# Enhanced dissipation and Hörmander’s hypoellipticity

Dallas Albritton  Courant Institute of Mathematical Sciences, New York
University, New York, NY 10012<EMAIL_ADDRESS>,  Rajendra Beekie
Courant Institute of Mathematical Sciences, New York University, New York, NY
10012<EMAIL_ADDRESS>and  Matthew Novack  Courant Institute of
Mathematical Sciences, New York University, New York, NY 10012
<EMAIL_ADDRESS>
###### Abstract.

We examine the phenomenon of enhanced dissipation from the perspective of
Hörmander’s classical theory of second order hypoelliptic operators [31].
Consider a passive scalar in a shear flow, whose evolution is described by the
advection–diffusion equation

$\partial_{t}f+b(y)\partial_{x}f-\nu\Delta f=0\text{ on
}\mathbb{T}\times(0,1)\times\mathbb{R}_{+}$

with periodic, Dirichlet, or Neumann conditions in $y$. We demonstrate that
decay is enhanced on the timescale $T\sim\nu^{-(N+1)/(N+3)}$, where $N-1$ is
the maximal order of vanishing of the derivative $b^{\prime}(y)$ of the shear
profile and $N=0$ for monotone shear flows. In the periodic setting, we
recover the known timescale of Bedrossian and Coti Zelati [8]. Our results are
new in the presence of boundaries.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Bracket inequalities
  4. 4 Interpolation inequality
  5. 5 Proofs of Theorems 1.1 and 1.2
  6. A Proof of sample interpolation inequality

##  1\. Introduction

The advection–diffusion equation

$\displaystyle\partial_{t}f+\boldsymbol{u}\cdot\nabla f-\nu\Delta f$
$\displaystyle=0$ (1.1)

describes the evolution of a passive scalar $f$ which is transported by a
divergence-free velocity field $\boldsymbol{u}$ and simultaneously diffuses.
It is well known that the interplay between transport
$\boldsymbol{u}\cdot\nabla$ and weak diffusion $\nu\Delta$ ($0<\nu\ll 1$) may
cause solutions of (1.1) to decay more rapidly than by diffusion alone. This
phenomenon is known as _enhanced dissipation_.

Our goal is to clarify the relationship between enhanced dissipation and
Hörmander’s classical theory of hypoelliptic second order operators, developed
in [31]. We restrict our attention to the class of _shear flows_
$\boldsymbol{u}=(b(y),0)$. Enhanced dissipation within this class has already
been studied by Bedrossian and Coti Zelati in [8]. We will sharpen and extend
their results, though perhaps the main interest of our work is in the method
we employ, which is a new application of the old ideas of Hörmander.

We consider (1.1) on the spacetime domain $\Omega\times(0,+\infty)$, where
$\Omega=\mathbb{T}\times D$ and $D=\mathbb{T}$ or $(0,1)$. By convention, we
identify functions on the torus $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ with
functions satisfying periodic boundary conditions on $(0,1)$. When $D=(0,1)$,
we impose homogeneous Dirichlet or Neumann conditions on $\\{y=0,1\\}$. The
shear profile $b\in C^{\infty}(\bar{D})$ is assumed to have precisely $M\geq
0$ critical points. When $M\geq 1$, we denote these by $y_{i}$, $1\leq i\leq
M$. We further require that the maximal order of vanishing
$N_{i}\in\mathbb{N}$ at $y_{i}$ is finite:

$\displaystyle b^{(k)}(y_{i})=0\text{ for all }1\leq k\leq N_{i}$ (1.2a)
$\displaystyle b^{(N_{i}+1)}(y_{i})\neq 0\,.$ (1.2b)

Let $N=\max_{1\leq i\leq M}N_{i}$, or $N=0$ when $M=0$.

Since $\boldsymbol{u}\cdot\nabla=b(y)\partial_{x}$ in (1.1), the $x$-averages

$\langle f\rangle(y,t)=\int_{\mathbb{T}}f(x,y,t)\,dx$

satisfy the one-dimensional heat equation

$\partial_{t}\langle f\rangle-\nu\partial_{yy}\langle f\rangle=0\,$

and in general decay only on the diffusive timescale $\sim\nu^{-1}$. To
extract an enhanced dissipation timescale, it is necessary to subtract
$\langle f\rangle$. Without loss of generality, we thus require that the
initial data $f_{\rm in}$ satisfies

$\langle f_{\rm in}\rangle(y):=\int_{\mathbb{T}}f_{\rm in}(x,y)\,dx\equiv
0\,.$

It is instructive to extract the dissipation timescale in the Couette example
$b(y)=y$ posed for $(x,y)\in(\mathbb{R}/2\pi\mathbb{Z})\times\mathbb{R}$. In
this case, (1.1) is explicitly solvable, as was already known to Kelvin in
[35]. Upon taking the Fourier transform, we have

$\partial_{t}\widehat{f}-k\partial_{\eta}\widehat{f}+\nu(|k|^{2}+|\eta|^{2})\widehat{f}=0\,.$

The solution is

$\widehat{f}(k,\eta,t)=e^{-\nu k^{2}t}e^{-\nu t(\eta^{2}+\eta
kt+k^{2}t^{2}/3)}\widehat{f_{\rm in}}(k,\eta+kt)\,.$ (1.3)

The factor $e^{-\nu|k|^{2}t^{3}/3}$ dominates for large times and yields the
enhancement timescale $T\sim\nu^{-1/3}$, since $|k|\gtrsim 1$ when we
disregard the $k=0$ modes. The fundamental solution on
$\mathbb{R}^{2}\times\mathbb{R}_{+}$ was also famously computed by Kolmogorov
[37] (see also Hörmander’s introduction [31]).

