leq\frac{a_{2}}{4\delta}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}+\delta
a_{2}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}$ (4.12)

for $\delta>0$; in particular, as long as

$\frac{a_{2}}{\delta}\leq a_{1},\quad\delta a_{2}\leq\frac{a_{3}}{4},$ (4.13)

for some choice of $\delta$, we have that $\Phi$ satisfies

$\displaystyle\Phi(t)$ $\displaystyle\geq\frac{1}{2}\lVert
f\rVert_{L^{2}}^{2}+\frac{a_{1}}{4}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}+\frac{a_{3}}{4}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}$ (4.14)
$\displaystyle\Phi(t)$ $\displaystyle\leq\frac{1}{2}\lVert
f\rVert_{L^{2}}^{2}+\frac{3a_{1}}{4}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}+\frac{3a_{3}}{4}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}\,.$ (4.15)

We aim to use (4.14) and (4.15) to bound $\partial_{t}\Phi(t)$ in terms of
$\Phi(t)$.

The time derivatives of each of the four terms of $\Phi$ can be shown to
satisfy the following equations:

$\displaystyle\frac{1}{2}\partial_{t}\lVert f\rVert_{L^{2}}^{2}$
$\displaystyle=-\nu\lVert\nabla_{p}f\rVert_{L^{2}}^{2}$ (4.16)
$\displaystyle\frac{1}{2}\partial_{t}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}$
$\displaystyle=-\nu\lVert\Delta_{p}f\rVert_{L^{2}}^{2}-\langle({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f,\nabla_{p}f\rangle$ (4.17)
$\displaystyle\partial_{t}\langle({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f,\nabla_{p}f\rangle$
$\displaystyle=-\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}$ (4.18)
$\displaystyle\qquad-2\nu\langle({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\cdot(\nabla_{p}f),\Delta_{p}f\rangle$
$\displaystyle\qquad+(d-1)\nu\langle\bm{p}\cdot\nabla_{x}f,\Delta_{p}f\rangle$
$\displaystyle\frac{1}{2}\partial_{t}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}$
$\displaystyle=-\nu\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\nabla_{p}f\rVert_{L^{2}}^{2}$ (4.19)
$\displaystyle\qquad+\nu\langle\bm{p}\cdot\nabla_{x}f,({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\cdot(\nabla_{p}f)\rangle$
$\displaystyle\qquad+\nu\langle({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f,\bm{p}\cdot\nabla_{x}(\nabla_{p}f)\rangle\,.$

The first equation (4.16) follows immediately from (4.10), while the second
(4.17), third (4.18), and fourth (4.19) rely, respectively, on the following
commutators:

$\displaystyle[\nabla_{p},\bm{p}\cdot\nabla_{x}]$ $\displaystyle=({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}$ (4.20)
$\displaystyle[(\nabla_{p})_{i},\big{(}({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\big{)}_{i}]$
$\displaystyle=-(d-1)\bm{p}\cdot\nabla_{x}$
$\displaystyle[(\nabla_{p})_{i},\big{(}({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\big{)}_{j}]$
$\displaystyle=(-\delta_{ij}+2\bm{p}_{i}\bm{p}_{j})\bm{p}\cdot\nabla_{x}-\bm{p}_{i}\nabla_{x_{j}}\,.$

Here, for any function $h(\bm{p})$, the notation $(\nabla_{p})_{i}h$ is used
to denote the $i$-th component of the vector field $\nabla_{p}h$. Throughout,
we use that $\nabla_{p}$ can be computed by extending $h$ arbitrarily to a
function $\widetilde{h}$ in a neighborhood of the sphere, computing the flat
gradient, and projecting back to the sphere: $\nabla_{p}h=({\bf
I}-\bm{p}\otimes\bm{p})\nabla\widetilde{h}$.

Combining equations (4.16)–(4.19), $\partial_{t}\Phi$ then satisfies

$\displaystyle\partial_{t}\Phi(t)$
$\displaystyle\leq-\nu\lVert\nabla_{p}f\rVert_{L^{2}}^{2}-\nu
a_{1}\lVert\Delta_{p}f\rVert_{L^{2}}^{2}-a_{2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}$ (4.21)
$\displaystyle\quad-\nu a_{3}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\nabla_{p}f\rVert_{L^{2}}^{2}+a_{1}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}\lVert\nabla_{p}f\rVert_{L^{2}}$
$\displaystyle\quad+2\nu a_{2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\cdot(\nabla_{p}f)\rVert_{L^{2}}\lVert\Delta_{p}f\rVert_{L^{2}}$
$\displaystyle\quad+(d-1)\nu
a_{2}\lVert\bm{p}\cdot\nabla_{x}f\rVert_{L^{2}}\lVert\Delta_{p}f\rVert_{L^{2}}$
$\displaystyle\quad+\nu
a_{3}\lVert\bm{p}\cdot\nabla_{x}f\rVert_{L^{2}}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\cdot(\nabla_{p}f)\rVert_{L^{2}}$
$\displaystyle\quad+\nu a_{3}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}\lVert\bm{p}\cdot\nabla_{x}(\nabla_{p}f)\rVert_{L^{2}}$
$\displaystyle\leq-(\nu-\delta_{0}a_{1})\lVert\nabla_{p}f\rVert_{L^{2}}^{2}-\nu\bigg{(}a_{1}-a_{2}\delta_{1}-(d-1)a_{2}\delta_{2}\bigg{)}\lVert\Delta_{p}f\rVert_{L^{2}}^{2}$
$\displaystyle\quad-\bigg{(}a_{2}-\frac{a_{1}}{4\delta_{0}}-\nu\frac{a_{3}}{4\delta_{4}}\bigg{)}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}$
$\displaystyle\quad-\nu\bigg{(}a_{3}-\delta_{3}a_{3}-\frac{a_{2}}{4\delta_{1}}\bigg{)}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\nabla_{p}f\rVert_{L^{2}}^{2}$
$\displaystyle\quad+\nu\bigg{(}(d-1)\frac{a_{2}}{4\delta_{2}}+\frac{a_{3}}{4\delta_{3}}\bigg{)}\lVert\bm{p}\cdot\nabla_{x}f\rVert_{L^{2}}^{2}+\nu\delta_{4}a_{3}\lVert\bm{p}\cdot\nabla_{x}(\nabla_{p}f)\rVert_{L^{2}}^{2}\,,$

where each $\delta_{j}>0$ is also yet to be determined.

From here, we work mode-by-mode in $\bm{x}$. We consider a single spatial mode
$\bm{k}$ and take $k=\left\lvert\bm{k}\right\rvert$. The chosen constants
$a_{j}$ and $\delta_{j}$ will depend on both $\nu$ and $k$ in such a way that
the right-hand side of (4.21) is bounded by a multiple of $\Phi(t)$.

We make use of the following Poincaré-type inequality on $\mathbb{T}^{d}$, cf.
[6, Lemma 3.8].

######  Lemma 4.3.

For $f(\bm{p},\bm{x})=f_{k}(\bm{p})e^{i\bm{k}\cdot\bm{x}}$ with
$\nabla_{p}f\in L^{2}$ and $({\bf I}-\bm{p}\otimes\bm{p})\nabla_{x}f\in
L^{2}$, whenever $0<\nu<k$, we have

$\lVert\nabla_{x}f\rVert_{L^{2}}^{2}\lesssim
k^{3/2}\nu^{1/2}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}+k^{1/2}\nu^{-1/2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}\,.$ (4.22)

The proof of Lemma 4.3 appears in Appendix C.

To show the bound (4.4), we need each term of $\Phi(t)$, defined by (4.11), to
appear on the right-hand side of the inequality (4.21) with an appropriate
sign. Note that we are still missing a term proportional to $\lVert
f\rVert_{L^{2}}^{2}$. We may use Lemma 4.3 to insert this term: in particular,
for $a_{4}>0$, we have, for an appropriate constant $c_{0}$,

$\displaystyle-a_{4}k^{2}\lVert
f\rVert_{L^{2}}^{2}+a_{4}c_{0}\bigg{(}k^{3/2}\nu^{1/2}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}$
(4.23) $\displaystyle\hskip 142.26378pt+k^{1/2}\nu^{-1/2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}\bigg{)}\geq 0\,.$

Choosing the coefficients $a_{j}$ and $\delta_{j}$ in (4.21) and (4.23) to
satisfy

$\displaystyle a_{1}$ $\displaystyle=\overline{a}_{1}\nu^{1/2}k^{-1/2},\quad
a_{2}=\overline{a}_{2}k^{-1},\quad a_{3}=\overline{a}_{3}\nu^{-1/2}k^{-3/2},$
(4.24) $\displaystyle a_{4}$
$\displaystyle=\overline{a}_{4}\nu^{1/2}k^{-3/2},\quad\delta_{0}=\overline{\delta}_{0}\nu^{1/2}k^{1/2},\quad\delta_{1}=\overline{\delta}_{1}\nu^{1/2}k^{1/2},$
$\displaystyle\delta_{2}$
$\displaystyle=\overline{\delta}_{2}\nu^{1/2}k^{1/2},\quad\delta_{3}=\overline{\delta}_{3},\quad\delta_{4}=\overline{\delta}_{4}\nu^{1/2}k^{-1/2}\,,$

and using Lemma 4.3 again to absorb the final two terms on the right-hand side
of (4.21) into previous terms, we obtain (increasing $c_{0}$ if necessary)

$\displaystyle\partial_{t}\Phi(t)$ (4.25)
$\displaystyle\leq-\nu^{1/2}k^{1/2}\overline{a}_{4}\lVert
f\rVert_{L^{2}}^{2}-\nu\bigg{(}1-\overline{a}_{1}\overline{\delta}_{0}-(d-1)c_{0}\frac{\overline{a}_{2}}{4\overline{\delta}_{2}}-c_{0}\frac{\overline{a}_{3}}{4\overline{\delta}_{3}}-c_{0}\overline{a}_{4}\bigg{)}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}$
$\displaystyle\quad-\nu^{3/2}k^{-1/2}\bigg{(}\overline{a}_{1}-\overline{a}_{2}\overline{\delta}_{1}-(d-1)\overline{a}_{2}\overline{\delta}_{2}-c_{0}\overline{\delta}_{4}\overline{a}_{3}\bigg{)}\lVert\Delta_{p}f\rVert_{L^{2}}^{2}$
$\displaystyle\quad-k^{-1}\bigg{(}\overline{a}_{2}-\frac{\overline{a}_{1}}{4\overline{\delta}_{0}}-\frac{\overline{a}_{3}}{4\overline{\delta}_{4}}-(d-1)c_{0}\frac{\overline{a}_{2}}{4\overline{\delta}_{2}}-c_{0}\frac{\overline{a}_{3}}{4\overline{\delta}_{3}}-c_{0}\overline{a}_{4}\bigg{)}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}$
$\displaystyle\quad-\nu^{1/2}k^{-3/2}\bigg{(}\overline{a}_{3}-\overline{\delta}_{3}\overline{a}_{3}-\frac{\overline{a}_{2}}{4\overline{\delta}_{1}}-c_{0}\overline{\delta}_{4}\overline{a}_{3}\bigg{)}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\nabla_{p}f\rVert_{L^{2}}^{2}\,.$

We now choose $\overline{a}_{j}$ and $\overline{\delta}_{j}$ such that the
coefficients of each of the terms in (4.25) are strictly negative. Recalling
that we also need to satisfy the condition (4.13), we take

$\displaystyle\overline{a}_{1}$
$\displaystyle=\frac{c_{0}}{64c_{0}^{2}+1},\quad\overline{a}_{2}=\frac{\overline{a}_{1}}{16c_{0}}=\frac{1}{16(64c_{0}^{2}+1)},$
(4.26) $\displaystyle\overline{a}_{3}$
$\displaystyle=\frac{\overline{a}_{1}}{64c_{0}^{2}}=\frac{1}{64c_{0}(64c_{0}^{2}+1)},\quad\overline{a}_{4}=\frac{\overline{a}_{1}}{128c_{0}^{2}}=\frac{1}{128c_{0}(64c_{0}^{2}+1)},$
$\displaystyle\overline{\delta}_{0}$
$\displaystyle=64c_{0},\quad\overline{\delta}_{1}=4c_{0},\quad\overline{\delta}_{2}=(d-1)c_{0},\quad\overline{\delta}_{3}=\frac{1}{4},\quad\overline{\delta}_{4}=\frac{1}{4c_{0}}\,.$

Note that (4.13) holds for $\delta=\frac{1}{16c_{0}}\nu^{-1/2}k^{-1/2}$. We
thus have that $\partial_{t}\Phi$ satisfies

$\displaystyle\partial_{t}\Phi(t)$
$\displaystyle\leq-\nu^{1/2}k^{1/2}\overline{a}_{4}\lVert
f\rVert_{L^{2}}^{2}-\nu\frac{123}{128c_{0}}\overline{a}_{1}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}$
(4.27) $\displaystyle\quad-k^{-1}\frac{c_{0}}{4}\overline{a}_{3}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}-\nu^{3/2}k^{-1/2}31c_{0}^{2}\overline{a}_{3}\lVert\Delta_{p}f\rVert_{L^{2}}^{2}$
$\displaystyle\quad-\nu^{1/2}k^{-3/2}\frac{\overline{a}_{3}}{4}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}\nabla_{p}f\rVert_{L^{2}}^{2}$
$\displaystyle\leq-2\overline{a}_{4}\nu^{1/2}k^{1/2}\Phi(t)\,.$

Here we have used the upper bound (4.15) for $\Phi$ to bound the first three
terms. Noting that
$2\overline{a}_{4}=\overline{a}_{3}=\frac{1}{64c_{0}(64c_{0}^{2}+1)}$, we
obtain the estimate (4.4) of Theorem 4.1.

We next show that the estimate (4.5) holds for $L^{2}$ initial data. The
strategy of the proof can be found in, e.g., [20].

We begin by noting that, from (4.16), $\lVert f\rVert_{L^{2}}$ satisfies

$\lVert f(\cdot,t)\rVert_{L^{2}}^{2}=\lVert f^{\rm
in}\rVert_{L^{2}}^{2}-2\nu\int_{0}^{t}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}\,,$
(4.28)

and, in particular,

$\lVert f(\cdot,t)\rVert_{L^{2}}^{2}\leq\lVert f^{\rm
in}\rVert_{L^{2}}^{2}\,.$ (4.29)

Notice that for $0\leq t\leq
T_{\nu,k}:=\frac{1+\left\lvert\log\nu\right\rvert+\log
k}{\alpha_{0}\nu^{1/2}k^{1/2}}$, (4.29) automatically implies (4.5).

For $t\geq T_{\nu,k}$, we first note that by (4.28) and the mean value
theorem, there exists
$t^{*}\in\big{(}0,\frac{1}{\alpha_{0}\nu^{1/2}k^{1/2}}\big{)}$ such that

$2\nu\lVert\nabla_{p}f(\cdot,t^{*})\rVert_{L^{2}}^{2}=\alpha_{0}\nu^{1/2}k^{1/2}\left(\lVert
f^{\rm in}\rVert_{L^{2}}^{2}-\lVert
f(\cdot,t)\rVert_{L^{2}}^{2}\right)\leq\alpha_{0}\nu^{1/2}k^{1/2}\lVert f^{\rm
in}\rVert_{L^{2}}^{2}\,.$ (4.30)

Furthermore, using (4.24), we may rewrite the upper bound (4.15) as

$\Phi(t)\leq\frac{1}{2}\lVert
f\rVert_{L^{2}}^{2}+\frac{3}{4}\overline{a}_{1}\nu^{1/2}k^{-1/2}\lVert\nabla_{p}f\rVert_{L^{2}}^{2}+\frac{3}{4}\overline{a}_{3}\nu^{-1/2}k^{-3/2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f\rVert_{L^{2}}^{2}\,.$ (4.31)

Using (4.30) in (4.31), we then have

$\displaystyle\Phi(t^{*})$ $\displaystyle\leq\frac{1}{2}\lVert
f(\cdot,t^{*})\rVert_{L^{2}}^{2}+\frac{3}{8}\alpha_{0}\overline{a}_{1}\lVert
f^{\rm
in}\rVert_{L^{2}}^{2}+\frac{3}{4}\overline{a}_{3}\nu^{-1/2}k^{-3/2}\lVert({\bf
I}-\bm{p}\otimes\bm{p})\nabla_{x}f(\cdot,t^{*})\rVert_{L^{2}}^{2}$ (4.32)
$\displaystyle\leq\frac{1}{2}\lVert
f(\cdot,t^{*})\rVert_{L^{2}}^{2}+\frac{3}{8}\alpha_{0}\overline{a}_{1}\lVert
f^{\rm
in}\rVert_{L^{2}}^{2}+\frac{3}{4}\overline{a}_{3}\nu^{-1/2}k^{1/2}\lVert
f(\cdot,t^{*})\rVert_{L^{2}}^{2}$
$\displaystyle\leq\bigg{(}\frac{1}{2}+\frac{3}{8}\alpha_{0}\overline{a}_{1}+\frac{3}{4}\overline{a}_{3}\bigg{)}\nu^{-1/2}k^{1/2}\lVert
f^{\rm in}\rVert_{L^{2}}\,,$

by (4.29). For $t\geq T_{\nu,k}$, we thus obtain

$\displaystyle\lVert f(\cdot,t)\rVert_{L^{2}}^{2}$ $\displaystyle\leq
2\Phi(t)$ (4.33) $\displaystyle\leq
2e^{-\alpha_{0}\nu^{1/2}k^{1/2}(t-t^{*})}\Phi(t^{*})$
$\displaystyle\leq\bigg{(}1+\frac{3}{4}\alpha_{0}\overline{a}_{1}+\frac{3}{2}\overline{a}_{3}\bigg{)}e^{\alpha_{0}\nu^{1/2}k^{1/2}t^{*}}\nu^{-1/2}k^{1/2}e^{-\alpha_{0}\nu^{1/2}k^{1/2}t}\lVert
f^{\rm in}\rVert_{L^{2}}^{2}$ $\displaystyle\leq
e\bigg{(}1+\frac{3}{4}\alpha_{0}\overline{a}_{1}+\frac{3}{2}\overline{a}_{3}\bigg{)}\nu^{-1/2}k^{1/2}e^{-\alpha_{0}\nu^{1/2}k^{1/2}t}\lVert
f^{\rm in}\rVert_{L^{2}}^{2}\,,$

where we have used that $t^{*}\leq\frac{1}{\alpha_{0}\nu^{1/2}k^{1/2}}$. Now,
since $t\geq T_{\nu,k}$, the following bound holds:

$\nu^{-1/2}k^{1/2}e^{-\alpha_{0}\nu^{1/2}k^{1/2}t}\leq
e^{-\frac{\alpha_{0}\nu^{1/2}k^{1/2}}{1+\left\lvert\log\nu\right\rvert+\log
k}t}\,.$ (4.34)

Inserting (4.34) in (4.33) yields

$\lVert f(\cdot,t)\rVert_{L^{2}}^{2}\leq
e\bigg{(}1+\frac{3}{4}\alpha_{0}\overline{a}_{1}+\frac{3}{2}\overline{a}_{3}\bigg{)}\lVert
f^{\rm
in}\rVert_{L^{2}}^{2}e^{-\alpha_{0}\frac{\nu^{1/2}k^{1/2}}{1+\left\lvert\log\nu\right\rvert+\log
k}t}\,,$ (4.35)

and, defining
$\beta_{0}=e(1+\frac{3}{4}\alpha_{0}\overline{a}_{1}+\frac{3}{2}\overline{a}_{3})$,
we obtain (4.5). ∎

####  4.1.2. Enhancement for small $\overline{\psi}$

We now consider the full linearized system (4.1)–(4.2) with small
$\overline{\psi}>0$.

###### Proof of Corollary 4.2.

We begin by bounding $\nabla_{x}\bm{u}$ in terms of $f$. Upon multiplying
(4.2) and integrating by parts, we have

$\lVert\nabla_{x}\bm{u}\rVert_{L^{2}}^{2}=-\iota\langle\nabla_{x}\bm{u},\bm{p}\otimes\bm{p}f\rangle\leq\lVert\nabla_{x}\bm{u}\rVert_{L^{2}}\lVert
f\rVert_{L^{2}}\leq\frac{1}{2}\lVert\nabla_{x}\bm{u}\rVert_{L^{2}}^{2}+\frac{1}{2}\lVert
f\rVert_{L^{2}}^{2}\,,$ (4.36)

from which we obtain the bound

$\lVert\nabla_{x}\bm{u}\rVert_{L^{2}}\leq\lVert f\rVert_{L^{2}}\,.$ (4.37)

We write $f$ satisfying (4.1)–(4.2) using Duhamel’s formula:

$f(\cdot,t)=e^{\mathcal{L}_{\nu}t}f^{\rm
in}+d\overline{\psi}\int_{0}^{t}e^{\mathcal{L}_{\nu}(t-s)}\nabla_{x}\bm{u}:\bm{p}\otimes\bm{p}\,ds\,.$
(4.38)

Note that, using (4.37) along with the semigroup estimate (4.7), we have

$\lVert\int_{0}^{t}e^{\mathcal{L}_{\nu}(t-s)}\nabla_{x}\bm{u}:\bm{p}\otimes\bm{p}\,ds\rVert_{L^{2}}\leq\beta_{0}^{1/2}\int_{0}^{t}e^{-\frac{\alpha_{0}}{2}\lambda_{\nu}(t-s)}\lVert
f(\cdot,t)\rVert_{L^{2}}\,ds\,.$ (4.39)

Then, multiplying (4.38) by the time weight
$e^{\frac{\alpha_{0}}{4}\lambda_{\nu}t}$ and taking the $L^{2}$ norm in
$\bm{x}$ and $\bm{p}$, we may estimate the integral term as

$\displaystyle e^{\frac{\alpha_{0}}{4}\lambda_{\nu}t}\lVert
f(\cdot,t)\rVert_{L^{2}}$ $\displaystyle\leq
e^{-\frac{\alpha_{0}}{4}\lambda_{\nu}t}\lVert f^{\rm
in}\rVert_{L^{2}}+d\overline{\psi}\beta_{0}^{1/2}\int_{0}^{t}e^{-\frac{\alpha_{0}}{4}\lambda_{\nu}(t-s)}e^{\frac{\alpha_{0}}{4}\lambda_{\nu}s}\lVert
f(\cdot,s)\rVert_{L^{2}}\,ds$ (4.40) $\displaystyle\leq
e^{-\frac{\alpha_{0}}{4}\lambda_{\nu}t}\lVert f^{\rm
in}\rVert_{L^{2}}+\frac{\overline{\psi}}{\lambda_{\nu}}\frac{4d\beta_{0}^{1/2}}{\alpha_{0}}\sup_{t\in(0,T)}\left(e^{\frac{\alpha_{0}}{4}\lambda_{\nu}t}\lVert
f(\cdot,t)\rVert_{L^{2}}\right)\,.$

Thus, if we take $\overline{\psi}$ to satisfy

$\overline{\psi}\leq\lambda_{\nu}\frac{\alpha_{0}}{8d\beta_{0}^{1/2}}\,,$
(4.41)

we may absorb the $\lVert f(\cdot,t)\rVert_{L^{2}}$ term on the right-hand
side of (4.40) into the left-hand side to obtain

$\sup_{t\in(0,T)}\left(e^{\frac{\alpha_{0}}{4}\lambda_{\nu}t}\lVert
f(\cdot,t)\rVert_{L^{2}}\right)\leq
2\sup_{t\in(0,T)}\left(e^{-\frac{\alpha_{0}}{4}\lambda_{\nu}t}\right)\lVert
f^{\rm in}\rVert_{L^{2}}\leq 2\lVert f^{\rm in}\rVert_{L^{2}}\,.$ (4.42)

From this we obtain (4.9). ∎

Finally, we demonstrate a smoothing estimate which will be necessary in the
nonlinear argument:

######  Lemma 4.4 (Smoothing-in-$\bm{p}$).

(i) Let $f$ be the solution of the linearized PDE on $\mathbb{T}^{d}\times
S^{d-1}$ with $\overline{\psi}\ll\lambda_{\nu}$ and $f(\cdot,0)=f^{\rm in}\in
L^{2}$. Then

$\nu\int_{0}^{+\infty}e^{2c\lambda_{\nu}t}\int|\nabla_{p}f|^{2}\,d\bm{x}\,d\bm{p}\,dt\lesssim\|f^{\rm
in}\|_{L^{2}}^{2}\,.$ (4.43)

(ii) Let $h$ be the solution of the linearized PDE

$\partial_{t}h+\bm{p}\cdot\nabla_{x}h-d\overline{\psi}\nabla\bm{u}:\bm{p}\otimes\bm{p}=\nu\Delta_{p}h+\kappa\Delta_{x}h+\operatorname{div}_{p}\bm{g}$
(4.44)

on $\mathbb{T}^{d}\times S^{d-1}$ for some $\bm{g}\in
L^{2}(\mathbb{T}^{d}\times S^{d-1}\times\mathbb{R}_{+})$, with
$\overline{\psi}\ll\lambda_{\nu}$ and $h(\cdot,0)=0$. Then

$\|h(\cdot,t)\|_{L^{2}}^{2}\lesssim\nu^{-1}\int_{0}^{t}e^{-2c\lambda_{\nu}(t-s)}\int|\bm{g}|^{2}\,d\bm{x}\,d\bm{p}\,ds\,.$
(4.45)

###### Proof.

To prove (i), we extract the following from the basic energy estimate:

$\displaystyle\nu\int_{t_{1}}^{t_{2}}\int|\nabla_{p}f|^{2}\,d\bm{x}\,d\bm{p}\,dt$
$\displaystyle\leq\frac{1}{2}\|f(\cdot,t_{1})\|_{L^{2}}^{2}+C\overline{\psi}\int_{t_{1}}^{t_{2}}\int|f|^{2}\,d\bm{x}\,d\bm{p}\,dt\,$
(4.46) $\displaystyle\leq
Ce^{-2c\lambda_{\nu}t_{1}}\|f(\cdot,t_{1})\|_{L^{2}}^{2}+C\overline{\psi}\int_{t_{1}}^{+\infty}e^{-2c\lambda_{\nu}t}\,dt\,\|f(\cdot,t_{1})\|_{L^{2}}^{2}$
$\displaystyle\leq
Ce^{-2c\lambda_{\nu}t_{1}}\|f(\cdot,t_{1})\|_{L^{2}}^{2}\,,$

since $\overline{\psi}\ll\lambda_{\nu}$. We choose
$t_{2}=t_{1}+\lambda_{\nu}^{-1}$ and sum the estimate over
$t_{1}=k\lambda_{\nu}^{-1}$, $k=0,1,\ldots$. Finally, (ii) follows from a
duality argument as in Corollary 3.2. ∎

###  4.2. Nonlinear enhancement

Finally, we consider the full nonlinear equation (1.1) for
$\psi=\overline{\psi}+f$ on $\mathbb{T}^{d}$. Recalling the definition (3.20)
of the linear operator $L_{\nu,\kappa}^{\overline{\psi},\iota}$, we write
(1.1) as an equation for the perturbation $f$:

$\partial_{t}f-L_{\nu,\kappa}^{\overline{\psi},\iota}f+\bm{u}\cdot\nabla_{x}f+{\rm{div}}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla\bm{u}[f]\bm{p})f]=0\,.$ (4.47)

We show that for sufficiently small $f^{\rm in}$ and $\overline{\psi}$, the
nonlinear evolution of $f$ satisfies the same enhanced dissipation as the
linear evolution (Corollary 4.2).