In [8], Bedrossian and Coti Zelati identified the following enhanced
dissipation timescales depending on the shear profile:

$T\sim\nu^{-\frac{N+1}{N+3}}$ (1.4)

in the periodic-in-$y$ setting, up to a factor $(1+\log\nu^{-1})^{2}$ which
was removed by Wei [44]. The enhancement is measured in terms of the
‘flatness’ of the shear profile at its critical points. Since the Couette flow
has no critical points, the above timescale for $N=0$ matches the timescale
extracted from (1.3). These timescales are essentially sharp, as demonstrated
by Drivas and Coti Zelati in [18]. In the Neumann setting, Bedrossian and Coti
Zelati obtain the same timescale except in the monotone case $N=0$, in which
the timescale $T\sim\nu^{-1/2}$ (up to a logarithmic correction) is obtained
instead of $T\sim\nu^{-1/3}$. We demonstrate that the enhanced dissipation
timescale is, at worst, (1.4) in the Dirichlet and Neumann settings for all
$N\in\mathbb{N}_{0}$.

To better clarify the connections to hypoellipticity, we consider the
_hypoelliptic_ advection-diffusion equation proposed in [8]:

$\displaystyle\partial_{t}f+b(y)\partial_{x}f-\nu\partial_{yy}f$
$\displaystyle=0.$ (1.5)

The PDEs (1.1) and (1.5) are actually _equivalent_ in the following sense: If
$f$ is a solution of (1.5) with initial data $f_{\rm in}$, then $g=e^{\nu
t\partial_{xx}}f$ is a solution of (1.1). In the sequel, we focus only on
smooth solutions of (1.5): For each $f_{\rm in}\in C^{\infty}(\bar{\Omega})$
satisfying the periodic, Dirichlet, or Neumann boundary conditions, there
exists a unique smooth solution $f\in C([0,+\infty);L^{2}(\Omega))\cap
C^{\infty}(\bar{\Omega}\times(0,+\infty))$ to (1.5) satisfying
$f(\cdot,0)=f_{\rm in}$ and the desired boundary conditions. Smoothness up to
$t=0$ would require additional compatibility conditions on $f_{\rm in}$. _We
assume that $\langle f\rangle=\langle f_{\rm in}\rangle\equiv 0$._

Our main theorems are the following:

######  Theorem 1.1 (Enhanced dissipation).

There exist $d_{0},\nu_{0}>0$ and $C>1$, depending only on $b$, such that the
solution $f$ of (1.5) with periodic, Dirichlet, or Neumann conditions
satisfies

$\left\|f(\cdot,t)\right\|_{L^{2}(\Omega)}\leq C\left\|f_{\rm
in}\right\|_{L^{2}(\Omega)}\exp\left(-d_{0}\nu^{\frac{N+1}{N+3}}t\right)$

for all $t>0$ and $\nu\in(0,\nu_{0})$.

######  Theorem 1.2 (Gevrey-in-$x$ smoothing).

There exist $d_{0},\nu_{0}>0$ depending only on $b$ such that the solution $f$
of (1.5) is instantly Gevrey-$p$ regular for all $p>\frac{N+3}{2}$ and
$\nu\in(0,\nu_{0})$:

$\left\lVert\exp\left(d_{0}\nu^{\frac{N+1}{N+3}}|\partial_{x}|^{\frac{1}{p}}t\right)f(\cdot,t)\right\rVert_{L^{2}(\Omega)}\lesssim_{b,p}\left\|f_{\rm
in}\right\|_{L^{2}(\Omega)}.$

######  Remark 1.3 (Improvements).

Theorems 1.1 and 1.2 improve the main theorems of Bedrossian and Coti Zelati
[8] and Wei [44] in the following ways:

  * •

We treat Dirichlet conditions in addition to periodic and Neumann conditions.

  * •

In the Neumann setting, we improve the timescale from $T\sim\nu^{-1/2}$ in [8]
to $T\sim\nu^{-1/3}$ when $N=0$, and we remove the additional logarithmic
factors when $N\in\mathbb{N}_{0}$.

In the periodic-in-$y$ setting, $p=\frac{N+3}{2}$ is also possible in Theorem
1.2, as demonstrated by Wei [44] using resolvent estimates and a
Gearhardt–Prüss-type theorem.

######  Remark 1.4 (Generalizations).

Our approach generalizes, with minimal effort, to divergence-form operators
$\partial_{y}(a(y)\partial_{y}\cdot)$ satisfying $a\in L^{\infty}(D)$ and
$a\geq c>0$. It is also possible to generalize to operators
$\operatorname{div}(A(y)\nabla\cdot)$. The norms in the ‘interpolation
inequality’, Proposition 4.5, would need to be adjusted to incorporate $x$
derivatives.111Roughly speaking, one should measure $\|\nabla f\|_{L^{2}}$ and
$\|X_{0}f\|_{L^{2}_{t}\dot{H}^{-1}_{x,y}}$ instead of
$\|\partial_{y}f\|_{L^{2}}$ and $\|X_{0}f\|_{L^{2}_{t,x}\dot{H}^{-1}_{y}}$.
Doing so would allow us to treat circular flows
$\boldsymbol{u}=r\Omega(r)e_{\theta}$ in an annulus $r_{0}<r<r_{1}$, where
$\Omega$ has finitely many critical points. It is also possible to generalize
to shear flows which vary on the enhancement timescale
$\nu^{-\frac{N+1}{N+3}}$, though the time translation symmetry is convenient
in the proof of Proposition 4.5. We view these generalizations as an
indication that, in a certain sense, Hörmander’s method is robust.