Noting that the $k=0$ spatial mode is not enhanced, we must treat the $k=0$
and $k\neq 0$ modes of $f$ separately. We define $\mathbb{P}_{0}$ to be the
projection onto the $k=0$ spatial mode,

$\mathbb{P}_{0}h(\bm{x},\bm{p},t)=\int_{\mathbb{T}^{d}}h(\bm{x},\bm{p},t)\,d\bm{x}\,,$
(4.48)

and define $\mathbb{P}_{\neq}={\rm Id}-\mathbb{P}_{0}$. We consider separately
the evolution of

$f_{0}:=\mathbb{P}_{0}f\quad\text{and}\quad f_{\neq}:=\mathbb{P}_{\neq}f\,.$
(4.49)

Note that $\mathbb{P}_{0}$ commutes with
$L_{\nu,\kappa}^{\overline{\psi},\iota}$ and that
$L_{\nu,\kappa}^{\overline{\psi},\iota}f_{0}=-\nu\Delta_{p}f_{0}$. Further
note that
$\mathbb{P}_{0}\big{(}\bm{u}\cdot\nabla_{x}f\big{)}=\mathbb{P}_{0}\big{(}\operatorname{div}_{x}(\bm{u}f)\big{)}=0$
and that, since $\mathbb{P}_{0}\nabla_{x}\bm{u}=0$, the only contribution to
the $k=0$ mode from the ${\rm{div}}_{p}$ term in (4.47) is due to interactions
between $\bm{u}[f_{\neq}]$ and $f_{\neq}$. The evolution of the zero mode
$f_{0}(\bm{p},t)$ thus satisfies a forced heat equation in $\bm{p}$ whose
forcing depends only on nonzero modes:

$\partial_{t}f_{0}-\nu\Delta_{p}f_{0}+\mathbb{P}_{0}\,{\rm{div}}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}[f_{\neq}]\bm{p})f_{\neq}]=0.$ (4.50)

Using that $\bm{u}[f_{0}]=0$, the nonzero modes $f_{\neq}(\bm{x},\bm{p},t)$
evolve via

$\displaystyle\partial_{t}f_{\neq}-L_{\nu,\kappa}^{\overline{\psi},\iota}f_{\neq}+\bm{u}[f_{\neq}]\cdot\nabla_{x}f_{\neq}$
$\displaystyle+\mathbb{P}_{\neq}{\rm{div}}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}[f_{\neq}]\bm{p})f_{\neq}]$ (4.51)
$\displaystyle+\mathbb{P}_{\neq}{\rm{div}}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}[f_{\neq}]\bm{p})f_{0}]=0.$

We are now equipped to prove the nonlinear enhancement result of Theorem 1.4.

###### Proof of Theorem 1.4.

We begin by defining our function spaces. For the zero mode evolution, we will
consider the space $X_{T}=\\{g\in C_{t}H^{2}_{x}L^{2}_{p}\,:\,\lVert
g\rVert_{X_{T}}<\infty\\}$; for the nonzero modes we consider $Y_{T}=\\{g\in
C_{t}H^{2}_{x}L^{2}_{p}\,:\,\lVert g\rVert_{Y_{T}}<\infty\\}$, where the norms
$\lVert\cdot\rVert_{X_{T}}$ and $\lVert\cdot\rVert_{Y_{T}}$ are given by

$\lVert\cdot\rVert_{X_{T}}=\sup_{t\in[0,T]}\,e^{\delta_{0}\nu
t}\lVert\cdot\rVert_{H^{2}_{x}L^{2}_{p}},\qquad\lVert\cdot\rVert_{Y_{T}}=\sup_{t\in[0,T]}\,e^{\delta_{\neq}\lambda_{\nu}t}\lVert\cdot\rVert_{H^{2}_{x}L^{2}_{p}}$
(4.52)

for constants $0<\delta_{0}<c_{0}$ and $0<\delta_{\neq}<c_{\neq}$, where
$c_{0}$ (defined in (4.55)) and $c_{\neq}=\alpha_{0}/4$ (defined in Corollary
4.2) are decay rates from the linear theory.

We again use a bootstrap argument to show Theorem 1.4. For some
$\varepsilon\leq\varepsilon_{0}$ and $C_{0}>2$ to be determined, we assume
that $\lVert f_{\neq}\rVert_{Y_{T}}\leq C_{0}\varepsilon$, and aim to show
that

$\lVert f_{\neq}\rVert_{Y_{T}}\leq\frac{C_{0}}{2}\varepsilon\,.$ (4.53)

To show (4.53), we begin by bounding the evolution of $f_{0}$ in (4.50), since
$f_{\neq}$ depends on $f_{0}$ via (4.51). Using Duhamel’s formula, we may
write $f_{0}$ as

$\displaystyle f_{0}(\bm{p},t)$ $\displaystyle=e^{t\nu\Delta_{p}}f^{\rm
in}_{0}(\bm{p})+B_{0}(f_{\neq},f_{\neq})(\bm{p},t)\,,$ (4.54) $\displaystyle
B_{0}(f,g)(\bm{p},t)$
$\displaystyle=-\mathbb{P}_{0}\int_{0}^{t}e^{(t-s)\nu\Delta_{p}}{\rm{div}}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}[g]\bm{p})f](\bm{x},\bm{p},s)\,ds\,.$

Note that by standard estimates for the heat equation on $S^{d-1}$, we have

$\lVert e^{t\nu\Delta_{p}}\rVert_{L^{2}_{p}\to L^{2}_{p}}\lesssim e^{-c_{0}\nu
t}\,,\qquad\lVert e^{t\nu\Delta_{p}}{\rm{div}}_{p}\rVert_{L^{2}_{p}\to
L^{2}_{p}}\lesssim(\nu t)^{-1/2}e^{-c_{0}\nu t}$ (4.55)

for some $c_{0}>0$. Using (4.55) and that $H^{2}_{x}$ is an algebra, we may
estimate the bilinear term $B_{0}$ in (4.54) as

$\displaystyle\lVert
B_{0}(f_{\neq},f_{\neq})(\cdot,t)\rVert_{H^{2}_{x}L^{2}_{p}}$ (4.56)
$\displaystyle\qquad\lesssim\nu^{-1/2}\int_{0}^{t}(t-s)^{-1/2}e^{-c_{0}\nu(t-s)}\lVert({\bf
I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}_{\neq}\bm{p})f_{\neq}\rVert_{H^{2}_{x}L^{2}_{p}}\,ds$
$\displaystyle\qquad\lesssim\nu^{-1/2}\int_{0}^{t}(t-s)^{-1/2}e^{-c_{0}\nu(t-s)}e^{-2\delta_{\neq}\lambda_{\nu}s}\,ds\lVert
f_{\neq}\rVert_{Y_{T}}^{2}$ $\displaystyle\qquad\lesssim\nu^{-1}e^{-c_{0}\nu
t}(C_{0}\varepsilon)^{2}\,.$

In particular, again using (4.55), $f_{0}$ satisfies

$\lVert f_{0}\rVert_{X_{T}}\lesssim\lVert f^{\rm
in}_{0}\rVert_{L^{2}_{p}}+\nu^{-1}C_{0}^{2}\varepsilon^{2}\,.$ (4.57)

We next consider the evolution of $f_{\neq}$. Recalling the definition of
$S(t)$ in (3.21) as the semigroup associated with
$L_{\nu,\kappa}^{\overline{\psi},\iota}$ from (3.20), we may write $f_{\neq}$
using Duhamel’s formula as

$f_{\neq}(\cdot,t)=S(t)f_{\neq}^{\rm
in}(\cdot)+B_{1}(f_{\neq},f_{\neq})(\cdot,t)+B_{2}(f_{\neq},f_{\neq})(\cdot,t)+B_{2}(f_{0},f_{\neq})(\cdot,t)$
(4.58)

where

$\displaystyle B_{1}(f,g)(\cdot,t)$
$\displaystyle=-\int_{0}^{t}S(t-s)\,\operatorname{div}_{x}(\bm{u}[g]f)(\cdot,s)\,ds$
(4.59) $\displaystyle B_{2}(f,g)(\cdot,t)$
$\displaystyle=-\mathbb{P}_{\neq}\int_{0}^{t}S(t-s)\,{\rm{div}}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}[g]\bm{p})f](\cdot,s)\,ds\,.$ (4.60)

We proceed to estimate each term in (4.58). In addition to the smoothing-
in-$\bm{p}$ estimate of Lemma 4.4, we will make use of the following more
standard semigroup estimates, which follow directly from Corollary 4.2 with
$c_{\neq}=\frac{\alpha_{0}}{4}$:

$\|S(\cdot,t)\|_{L^{2}_{x,p}\to L^{2}_{x,p}}\lesssim
e^{-c_{\neq}\lambda_{\nu}t}\,,\quad\|S(\cdot,t)\operatorname{div}_{x}\|_{L^{2}_{x,p}\to
L^{2}_{x,p}}\lesssim(\kappa t)^{-1/2}e^{-c_{\neq}\kappa
t}e^{-c_{\neq}\lambda_{\nu}t}\,.$ (4.61)

Note that the second bound in (4.61) uses the decomposition (3.22).

Using (4.61), we have

$\displaystyle\lVert
B_{1}(f_{\neq},f_{\neq})(\cdot,t)\rVert_{H^{2}_{x}L^{2}_{p}}$
$\displaystyle\leq\int_{0}^{t}\lVert
S(t-s)\,\operatorname{div}_{x}(\bm{u}[f_{\neq}]f_{\neq})(\cdot,s)\rVert_{H^{2}_{x}L^{2}_{p}}\,ds$
(4.62)
$\displaystyle\leq\int_{0}^{t}\kappa^{-1/2}(t-s)^{-1/2}e^{-c_{\neq}\lambda_{\nu}(t-s)}\lVert\bm{u}[f_{\neq}]f_{\neq}\rVert_{H^{2}_{x}L^{2}_{p}}\,ds$
$\displaystyle\lesssim\kappa^{-1/2}\int_{0}^{t}(t-s)^{-1/2}e^{-c_{\neq}\lambda_{\nu}(t-s)}e^{-2\delta_{\neq}\lambda_{\nu}s}\,ds\,\lVert
f_{\neq}\rVert_{Y_{T}}^{2}$
$\displaystyle\lesssim\kappa^{-1/2}\lambda_{\nu}^{-1/2}e^{-\delta_{\neq}\lambda_{\nu}t}(C_{0}\varepsilon)^{2},$

where we have also used
$\lVert\bm{u}[f_{\neq}]\rVert_{H^{3}_{x}L^{2}_{p}}\lesssim\lVert
f_{\neq}\rVert_{H^{2}_{x}L^{2}_{p}}$.

Next, applying Lemma 4.4 with $\bm{g}=[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla_{x}\bm{u}[f_{\neq}]\bm{p})f_{\neq}]$, we obtain

$\displaystyle\lVert
B_{2}(f_{\neq},f_{\neq})(\cdot,t)\rVert_{H^{2}_{x}L^{2}_{p}}^{2}$
$\displaystyle\lesssim\nu^{-1}\int_{0}^{t}e^{-2c_{\neq}\lambda_{\nu}(t-s)}\lVert\nabla_{x}\bm{u}[f_{\neq}]\,f_{\neq}\rVert_{H^{2}_{x}L^{2}_{p}}^{2}\,ds$
(4.63)
$\displaystyle\lesssim\nu^{-1}\int_{0}^{t}e^{-2c_{\neq}\lambda_{\nu}(t-s)}e^{-4\delta_{\neq}\lambda_{\nu}s}\,ds\lVert
f_{\neq}\rVert_{Y_{T}}^{4}$
$\displaystyle\lesssim\nu^{-1}\lambda_{\nu}^{-1}e^{-2\delta_{\neq}\lambda_{\nu}t}(C_{0}\varepsilon)^{4}\,.$

Finally, using Lemma 4.4 along with the estimate (4.57), we have

$\displaystyle\lVert
B_{2}(f_{0},f_{\neq})(\cdot,t)\rVert_{H^{2}_{x}L^{2}_{p}}^{2}$
$\displaystyle\lesssim\nu^{-1}\int_{0}^{t}e^{-2c_{\neq}\lambda_{\nu}(t-s)}\lVert\nabla_{x}\bm{u}[f_{\neq}]\,f_{0}\rVert_{H^{2}_{x}L^{2}_{p}}^{2}\,ds$
(4.64)
$\displaystyle\lesssim\nu^{-1}\int_{0}^{t}e^{-2c_{\neq}\lambda_{\nu}(t-s)}e^{-2\delta_{\neq}\lambda_{\nu}s}e^{-2\delta_{0}\nu
s}\,ds\lVert f_{\neq}\rVert_{Y_{T}}^{2}\lVert f_{0}\rVert_{X_{T}}^{2}$
$\displaystyle\lesssim\nu^{-1}\lambda_{\nu}^{-1}e^{-2\delta_{\neq}\lambda_{\nu}t}(C_{0}\varepsilon)^{2}(\lVert
f^{\rm in}_{0}\rVert_{L^{2}_{p}}+\nu^{-1}C_{0}^{2}\varepsilon^{2})^{2}\,.$

Combining the bounds (4.62), (4.63), and (4.64), and using (4.61) to bound the
evolution of $f_{\neq}^{\rm in}$, we thus obtain

$\displaystyle\lVert f_{\neq}\rVert_{Y_{T}}$ $\displaystyle\leq\lVert
f_{\neq}^{\rm in}\rVert_{H^{2}_{x}L^{2}_{p}}+\lVert
B_{1}(f_{\neq},f_{\neq})\rVert_{Y_{T}}+\lVert
B_{2}(f_{\neq},f_{\neq})\rVert_{Y_{T}}+\lVert
B_{2}(f_{0},f_{\neq})\rVert_{Y_{T}}$ (4.65)
$\displaystyle\lesssim\big{(}1+\kappa^{-1/2}\lambda_{\nu}^{-1/2}C_{0}^{2}\varepsilon+\nu^{-1/2}\lambda_{\nu}^{-1/2}C_{0}^{2}\varepsilon$
$\displaystyle\qquad+\nu^{-1/2}\lambda_{\nu}^{-1/2}C_{0}\big{(}\lVert f^{\rm
in}_{0}\rVert_{L^{2}_{p}}+\nu^{-1}C_{0}^{2}\varepsilon^{2}\big{)}\big{)}\,\varepsilon\,.$

To close the bootstrap argument (4.53), we need

$C_{0}^{-1}+\kappa^{-1/2}\lambda_{\nu}^{-1/2}C_{0}\varepsilon+\nu^{-1/2}\lambda_{\nu}^{-1/2}C_{0}\varepsilon+\nu^{-1/2}\lambda_{\nu}^{-1/2}\big{(}\lVert
f^{\rm in}_{0}\rVert_{L^{2}_{p}}+\nu^{-1}C_{0}^{2}\varepsilon^{2}\big{)}\ll
1\,,$ (4.66)

which can be satisfied if $C_{0}\gg 1$,
$\varepsilon\ll\min(\kappa^{1/2},\nu^{1/2})\lambda_{\nu}^{1/2}$, and $\lVert
f^{\rm in}_{0}\rVert_{L^{2}_{p}}\ll\nu^{1/2}\lambda_{\nu}^{1/2}$. ∎

##  Appendix A Nondimensionalization

To facilitate comparison with results from the applied and computational
literature, we comment here on our choice of nondimensionalization for
equations (1.1)-(1.3). The fully dimensional version of the kinetic model is
given by

$\displaystyle\partial_{t}\psi+V_{0}\bm{p}\cdot\nabla\psi+\bm{u}\cdot\nabla\psi+\operatorname{div}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})\nabla\bm{u}\bm{p}\,\psi]$ $\displaystyle=d_{\rm
r}\Delta_{p}\psi+d_{\rm t}\Delta_{x}\psi$ (A.1)
$\displaystyle-\mu\Delta\bm{u}+\nabla
q=\operatorname{div}{\bm{\Sigma}}\,,\;\operatorname{div}\bm{u}$
$\displaystyle=0$ (A.2)
$\displaystyle{\bm{\Sigma}}=\sigma_{0}\int_{S^{d-1}}\psi(\bm{x},\bm{p},t)\bm{p}\otimes\bm{p}$
$\displaystyle\,d\bm{p}\,,$ (A.3)

where $V_{0}$ is the average swimming speed of the particles, $d_{\rm r}$ and
$d_{\rm t}$ are the rotational and translational diffusion coefficients, $\mu$
is the fluid viscosity, and $\sigma_{0}$ is the (signed) active stress
magnitude. The above system is considered on a $d$-dimensional periodic box
with side length $L$, that is, $\mathbb{R}^{d}/L\mathbb{Z}^{d}$.