The most naïve approach to Theorem 1.1 is to try to generalize the $L^{2}$
energy estimate for the heat equation $\partial_{t}f=\nu\Delta f$ on
$\mathbb{T}^{2}\times\mathbb{R}_{+}$:

$\frac{1}{2}\frac{d}{dt}\|f\|_{L^{2}}^{2}=-\nu\|\nabla f\|_{L^{2}}^{2}\leq-
C_{P}^{-1}\nu\|f\|_{L^{2}}^{2}\,,$ (1.6)

where $C_{P}=4\pi^{2}$ is the Poincaré constant and $f$ has zero mean. This
differential inequality implies $\|f\|_{L^{2}}^{2}\leq e^{-\nu
t/C_{P}}\|f_{\rm in}\|_{L^{2}}^{2}$. For the hypoelliptic PDE (1.5),
$-\nu\|\partial_{y}f\|_{L^{2}}^{2}$ appears on the right-hand side of the
energy estimate, and we lack a Poincaré inequality for this quantity. The goal
of hypoellipticity is precisely to extract differentiability-in-$x$ and, thus,
a Poincaré inequality, from the $L^{2}$ energy estimate and the PDE. In this
way, we salvage the naïve approach.

The desire to reproduce (1.6) is also a motivation for hypocoercivity. We
compare the two approaches after we sketch the main idea.

In the classical terminology, a partial differential operator $P$ with smooth
coefficients on an open set $U\subset\mathbb{R}^{d}$ is _hypoelliptic_ if
distributional solutions of $Pu=f$ are smooth in $U$ for $f\in C^{\infty}(U)$.
Let $X_{0},X_{1},\ldots,X_{J}$ be smooth vector fields on $U$. In [31],
Hörmander characterized hypoellipticity for second order operators

$L=\sum_{j=1}^{J}X_{j}^{*}X_{j}+X_{0}\,.$

$L$ is hypoelliptic _if and only if_ the Lie algebra generated by
$\\{X_{j}\\}_{j=0}^{J}$ spans $\mathbb{R}^{d}$ at every point $x_{0}\in U$.
Perhaps the main observation in [31] is that regularity ‘along’ two vector
fields $X$ and $Y$ implies regularity along their commutator $[X,Y]$ and sum
$X+Y$, up to certain controllable errors.

Hörmander’s theorem implies that solutions to (1.5) on
$\mathbb{T}^{2}\times\mathbb{R}_{+}$ are smooth, but it does not evidently
imply Theorem 1.1. Rather, we revisit Hörmander’s scheme (i) in a more
quantitative manner, (ii) with boundary, and (iii) in the singular limit as
$\nu\to 0^{+}$. On the other hand, our problem admits technical
simplifications that, we hope, illuminate the original analysis in [31].

Our work is partially inspired by the recent work of Armstrong and Mourrat
[3], who applied Hörmander’s methods in the context of the kinetic
Fokker–Planck equation. Hörmander’s analysis was also also recently revisited
by Bedrossian and coauthors in [7, 9] in the context of stochastic
differential equations.

### Sketch

Let us provide an exposition of the main steps in our proof. Consider a smooth
solution $f\colon\Omega\times\mathbb{R}_{+}\to\mathbb{R}$ to (1.5) with
periodic or homogeneous Dirichlet or Neumann boundary conditions and smooth
initial data $f_{\rm in}$ satisfying $\langle f_{\rm in}\rangle\equiv 0$.
Using the boundary conditions, the basic energy estimate is

$\frac{1}{2}\|f(\cdot,t)\|_{L^{2}}^{2}+\nu\int_{0}^{t}\int_{\Omega}|\partial_{y}f|^{2}\,dx\,dy\,ds=\frac{1}{2}\|f_{\rm
in}\|_{L^{2}}^{2},\quad\forall t\geq 0\,.$

This energy estimate furnishes two crucial pieces of information, which, to
avoid more notation, we present here only for Dirichlet boundary conditions.
Evidently,

$\nu\|f\|_{L^{2}_{t,x}(H^{1}_{0})_{y}(\Omega\times\mathbb{R}_{+})}^{2}\lesssim\|f_{\rm
in}\|_{L^{2}}^{2}\,.$ (1.7)

Then (1.7) and the PDE itself
$X_{0}f:=(\partial_{t}+b(y)\partial_{x})f=\nu\partial_{yy}f$ give

$\nu^{-1}\|X_{0}f\|_{L^{2}_{t,x}H^{-1}_{y}(\Omega\times\mathbb{R}_{+})}^{2}\lesssim\|f_{\rm
in}\|_{L^{2}}^{2}\,.$ (1.8)

As in [3], we could define a norm $\|\cdot\|_{H^{1,\nu}_{\rm hyp}}$ based on
the quantities controlled in (1.7) and (1.8), though it is not strictly
necessary.222Membership in $H^{1,\nu}_{\rm hyp}$ is actually equivalent to
solving the PDE with RHS in $L^{2}_{t,x}H^{-1}_{y}$. Hypoellipticity is
subsequently extracted from _embedding theorems_ for $H^{1,\nu}_{\rm
hyp}$.333As a consequence, after the initial energy estimate, time no longer
plays a distinguished role. It is possible to distinguish the role of time in
a ‘parabolic’ Hörmander method; see, for example, Bedrossian and Liss [9].
Notably, we have

######  Proposition 1.5 (Subelliptic estimate and Poincaré inequality,
Dirichlet case).

$\nu^{\frac{N+1}{N+3}}\|f\|_{L^{2}(\Omega_{T})}^{2}+\nu^{\frac{N+1}{N+3}}\|f\|_{Q^{1/(N+3)}_{\partial_{x}}(\Omega_{T})}^{2}\lesssim_{b}\nu\|f\|_{L^{2}_{t,x}H^{1}_{0,y}(\Omega_{T})}^{2}+\nu^{-1}\|X_{0}f\|_{L^{2}_{t,x}H^{-1}_{y}(\Omega_{T})}^{2}\,,$
(1.9)

where $\Omega_{T}=\Omega\times(0,T)$, $T=T_{0}\nu^{-\frac{N+1}{N+3}}$,
$\nu\leq\nu_{0}$, and $\nu_{0},T_{0}>0$ depend only on the shear profile $b$.