We nondimensionalize (A.1)-(A.3) according to

$\bm{u}^{*}=\frac{1}{V_{0}}\bm{u}\,,\quad\bm{x}^{*}=\frac{2\pi}{L}\bm{x}\,,\quad
t^{*}=\frac{2\pi
V_{0}}{L}t\,,\quad\psi^{*}=\frac{L\left\lvert\sigma_{0}\right\rvert}{2\pi\mu
V_{0}}\psi\,.$ (A.4)

Here, the (nondimensionalized) particle number density

$\overline{\psi}=\frac{1}{L^{d}}\int_{\mathbb{T}^{d}}\int_{S^{d-1}}\psi^{*}\,d\bm{p}\,d\bm{x}$
(A.5)

is a free parameter. Note that we may write

$\overline{\psi}=\frac{L\left\lvert\sigma_{0}\right\rvert n_{\psi}}{2\pi\mu
V_{0}}\,,$ (A.6)

where $n_{\psi}$ is the dimensional particle number density, given by (A.5)
with $\psi$ in place of $\psi^{*}$. The dimensionless rotational and
translational diffusion coefficients $\nu$, $\kappa$ in (1.1) are given by

$\nu=\frac{d_{\rm r}L}{2\pi V_{0}}\,,\quad\kappa=\frac{d_{\rm
t}2\pi}{LV_{0}}\,.$ (A.7)

##  Appendix B Strong solution theory

Let $T>0$, $\Omega^{d}=\mathbb{T}^{d}$ or $\mathbb{R}^{d}$, and $d=2,3$. Let
$\nu>0$ and $\kappa\geq 0$.

######  Definition B.1 (Strong solution).

A non-negative function $\psi:\Omega^{d}\times
S^{d-1}\times[0,T]\to[0,+\infty)$ is a _strong solution_ to (1.1)-(1.3) on
$\Omega^{d}\times S^{d-1}\times(0,T)$ with initial data $\psi^{\rm in}\in
H^{2}_{x}L^{2}_{p}\cap L^{1}_{x}L^{1}_{p}(\Omega^{d}\times S^{d-1})$ if the
following requirements are satisfied. Namely,

$\text{(i)}\quad\psi\in C([0,T];H^{2}_{x}L^{2}_{p})\cap
L^{2}_{t}H^{2}_{x}H^{1}_{p}\cap C([0,T];L^{1}_{x}L^{1}_{p})\,,$ (B.1)

(ii) the PDE (1.1)-(1.3) is satisfied in the sense of distributions on
$\Omega^{d}\times S^{d-1}\times(0,T)$, and (iii) $\|\psi(\cdot,t)-\psi^{\rm
in}\|_{L^{2}}\to 0$ as $t\to 0^{+}$.

######  Theorem B.2 (Strong solution theory).

_(Existence)_. For all $0\leq\psi^{\rm in}\in H^{2}_{x}L^{2}_{p}\cap
L^{1}_{x}L^{1}_{p}$, there exists $\bar{T}=\bar{T}(\|\psi^{\rm
in}\|_{H^{2}_{x}L^{2}_{p}},\nu)>0$ (the _guaranteed existence time_) and a
strong solution $\psi$ to (1.1)-(1.3) on $\Omega^{d}\times
S^{d-1}\times(0,\bar{T})$ with initial data $\psi^{\rm in}$.

_(Uniqueness)_. If $\psi,\widetilde{\psi}$ are two strong solutions on
$\Omega^{d}\times S^{d-1}\times(0,T)$, then $\psi\equiv\widetilde{\psi}$.

###### Proof.

_(Existence)_. Let $\varepsilon>0$. For $0\leq\varphi\in
C^{\infty}_{0}(\mathbb{R}^{d})$ with $\int\varphi=1$, define the mollification
$(g)_{\varepsilon}:=\varepsilon^{-d}g\ast\varphi(\cdot/\varepsilon)$.

Consider the _mollified equations_

$\partial_{t}\psi+\bm{p}\cdot\nabla_{x}\psi+(\bm{u})_{\varepsilon}\cdot\nabla_{x}\psi+\operatorname{div}_{p}\left[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla(\bm{u})_{\varepsilon}\bm{p})\psi\right]=\nu\Delta_{p}\psi+\kappa\Delta_{x}\psi\,,$
(B.2)

subject to the constitutive law (1.2)-(1.3) for $\bm{u}$. We also mollify the
initial condition $\psi(\cdot,0)=(\psi^{\rm in})_{\varepsilon}$.

Define the _Picard iterates_ $\psi_{n}$ inductively: $\psi_{-1}=0$ and
$\psi_{n}$, $n=0,1,2,\dots$, is the solution to the following linear
advection-diffusion equation with spatially smooth coefficients:

$\partial_{t}\psi_{n}+\bm{p}\cdot\nabla_{x}\psi_{n}+\bm{u}[\psi_{n-1}]\cdot\nabla_{x}\psi_{n}+\operatorname{div}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})\nabla\bm{u}[\psi_{n-1}]\bm{p}\psi_{n}]=\nu\Delta_{p}\psi_{n}+\kappa\Delta_{x}\psi_{n}\,,$
(B.3)

with smooth initial condition $\psi_{n}(\cdot,0)=(\psi^{\rm
in})_{\varepsilon}$. Clearly, $\psi_{n}\geq 0$ and mass is conserved, i.e.,
$\int\psi_{n}(\bm{x},\bm{p},t)\,d\bm{x}\,d\bm{p}=\int\psi^{\rm
in}(\bm{x},\bm{p})\,d\bm{x}\,d\bm{p}$ for all $t>0$.

_1._A priori_ estimates._ To begin, we record the following _a priori_ energy
estimates for a smooth solution $\psi$ to (1.1):

$\displaystyle\frac{1}{2}\frac{d}{dt}\lVert\psi\rVert_{L^{2}_{x,p}}^{2}+\frac{1}{2}\int_{x,p}\operatorname{div}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla\bm{u}\bm{p})]\,\psi^{2}\,d\bm{x}d\bm{p}$ (B.4)
$\displaystyle\qquad=-\nu\lVert\nabla_{p}\psi\rVert_{L^{2}_{x,p}}^{2}-\kappa\lVert\nabla_{x}\psi\rVert_{L^{2}_{x,p}}^{2}\,,$
$\displaystyle\frac{1}{2}\frac{d}{dt}\lVert\nabla_{x}\psi\rVert_{L^{2}_{x,p}}^{2}+\int_{x,p}(\nabla\bm{u}\nabla_{x}\psi)\cdot\nabla_{x}\psi\,d\bm{x}d\bm{p}$
(B.5) $\displaystyle\qquad-\int_{x,p}({\bf
I}-\bm{p}\otimes\bm{p})(\nabla^{2}\bm{u}\bm{p}\psi):(\nabla_{p}\nabla_{x}\psi)\,d\bm{x}d\bm{p}$
$\displaystyle\qquad+\frac{1}{2}\int_{x,p}\operatorname{div}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla\bm{u}\bm{p})]\,\left\lvert\nabla_{x}\psi\right\rvert^{2}\,d\bm{x}d\bm{p}$
$\displaystyle\qquad\qquad=-\nu\lVert\nabla_{p}\nabla_{x}\psi\rVert_{L^{2}_{x,p}}^{2}-\kappa\lVert\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}^{2}\,,$
$\displaystyle\frac{1}{2}\frac{d}{dt}\lVert\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}^{2}+\int_{x,p}(\nabla^{2}\bm{u}\cdot\nabla_{x}\psi+\nabla\bm{u}\cdot\nabla_{x}^{2}\psi):\nabla_{x}^{2}\psi\,d\bm{x}d\bm{p}$
(B.6) $\displaystyle\qquad-\int_{x,p}({\bf
I}-\bm{p}\otimes\bm{p})[\nabla^{3}\bm{u}\bm{p}\psi+2\nabla^{2}\bm{u}\bm{p}\otimes\nabla_{x}\psi]:(\nabla_{p}\nabla_{x}^{2}\psi)\,d\bm{x}d\bm{p}$
$\displaystyle\qquad+\frac{1}{2}\int_{x,p}\operatorname{div}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})(\nabla\bm{u}\bm{p})]\,\left\lvert\nabla_{x}^{2}\psi\right\rvert^{2}\,d\bm{x}d\bm{p}$
$\displaystyle\qquad\qquad=-\nu\lVert\nabla_{p}\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}^{2}-\kappa\lVert\nabla_{x}^{3}\psi\rVert_{L^{2}_{x,p}}^{2}\,.$

The final term on the left-hand side of each of (B.4)-(B.6) arises from
integration by parts twice in $\bm{p}$.