The quantity $\|\cdot\|_{Q^{1/(N+3)}_{\partial_{x}}}$ is a Besov-type norm
measuring $1/(N+3)$-differentiability in the direction $\partial_{x}$, and it
controls $\|f\|_{L^{2}}$ when $\langle f\rangle\equiv 0$ as above. In the
terminology of [3], (1.9) is a _Hörmander inequality_. See Theorem 4.6 for the
periodic, Dirichlet, and Neumann settings.

With (1.9) in hand, we conclude the proof of exponential decay according to a
Grönwall-type argument along a discrete set of times (see Section 5). The
discrete-in-time proof is natural in the following sense. For the Couette
example (1.3), the differential inequality
$\nicefrac{{d}}{{dt}}\|f(\cdot,t)\|_{L^{2}}^{2}\leq-
C_{0}\nu^{\nicefrac{{1}}{{3}}}\|f(\cdot,t)\|_{L^{2}}^{2}$ would imply
$\|f(\cdot,t)\|_{L^{2}}^{2}\leq e^{-C_{0}\nu^{\nicefrac{{1}}{{3}}}t}\|f_{\rm
in}\|_{L^{2}}^{2}$, which is false, as we see from the solution formula (1.3).
Indeed, the superlinear decay in the term $e^{-\nu k^{2}t^{3}/3}$ indicates
that ‘nothing happens’ until the ‘crossover time’, which is proportional to
$\nu^{-\nicefrac{{1}}{{3}}}$. In light of this, the prefactor $C>1$ in Theorem
1.1 and the scaling of $T_{0}$ in Proposition 1.5 are _essential_.

### Sample Hörmander inequalities

To convey the main idea of the embedding (1.9), we demonstrate a simple
embedding on $\mathbb{R}^{2}$, which is an analogue of (1.9) for the PDE
$y\partial_{x}f-\nu\partial_{yy}f=g$. To quantify the fractional
differentiability, we define the following Besov-type seminorms based on
finite differences:

$\|f\|_{Q^{1}_{\partial_{y}}}:=\sup_{h\in\mathbb{R}\setminus\\{0\\}}h^{-1}\|f(x,y+h)-f(x,y)\|_{L^{2}}$
$\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}:=\sup_{h\in\mathbb{R}\setminus\\{0\\}}h^{-1}\|f(x+\operatorname{sgn}(h)h^{2}y,y)-f(x,y)\|_{L^{2}}$
$\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}:=\sup_{h\in\mathbb{R}\setminus\\{0\\}}h^{-1}\|f(x+h^{3},y)-f(x,y)\|_{L^{2}}\,.$

We evidently have
$\|f\|_{Q^{1}_{\partial_{y}}}\lesssim\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}$ (see
the characterization of Sobolev norms by finite differences in Evans [22]).
The quantity $\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}$, which measures
$\nicefrac{{1}}{{2}}$ derivative in the $y\partial_{x}$ direction, will be
controlled by the ‘interpolation inequality’ below. The analogous estimate for
the heat equation is $f\in L^{2}_{t}H^{1}_{x}\cap H^{1}_{t}H^{-1}_{x}\subset
H^{\nicefrac{{1}}{{2}}}_{t}L^{2}_{x}$ on $\mathbb{R}^{d}\times(0,T)$.

In the following, we require that the terms on the right-hand side of the
inequalities (1.10), (1.12) and (1.15) are _well defined and non-vanishing_
(which is necessary in the optimization steps). Ultimately, we will
demonstrate

######  Lemma 1.6 (Sample subelliptic estimate).

In the above notation,

$\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}\lesssim\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}^{\nicefrac{{2}}{{3}}}\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}^{\nicefrac{{1}}{{3}}}\
.$ (1.10)

The explains, in part, the emergence of the powers of $\nu$ in the main
theorem. If we suppose
$\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}\lesssim\nu^{-\nicefrac{{1}}{{2}}}$ and
$\|f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}\lesssim\nu^{\nicefrac{{1}}{{2}}}$, then we
recover
$\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}\lesssim\nu^{-\nicefrac{{1}}{{6}}}$,
analogous to the time-dependent subelliptic estimate (1.9) with $N=0$. If we
replace $y\partial_{x}f$ by $y^{N+1}\partial_{x}f$ in (1.10), then we would
expect

$\|f\|_{Q^{1/(N+1)}_{\partial_{x}}}\lesssim\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}^{\frac{N+2}{N+3}}\|y^{N+1}\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}^{\frac{1}{N+3}}\,,$
(1.11)

though we do not demonstrate this here (see Farkas and Lunardi [23] for
similar estimates).

The subelliptic estimate (1.10) follows from combining the ‘bracket
inequality’ and the ‘interpolation inequality’. The bracket inequality is
based on the identity $e^{-Bt}e^{-At}e^{Bt}e^{At}=e^{t^{2}[A,B]+O(t^{3})}$ for
vector fields $A$, $B$. In the proof, this inequality manifests in (1.14),
where the $O(t^{3})$ term vanishes due to the special structure of
$\partial_{y}$ and $y\partial_{x}$.