From (B.4)-(B.6), we obtain the following bounds:

$\frac{d}{dt}\lVert\psi\rVert_{L^{2}_{x,p}}^{2}\lesssim\lVert\nabla\bm{u}\rVert_{L^{\infty}_{x}}\lVert\psi\rVert_{L^{2}_{x,p}}^{2}-\nu\lVert\nabla_{p}\psi\rVert_{L^{2}_{x,p}}^{2}-\kappa\lVert\nabla_{x}\psi\rVert_{L^{2}_{x,p}}^{2}\,,$
(B.7) $\displaystyle\frac{d}{dt}\lVert\nabla_{x}\psi\rVert_{L^{2}_{x,p}}^{2}$
$\displaystyle\lesssim\lVert\nabla\bm{u}\rVert_{L^{\infty}_{x}}\lVert\nabla_{x}\psi\rVert_{L^{2}_{x,p}}^{2}+\lVert\nabla^{2}\bm{u}\rVert_{L^{2}_{x}}\lVert\psi\rVert_{L^{\infty}_{x}L^{2}_{p}}\lVert\nabla_{p}\nabla_{x}\psi\rVert_{L^{2}_{x,p}}$
(B.8)
$\displaystyle\quad-\nu\lVert\nabla_{p}\nabla_{x}\psi\rVert_{L^{2}_{x,p}}^{2}-\kappa\lVert\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}^{2}\,,$
$\displaystyle\frac{d}{dt}\lVert\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}^{2}$
$\displaystyle\lesssim\lVert\nabla^{2}\bm{u}\rVert_{L^{4}_{x}}\lVert\nabla_{x}\psi\rVert_{L^{4}_{x}L^{2}_{p}}\lVert\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}+\lVert\nabla^{3}\bm{u}\rVert_{L^{2}_{x}}\lVert\psi\rVert_{L^{\infty}_{x}L^{2}_{p}}\lVert\nabla_{p}\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}$
(B.9)
$\displaystyle\quad+\lVert\nabla\bm{u}\rVert_{L^{\infty}_{x}}\lVert\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}^{2}+\lVert\nabla^{2}\bm{u}\rVert_{L^{4}_{x}}\lVert\nabla_{x}\psi\rVert_{L^{4}_{x}L^{2}_{p}}\lVert\nabla_{p}\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}$
$\displaystyle\quad-\nu\lVert\nabla_{p}\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}^{2}-\kappa\lVert\nabla_{x}^{3}\psi\rVert_{L^{2}_{x,p}}^{2}\,.$

Furthermore, we note that $\nabla\bm{u}$ satisfies

$\lVert\nabla\bm{u}\rVert_{H^{2}_{x}}\lesssim\lVert\psi\rVert_{H^{2}_{x}L^{2}_{p}}\,.$
(B.10)

Using Sobolev embedding and the bound (B.10) for $\bm{u}$, and using Young’s
inequality to absorb the terms
$\lVert\nabla_{p}\nabla_{x}^{2}\psi\rVert_{L^{2}_{x,p}}$ into the rotational
diffusion term, we obtain

$\displaystyle\frac{d}{dt}\lVert\psi\rVert_{H^{2}_{x}L^{2}_{p}}^{2}+\nu\|\nabla_{p}\psi\|_{H^{2}_{x}L^{2}_{p}}^{2}$
$\displaystyle\lesssim_{\nu}\lVert\psi\rVert_{H^{2}_{x}L^{2}_{p}}^{3}+\lVert\psi\rVert_{H^{2}_{x}L^{2}_{p}}^{4}\,.$
(B.11)

By Gronwall’s inequality, there exists $\bar{T}=\bar{T}(\|\psi^{\rm
in}\|_{H^{2}_{x}L^{2}_{p}},\nu)>0$ with the property

$\|\psi\|_{L^{\infty}_{t}H^{2}_{x}L^{2}_{p}(\Omega^{d}\times
S^{d-1}\times(0,\bar{T}))}^{2}+\nu\|\nabla_{p}\psi\|_{L^{2}_{t}H^{2}_{x}L^{2}_{p}(\Omega^{d}\times
S^{d-1}\times(0,\bar{T}))}^{2}\lesssim_{\nu}\|\psi^{\rm
in}\|_{H^{2}_{x}L^{2}_{p}}^{2}\,$ (B.12)

While the above _a priori_ estimates were for smooth $\bm{u}$, analogous
computations for the Picard iterates $\psi_{n}$ to the mollified equations
produce the inequality

$\displaystyle\frac{d}{dt}\lVert\psi_{n}\rVert_{H^{2}_{x}L^{2}_{p}}^{2}+\nu\|\nabla_{p}\psi_{n}\|_{H^{2}_{x}L^{2}_{p}}^{2}$
$\displaystyle\lesssim_{\nu}(\lVert\psi_{n-1}\rVert_{H^{2}_{x}L^{2}_{p}}+\lVert\psi_{n-1}\rVert_{H^{2}_{x}L^{2}_{p}}^{2})\lVert\psi_{n}\rVert_{H^{2}_{x}L^{2}_{p}}^{2}$
(B.13)

and, consequently, the _a priori_ estimates

$\|\psi_{n}\|_{L^{\infty}_{t}H^{2}_{x}L^{2}_{p}(\Omega^{d}\times
S^{d-1}\times(0,\bar{T}))}^{2}+\nu\|\nabla_{p}\psi_{n}\|_{L^{2}_{t}H^{2}_{x}L^{2}_{p}(\Omega^{d}\times
S^{d-1}\times(0,\bar{T}))}^{2}\lesssim_{\nu}\|\psi^{\rm
in}\|_{H^{2}_{x}L^{2}_{p}}^{2}\,.$ (B.14)

_2\. Contraction_. Letting $\bm{u}_{n}=\bm{u}[\psi_{n}]$, the difference in
Picard iterates $\psi_{n}-\psi_{n-1}$ satisfies

$\displaystyle\partial_{t}($
$\displaystyle\psi_{n}-\psi_{n-1})+\bm{p}\cdot\nabla_{x}(\psi_{n}-\psi_{n-1})+(\bm{u}_{n-1}-\bm{u}_{n-2})\cdot\nabla_{x}\psi_{n}$
(B.15)
$\displaystyle+\bm{u}_{n-2}\cdot\nabla_{x}(\psi_{n}-\psi_{n-1})+\operatorname{div}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})\nabla(\bm{u}_{n-1}-\bm{u}_{n-2})\bm{p}\psi_{n}]$
$\displaystyle+\operatorname{div}_{p}[({\bf
I}-\bm{p}\otimes\bm{p})\nabla\bm{u}_{n-2}\bm{p}(\psi_{n}-\psi_{n-1})]$
$\displaystyle=\nu\Delta_{p}(\psi_{n}-\psi_{n-1})+\kappa\Delta_{x}(\psi_{n}-\psi_{n-1})\,.$

Multiplying by $\psi_{n}-\psi_{n-1}$ and integrating by parts, we have

$\displaystyle\frac{d}{dt}\lVert\psi_{n}-\psi_{n-1}\rVert_{L^{2}_{x,p}}^{2}\lesssim\left|\int_{x,p}(\bm{u}_{n-1}-\bm{u}_{n-2})\cdot\nabla_{x}\psi_{n}(\psi_{n}-\psi_{n-1})\,d\bm{x}\,d\bm{p}\right|$
(B.16)
$\displaystyle\quad+\lVert\nabla\bm{u}_{n-2}\rVert_{L^{\infty}_{x}}\lVert\psi_{n}-\psi_{n-1}\rVert_{L^{2}_{x,p}}^{2}$
$\displaystyle\quad+\lVert\nabla(\bm{u}_{n-1}-\bm{u}_{n-2})\rVert_{L^{2}_{x}}\lVert\psi_{n}\rVert_{L^{\infty}_{x}L^{2}_{p}}\lVert\nabla_{p}(\psi_{n}-\psi_{n-1})\rVert_{L^{2}_{x,p}}$
$\displaystyle\quad-\nu\lVert\nabla_{p}(\psi_{n}-\psi_{n-1})\rVert_{L^{2}_{x,p}}^{2}-\kappa\lVert\nabla_{x}(\psi_{n}-\psi_{n-1})\rVert_{L^{2}_{x,p}}^{2}\,.$

In dimension three and on $\mathbb{T}^{2}$, we have

$\displaystyle\left|\int_{x,p}(\bm{u}_{n-1}-\bm{u}_{n-2})\cdot\nabla_{x}\psi_{n}(\psi_{n}-\psi_{n-1})\,d\bm{x}\,d\bm{p}\right|$
(B.17)
$\displaystyle\qquad\lesssim\|\bm{u}_{n-1}-\bm{u}_{n-2}\|_{L^{6}_{x}}\|\nabla_{x}\psi_{n}\|_{L^{3}}\|\psi_{n}-\psi_{n-1}\|_{L^{2}}\,.$

The approach on $\mathbb{R}^{2}$ is more subtle, and we return to it later. By
the _a priori_ bound (B.14), we have that
$\lVert\psi_{n}\rVert_{H^{2}_{x}L^{2}_{p}}$ is controlled for sufficiently
small time. Furthermore, by (B.10), we may bound
$\lVert\nabla(\bm{u}_{n-1}-\bm{u}_{n-2})\rVert_{L^{2}_{x}}$ by
$\lVert\psi_{n-1}-\psi_{n-2}\rVert_{L^{2}_{x}}$. Using Young’s inequality to
absorb the $\lVert\nabla_{p}(\psi_{n}-\psi_{n-1})\rVert_{L^{2}_{x,p}}$ term
into the rotational diffusion term, we obtain

$\frac{d}{dt}\lVert\psi_{n}-\psi_{n-1}\rVert_{L^{2}_{x,p}}^{2}\lesssim_{\nu}\lVert\psi_{n-1}-\psi_{n-2}\rVert_{L^{2}_{x,p}}^{2}+\lVert\psi_{n}-\psi_{n-1}\rVert_{L^{2}_{x,p}}^{2}\,.$
(B.18)

This is a differential inequality of the form

$\frac{d}{dt}A_{n}\leq MA_{n-1}+MA_{n}\,,$ (B.19)

where $M>0$ is independent of $n$ and each $A_{n}(t)\geq 0$. By Grönwall’s
inequality,

$A_{n}(t)\leq M\int_{0}^{t}e^{M(t-s)}A_{n-1}(s)\,ds\,.$ (B.20)

In particular, for sufficiently small $T$, we have
$\sup_{t\in(0,T)}A_{n}\leq\frac{1}{2}\sup_{t\in(0,T)}A_{n-1}$. Thus, we have
strong convergence of the Picard iterates $\psi_{n}\to\psi_{\infty}$ in
$C_{t}L^{2}_{x}L^{2}_{p}$ for short times. By lower semicontinuity, the _a
priori_ estimates (B.14) persist as $n\to+\infty$. Then we allow the
mollification parameter $\varepsilon\to 0^{+}$. That the solutions belong to
$C([0,T];H^{2}_{x}L^{2}_{p}\cap L^{1}_{x}L^{1}_{p})$ can be justified after
the fact via the linear theory. Looking ahead, once uniqueness is known, we
can extend the solution to its maximal time of existence, for which a lower
bound is $\bar{T}$ from Step 1.

_2b. Contraction in $\mathbb{R}^{2}$_. We now address the two-dimensional
setting. We consider also estimates satisfied by
$\|\psi_{n}-\psi_{n-1}\|_{L^{1}}$, namely (with $q=2+\varepsilon$,
$0<\varepsilon\ll 1$),

$\displaystyle\frac{d}{dt}\|\psi_{n}-\psi_{n-1}\|_{L^{1}}$ (B.21)
$\displaystyle\quad\leq\|\bm{u}_{n-1}-\bm{u}_{n-2}\|_{L^{q}}\|\nabla_{x}\psi_{n}\|_{L^{q^{\prime}}}+\|\operatorname{div}_{p}[(\mathbf{I}-\bm{p}\otimes\bm{p})\nabla(\bm{u}_{n-1}-\bm{u}_{n-2})\bm{p}\psi_{n}]\|_{L^{1}}$
$\displaystyle\quad\lesssim(\|\psi_{n-1}-\psi_{n-2}\|_{L^{1}}+\|\psi_{n-1}-\psi_{n-2}\|_{L^{2}})\|\psi_{n}\|_{L^{1}_{x}L^{1}_{p}\cap
H^{2}_{x}L^{2}_{p}}$
$\displaystyle\quad\quad+\|\psi_{n-1}-\psi_{n-2}\|_{L^{2}}\|\psi_{n}\|_{L^{2}_{x}H^{1}_{p}}\,.$

To complete the $L^{2}$ estimate, we write

$\displaystyle\left|\int_{x,p}(\bm{u}_{n-1}-\bm{u}_{n-2})\cdot\nabla_{x}\psi_{n}(\psi_{n}-\psi_{n-1})\,d\bm{x}\,d\bm{p}\right|$
(B.22)
$\displaystyle\quad\lesssim\|\bm{u}_{n-1}-\bm{u}_{n-2}\|_{L^{4}_{x}}\|\nabla_{x}\psi_{n}\|_{L^{4}}\|\psi_{n}-\psi_{n-1}\|_{L^{2}}$
$\displaystyle\quad\lesssim\|\nabla_{x}\psi_{n}\|_{L^{4}}(\|\psi_{n-1}-\psi_{n-2}\|_{L^{1}}+\|\psi_{n-1}-\psi_{n-2}\|_{L^{2}})\|\psi_{n}-\psi_{n-1}\|_{L^{2}}\,.$

We now conclude via the differential inequality for
$\|\psi_{n}-\psi_{n-1}\|_{L^{1}}^{2}+\|\psi_{n}-\psi_{n-1}\|_{L^{2}}^{2}$.

_(Uniqueness)_. Given two strong solutions $\psi$, $\widetilde{\psi}$ on
$\Omega^{d}\times S^{d-1}\times(0,T)$, we consider again the differential
inequality satisfied by $\|\psi-\widetilde{\psi}\|_{L^{2}}^{2}$ (or, when
$\Omega^{d}=\mathbb{R}^{2}$,
$\|\psi-\widetilde{\psi}\|_{L^{1}}^{2}+\|\psi-\widetilde{\psi}\|_{L^{2}}^{2}$).
Estimates analogous to (B.18) grant uniqueness. ∎

##  Appendix C Proof of Lemma 4.3

Here, we prove the Poincaré-type inequality (4.22).

###### Proof.

Without loss of generality, we may choose $\bm{k}=k\bm{e}_{1}$ with
$k=|\bm{k}|$ and define
$\bm{p}=p_{1}\bm{e}_{1}+p_{2}\bm{e}_{2}+p_{3}\bm{e}_{3}$ with
$p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1$.

Away from $p_{1}^{2}=1$, we may bound the full spatial gradient $k\lVert
f_{k}\rVert_{L^{2}}$ in terms of the spatial gradient projected off the unit
sphere, i.e.,
$k\lVert(\bm{e}_{1}-p_{1}\bm{p})f_{k}\rVert_{L^{2}}=k\lVert\sqrt{1-p_{1}^{2}}f_{k}\rVert_{L^{2}}$.
Near $p_{1}^{2}=1$, we must instead control $k\lVert f_{k}\rVert_{L^{2}}$
using the orientational gradient $\lVert\nabla_{p}f_{k}\rVert_{L^{2}}$.

We thus define a cutoff function on the unit sphere:

$\varphi_{\delta}=\begin{cases}1,&1-p_{1}^{2}<\delta\\\
0,&1-p_{1}^{2}>2\delta\end{cases}$ (C.1)

for some $0<\delta<\frac{1}{2}$, with smooth decay between. Then, away from
$p_{1}^{2}=1$, we have

$\lVert
f_{k}(1-\varphi_{\delta})\rVert_{L^{2}}^{2}\leq\frac{1}{\delta}\lVert\sqrt{1-p_{1}^{2}}f_{k}\rVert_{L^{2}}^{2}\,.$
(C.2)

Near $p_{1}^{2}=1$, we have

$\displaystyle\lVert f_{k}\varphi_{\delta}\rVert_{L^{2}}^{2}$
$\displaystyle\leq
c\delta\lVert\nabla_{p}(f_{k}\varphi_{\delta})\rVert_{L^{2}}^{2}\leq
c\delta\bigg{(}\lVert(\nabla_{p}f_{k})\varphi_{\delta}\rVert_{L^{2}}^{2}+\lVert
f_{k}(\nabla_{p}\varphi_{\delta})\rVert_{L^{2}}^{2}\bigg{)}$ (C.3)
$\displaystyle\leq
c\delta\bigg{(}\lVert\nabla_{p}f_{k}\rVert_{L^{2}}^{2}+\frac{1}{\delta^{2}}\lVert\sqrt{1-p_{1}^{2}}f_{k}\rVert_{L^{2}}^{2}\bigg{)}\,,$

where we have used that
$|\nabla_{p}\varphi_{\delta}|\lesssim\frac{1}{\delta^{1/2}}$ is supported
within the strip $\delta\leq 1-p_{1}^{2}\leq 2\delta$. Together, we obtain

$k^{2}\lVert
f_{k}\rVert_{L^{2}}^{2}\lesssim\frac{k^{2}}{\delta}\lVert\sqrt{1-p_{1}^{2}}f_{k}\rVert_{L^{2}}^{2}+\delta
k^{2}\lVert\nabla_{p}f_{k}\rVert_{L^{2}}^{2}\,,$ (C.4)

and, choosing $\delta=\frac{1}{4}\nu^{1/2}k^{-1/2}$, we obtain Lemma 4.3. ∎

### Acknowledgments

DA was supported by NSF Postdoctoral Fellowship Grant No. 2002023 and Simons
Foundation Grant No. 816048. LO acknowledges support from NSF Postdoctoral
Fellowship DMS-2001959.

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