######  Lemma 1.7 (Sample bracket inequality).

In the above notation,

$\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}\lesssim\|f\|_{Q^{1}_{\partial_{y}}}^{\nicefrac{{1}}{{3}}}\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}^{\nicefrac{{2}}{{3}}}\,.$
(1.12)

###### Proof.

Let $h\in\mathbb{R}$. If we demonstrate

$\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}\lesssim\|f\|_{Q^{1}_{\partial_{y}}}+\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}\,,$
(1.13)

then we consider $f_{\lambda}(x,y)=f(\lambda x,y)$ and optimize $\lambda>0$ to
complete the proof. We write

$\displaystyle f(x+h^{3},y)-f(x,y)$
$\displaystyle=f(x+h^{3},y)-f(x+(y+h)h^{2},y)$ (1.14)
$\displaystyle+f(x+(y+h)h^{2},y)-f(x+(y+h)h^{2},y+h)$
$\displaystyle+f(x+(y+h)h^{2},y+h)-f(x,y+h)$
$\displaystyle+f(x,y+h)-f(x,y)\,.$

First, we treat the terms based on ‘flowing’ in the direction
$\mp\partial_{y}$:

$\|f(x+(y+h)h^{2},y)-f(x+(y+h)h^{2},y+h)\|_{L^{2}}=\|f(x,y+h)-f(x,y)\|\lesssim
h\|f\|_{Q^{1}_{\partial_{y}}}\,.$

Second, we treat the terms based on ‘flowing’ in the direction $\mp
y\partial_{x}$:

$\|f(x+h^{3},y)-f(x+(y+h)h^{2},y)\|_{L^{2}}=\|f(x+(y+h)h^{2},y+h)-f(x,y+h)\|_{L^{2}}\lesssim
h\|f\|_{Q^{\nicefrac{{1}}{{2}}}}\,.$

Combining the estimates completes the proof of (1.13). ∎

The interpolation inequality is more difficult (see Section 5 in Hörmander’s
paper [31]), and we delay the proof to Appendix A:

######  Lemma 1.8 (Example interpolation inequality).

In the above notation,

$\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}\lesssim\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}^{\nicefrac{{1}}{{2}}}\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}^{\nicefrac{{1}}{{2}}}\,.$
(1.15)

The main difficulty in (1.15) can be seen in the following way. Let
$X_{0}=y\partial_{x}$ and $e^{h^{2}X_{0}}f=f(x+h^{2}y,y)$. To achieve
$\|e^{h^{2}X_{0}}f-f\|_{2}^{2}\lesssim h^{2}$, one differentiates the left-
hand side in $h$. Then

$\displaystyle\frac{d}{dh}\|e^{h^{2}X_{0}}f-f\|_{2}^{2}=4h\langle
e^{h^{2}X_{0}}f-f,X_{0}f\rangle\,,$

which is estimated by duality: $X_{0}f\in L^{2}_{x}\dot{H}^{-1}_{y}$ and
$e^{h^{2}X_{0}}f-f\in L^{2}_{x}\dot{H}^{1}_{y}$. However,
$\|\partial_{y}f\|_{L^{2}}$ is not generally preserved under the flow of
$X_{0}$, as we must commute $\partial_{y}$ with $e^{h^{2}X_{0}}$. The chain
rule gives
$\partial_{y}e^{h^{2}X_{0}}f=h^{2}\partial_{x}e^{h^{2}X_{0}}f+e^{h^{2}X_{0}}\partial_{y}f$,
and the situation looks bleak because $\partial_{x}f$ is uncontrolled in
$L^{2}$. Hörmander’s method involves the above computation for a smooth
function $S_{\ell}f$, obtained by mollifying $f$ in the direction
$[Y,X_{0}]=\partial_{x}$ at scale $\ell>0$. The additional mollification makes
it possible to exploit the small prefactor $h^{2}$. The complete proof is
contained in Appendix A.

### Further difficulties

Foremost is the Taylor remainder term, which vanished in the calculation
(1.14). In our setting, the left-hand side of the analogous bracket
calculation contains $f(x+(b(y+h)-b(y))h^{2},y)$ rather than
$f(x+b^{\prime}(y)h^{3},y)$. This error is estimated in terms of further
commutators. That is, to control $\nicefrac{{1}}{{3}}$ derivative in the
direction $b^{\prime}(y)\partial_{x}$, we already require control of
$\nicefrac{{1}}{{4}}$ derivative in the direction
$b^{\prime\prime}(y)\partial_{x}$, $\nicefrac{{1}}{{5}}$ derivative in the
direction $b^{(3)}\partial_{x}$, etc. Eventually, these errors can be absorbed
by applying the trick $[Y,X_{0}]=[\varepsilon^{-1}Y,\varepsilon X_{0}]$ to
introduce small constants.

The interpolation inequality also exhibits a ‘cascading effect’ and requires
mollification along the vector fields $[Y,X_{0}]$, $[Y,[Y,X_{0}]]$, etc.
Finally, the boundaries in $y$ when $D=(0,1)$ and in the time $t$ restrict the
‘admissible finite differences’ $h\in\mathbb{R}\setminus\\{0\\}$. One requires
$|h|\leq h_{0}$ with judicious $\nu$-dependent choices of $h_{0}$. We
emphasize that the more tedious calculations present in this paper essentially
result from either Taylor remainders, interpolation, or admissible finite
differences. The scaling with respect to $\nu$ which produces the enhanced
dissipation timescale is already visible in the discussion following Lemma 1.6
and is unrelated to these technicalities.

### Comparison with hypocoercivity

A commonly used method to prove enhanced dissipation [8] is _hypocoercivity_ ,
whose mathematical framework was developed by Villani in [43]. Hypocoercivity
has roots in kinetic theory444See also the weighted energy method of Guo in
[27]. and was later adapted to PDEs of fluid dynamics in [24, 6]. Its premise
is to design a new energy functional $\Phi$ which satisfies the differential
inequality $\dot{\Phi}\leq-C_{0}\Phi$. Commonly, $\Phi$ is equivalent to
$\|\cdot\|_{H^{1}_{x,y}}^{2}$ and is ‘augmented’ by certain commutators; for
example,

$\Phi[f]:=\|f\|_{2}^{2}+a\nu^{2/3}\|\partial_{y}f\|_{2}^{2}+2b\nu^{1/3}\langle\partial_{x}f,\partial_{y}f\rangle+c\|\partial_{x}f\|_{2}^{2}$

in the Couette case with suitable weights $0<c\ll b\ll a\ll 1$. We should
interpret $\partial_{x}$ as the commutator $[\partial_{y},y\partial_{x}]$
above. Perhaps the main advantage of hypocoercivity is that it is elementary,
consisting only of well-chosen energy estimates and integration by parts.

Hypocoercivity naturally lives at higher regularity $\approx
L^{\infty}_{t}H^{1}_{x,y}$ than Hörmander’s method, which lives at the level
of the basic energy estimate $\approx L^{\infty}_{t}L^{2}_{x,y}\cap
L^{2}_{t,x}H^{1}_{y}$. As a consequence, hypocoercive methods encounter two
technical difficulties that are absent from our approach:

  * •

$L^{2}$ stability estimates require further arguments that exploit the
smoothing. In [24, 8], this is addressed by passing through the resolvent
formalism: The hypocoercivity method estimates the pseudospectrum and,
therefore, the semigroup. It is this step which contributes the logarithmic
correction, which can be removed by the Gearhardt–Prüss-type theorem of Wei in
[44].555Notice, however, that resolvent-based methods may not generalize
easily to unsteady flows. Our approach can be easily adapted to shear flows
that vary on the timescale $O(\nu^{-(N+1)/(N+3)})$. In the monotone shear
case, one can use the smoothing more directly [15]. A different choice,
popular in nonlinear problems, is simply to consider stability in more regular
spaces [6].

  * •

Hypocoercivity may not always generalize well to boundaries, which is why [8]
does not achieve the optimal timescales in the Neumann case.

Finally, we mention that hypocoercivity’s main purpose is not to capture
hypoelliptic regularization, although in the shear flow case it can be
recovered by studying the equation mode-by-mode in $x$; see (2.6) in [8] for a
definition of the corresponding functional $\Phi_{k}$. In our method, the
enhanced dissipation is essentially a _consequence_ of the hypoelliptic
regularization, in the sense that the subelliptic estimate (1.9) controls a
norm measuring fractional regularity in $x$ using the energy estimates (1.7)
and (1.8) and leads directly to the timescale (1.4) by a Poincaré inequality
in $x$.

### Existing literature

_1\. Enhanced dissipation_. In [14], Constantin, Kiselev, Ryzhik, and Zlatoš
investigated enhanced dissipation in broad generality in terms of spectral
properties of the operator $\boldsymbol{u}\cdot\nabla$. This work was recently
revisited and quantified by Coti Zelati, Delgadino, and Elgindi in [16] and
Wei in [44]. Shear flows are not relaxation enhancing in the sense of [14].

Shear flows are steady solutions of the two-dimensional Euler equations and
(forced) Navier-Stokes equations. The corresponding linearized equations are

$\partial_{t}\omega+b(y)\partial_{x}\omega-b^{\prime\prime}(y)\partial_{x}\psi=\nu\Delta\omega,\quad\Delta\psi=\omega,$
(1.16)

where $\omega$ is the vorticity perturbation. Compared to (1.1), (1.16)
contains the additional non-local term $-b^{\prime\prime}(y)\partial_{x}\psi$
which complicates the analysis. This term vanishes for the Couette flow
$b(y)=y$, and we can extract the inviscid damping and enhanced dissipation
directly from the solution formula in (1.3). Notable papers on nonlinear
stability of the Couette flow include [10, 11, 12, 34]. Other important flows
are the Poiseuille (pipe/channel) flow $b(y)=y(1-y)$ [19], Kolmogorov
(sinusoidal) flow [6, 45, 46], and Oseen vortices [24, 21, 20, 39, 25].

When considering passive scalars in an infinite pipe, one refers to
_Taylor–Aris dispersion_ [42, 2]. This topic was recently reexamined via
center manifolds and hypocoercivity in [5].

Passive scalars in power law circular flows were considered by Dolce and Coti
Zelati in [17].

Passive scalars in shear flows with non-local diffusion were recently
investigated by He [29] by resolvent methods.

_2\. Hypoellipticity_. Hörmander’s theorem was reproved using
pseudodifferential operators by Kohn in [36], and it is Kohn’s proof which
Hörmander includes in his monograph [32], see Chapter 22. We mention also the
stochastic proof reviewed in [28]. Otherwise, the literature is immense, and
we refer to [13] for a survey on Hörmander’s vector fields and the books [30,
38] on microlocal analysis.

_3\. Kinetic equations and regularity techniques_. Recent work on kinetic
equations has illuminated connections between Hörmander’s hypoellipticity and
regularity techniques in the style of De Giorgi–Nash–Moser. In [41], Silvestre
derived _a priori_ Hölder estimates for the Boltzmann equation, and follow-up
work of Silvestre and Imbert [33] obtained weak Harnack and Hölder estimates
for a large class of integro-differential equations, including the Boltzmann
equation without cutoff. Golse, Imbert, Mouhot, and Vasseur [26] studied the
Harnack inequality for kinetic Fokker-Planck and Landau equations. For open
questions and a presentation of these and related programs of research, we
refer to Mouhot [40].

### Acknowledgements

DA was supported by the NSF Postdoctoral Fellowship Grant No. 2002023. RB was
supported by the NSF Graduate Fellowship Grant No. 1839302. MN was supported
by the National Science Foundation under Grant No. DMS-1928930 while
participating in a program hosted by the Mathematical Sciences Research
Institute in Berkeley, California, during the spring 2021 semester. The
authors also thank Vlad Vicol, Scott Armstrong, Jean–Christophe Mourrat, and
Jacob Bedrossian for valuable discussions.

##  2\. Preliminaries

Following the convention specified in the introduction, the torus $\mathbb{T}$
may be regarded as the set $(0,1)$ with periodic boundary conditions.
Furthermore, unless specified otherwise, functions $f$ and $u$ are
_qualitatively smooth_ , though they are only measured in low regularity.

######  Definition 2.1 (Flow Maps and Pullbacks).

Let $E\subseteq D$ be open and $(t_{1},t_{2})$ be given. Let $X$ be a
Lipschitz vector field defined on $\mathbb{T}\times E\times(t_{1},t_{2})$. For
$\widetilde{E}\subseteq E$ open,
$(\widetilde{t}_{1},\widetilde{t}_{2})\subseteq(t_{1},t_{2})$, and
$\sigma\in(\sigma_{1},\sigma_{2})$, consider the function

$\phi(\sigma,x,y,t)\colon(\sigma_{1},\sigma_{2})\times\mathbb{T}\times\widetilde{E}\times(\widetilde{t}_{1},\widetilde{t}_{2})\to\mathbb{T}\times
E\times(t_{1},t_{2})\,$

defined as the solution to the ordinary differential equation666In making this
definition, we have implicitly assumed that the flow remains contained in
$\mathbb{T}\times E\times(t_{1},t_{2})$, a condition which will be satisfied
for every vector field considered in this paper.

$\partial_{\sigma}\phi=X(\phi),\qquad\phi(0,x,y,t)=(x,y,t)\,.$

Then for each $\sigma\in(\sigma_{1},\sigma_{2})$ and any measurable function
$u:\Omega\times(0,1)\rightarrow\mathbb{R}$, we may define the action of the
associated pull back $\exp(\sigma X)$ on $u$ by

$\exp(\sigma X)u(x,y,t)=u\left(\exp(\sigma
X)(x,y,t)\right)=u\left(\phi(\sigma,x,y,t)\right)\,.$

Throughout the paper, we shall use the following notation and shorthand for
particular vector fields.

######  Definition 2.2 (Vector Fields).

We fix notation for the following vector fields. $A$ and $B$ are allowed to be
arbitrary Lipschitz vector fields as defined in Definition 2.1, and $b\in
C^{\infty}(D)$ satisfies (1.2).

$\displaystyle[A,B]$ $\displaystyle=(\textnormal{ad}A)B=AB-BA$ (2.1a)
$\displaystyle Y$ $\displaystyle=\nu^{\nicefrac{{1}}{{2}}}\partial_{y}$ (2.1b)
$\displaystyle X_{0}$ $\displaystyle=\partial_{t}+b(y)\partial_{x}$ (2.1c)
$\displaystyle Z_{1}$
$\displaystyle=[Y,X_{0}]=\nu^{\nicefrac{{1}}{{2}}}b^{\prime}(y)\partial_{x}$
(2.1d) $\displaystyle Z_{k}$
$\displaystyle=[Y,Z_{k-1}]=\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y)\partial_{x}\,,\qquad
k\geq 2\,.$ (2.1e)

We shall frequently refer to the flow maps and pull backs induced by the
vector fields defined above. For example, in the case of $Y$ and $X_{0}$,
these flows maps are simply translations along the trajectories of $Y$ and
$X_{0}$, defined by

$\exp(\sigma
Y)u(x,y,t)=u(x,y+\sigma\nu^{\nicefrac{{1}}{{2}}},t),\qquad\exp(\sigma
X_{0})u(x,y,t)=u(x+\sigma b(y),y,t+\sigma)\,.$

######  Definition 2.3 (Parameters).

For $0\leq k\leq N+2$, we shall use the notation

$s_{k}=\frac{1}{k+2},\qquad m_{k}=\frac{1}{s_{k}}\,.$ (2.2)

Note that by the definition of $m_{k}$, we have $m_{k+1}=1+m_{k}$. In
addition, for $0\leq k\leq N+1$, we define an increasing sequence of positive
numbers

$0<c_{0}<c_{1}<\cdots<c_{k-1}<c_{k}<c_{k+1}<\cdots<c_{N+1}<\infty\,.$ (2.3)

While we shall specify the exact values of the $c_{k}$ later,777See conditions
(2.8), (3.4), (3.5), (3.6), and (3.7), which culminate in the choice of the
$c_{k}$’s in Corollary 3.3.  each $c_{k}$ satisfies
$c_{k}^{2}\approx_{b,k}\nu^{-\frac{N+1}{N+3}}$, and
$c_{0}^{2}=\nu^{-\frac{N+1}{N+3}}$.

We shall use $s_{k}$, $m_{k}$, and $c_{k}$ to define a number of Besov-type
norms. The parameter $s_{k}$ specifies an _index of differentiability_ , while
$c_{k}$ quantifies the _magnitudes_ of the finite differences allowed in each
Besov-type norm. These spaces are complicated by (1) the fact that they are
$B^{s}_{p,q}$-type Besov norms with $q=+\infty$, (2) finite differences must
stay within the space-time domain under consideration, and (3) an extension
procedure would be non-trivial (for example, $X_{0}u$ cannot be extended by a
simple even reflection in time).

######  Definition 2.4 (Besov Spaces).

We define the following fractional Besov-type seminorm along the vector fields
$X_{0}$:

$\displaystyle\left\|u\right\|_{\mathbf{X}}=\sup_{\sigma\in(0,\nicefrac{{c_{0}}}{{\sqrt{2}}})}\sigma^{-1}\big{(}$
$\displaystyle\left\|\exp(\sigma^{2}X_{0})u-u\right\|_{L^{2}(\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{{2}}}))}$
$\displaystyle\qquad+\left\|\exp(-\sigma^{2}X_{0})u-u\right\|_{L^{2}(\Omega\times(\nicefrac{{c_{0}^{2}}}{{2}},c_{0}^{2}))}\big{)}$

For $u\in L^{2}_{t,x}H^{1}_{y}(\Omega\times(0,c_{0}^{2}))$ (qualitative
assumption), we define the seminorm

$\|u\|_{\mathbf{Y}}=\nu^{\nicefrac{{1}}{{2}}}\|\partial_{y}u\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\,.$

This norm is appropriate in Dirichlet, periodic, or Neumann settings, and by
the analysis in Evans p. 292–293 [22] is equivalent to

$\displaystyle\widetilde{\|}u\|_{\mathbf{Y}}=\sup_{\sigma\in\left(0,\frac{\nu^{-\nicefrac{{1}}{{2}}}}{2}\right)}\sigma^{-1}\big{(}$
$\displaystyle\left\|\exp(\sigma
Y)u-u\right\|_{L^{2}(\mathbb{T}\times(0,\nicefrac{{1}}{{2}})\times(0,c_{0}^{2}))}$
$\displaystyle\qquad+\left\|\exp(-\sigma
Y)u-u\right\|_{L^{2}(\mathbb{T}\times(\nicefrac{{1}}{{2}},1)\times(0,c_{0}^{2}))}\big{)}\,.$

That is, $\|u\|_{\mathbf{Y}}\approx\widetilde{\|}u\|_{\mathbf{Y}}$. For $g\in
L^{2}_{t,x}(H^{1})^{*}_{y}(\Omega\times(0,c_{0}^{2}))$
($H^{-1}_{y}:=(H^{1}_{0})^{*}_{y}$ in the Dirichlet setting), we define the
dual seminorm

$\left\|g\right\|_{\mathbf{Y}^{*}(\Omega\times(0,c_{0}^{2}))}=\sup_{\left\|u\right\|_{\mathbf{Y}}=1}\int_{\Omega\times(0,c_{0}^{2})}gu\,dx\,dy\,dt<\infty\,$

where $u\in L^{2}_{t,x}H^{1}_{y}(\Omega\times(0,c_{0}^{2}))$
($(H^{1}_{0})_{y}$ in the Dirichlet setting). This definition is suitable for
the integration by parts in Section 5.

Consider a vector field $Z=c(y)\partial_{x}$ and $s>0$. For a free parameter
$\sigma_{Z}>0$, we define

$\left\|u\right\|_{Q^{s}_{Z}}=\sup_{\sigma\in(0,\sigma_{Z})}\sigma^{-1}\left\|\exp\sigma^{\nicefrac{{1}}{{s}}}Zu-u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\,.$
(2.4)

If $c(y)\equiv 1$, we set $\sigma_{\partial_{x}}=\infty$888Due to the
periodicity in $x$, finite differences for $\sigma>1$ cannot contribute to the
supremum. so that

$\left\|u\right\|_{Q^{s}_{\partial_{x}}}=\sup_{\sigma\in(0,\infty)}\sigma^{-1}\left\|\exp(\sigma^{\nicefrac{{1}}{{s}}}\partial_{x})u-u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\,.$
(2.5)

For $k\geq 1$, set $\sigma_{Z_{k}}=c_{k}$, the precise value of which will be
determined later. Then define

$\left\|u\right\|_{\mathbf{Z}_{k}}=\left\|u\right\|_{Q^{s_{k}}_{Z_{k}}(\Omega\times(0,c_{0}^{2}))}=\sup_{(0,c_{k})}\sigma^{-1}\left\|\exp(\sigma^{m_{k}}Z_{k})u-u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\,.$

The $Q$ stands for ‘quotient’. Let us make an analogy for such spaces in the
periodic single variable case $x\in\mathbb{T}$. We have that
$Q^{1}_{\partial_{x}}$ and $\dot{H}^{1}$ are equivalent [22]. When $s\in(0,1)$
is not an integer, we have that $Q^{s}_{\partial_{x}}$ and
$\dot{B}^{s}_{2,\infty}$ are equivalent.

Before recording a few simple properties of these spaces, let us first set
some notation. Recalling (1.2), let $U_{i}$ be a sufficiently small
neighborhood of $y_{i}$ so that for $k\geq 2$,

$B_{k}=\inf_{y\in\mathcal{U}_{k}}\left|b^{(k)}(y)\right|>0\,,\qquad\mathcal{U}_{k}=\bigcup_{\\{i:N_{i}=k-1\\}}U_{i}\,,$

and

$B_{1}=\inf_{y\in\mathcal{U}_{1}}\left|b^{\prime}(y)\right|>0\,,\qquad\mathcal{U}_{1}=(0,1)\setminus\bigcup_{i=1}^{M}U_{i}\,.$

The next lemma asserts that $\mathcal{U}_{k}$ is the set of $y$-values on
which $Z_{k}$ is coercive in the $x$-direction, and $B_{k}$ is the lower bound
on the coercivity.

######  Lemma 2.5 (Comparable norms).

Let $A(y):D\rightarrow\mathbb{R}$ be a smooth function, and let
$\mathcal{U}_{k}$ and $B_{k}$ be defined as above.

  1. (1)

If $A(y)$ satisfies $0\leq A\leq A_{1}$, then for any value of
$\sigma_{A\partial_{x}}$, we have that

$\left\|u\right\|_{Q_{A\partial_{x}}^{s}}\lesssim_{s}A_{1}^{s}\left\|u\right\|_{Q^{s}_{\partial_{x}}}\,.$
(2.6)

If in addition ${\sigma_{A\partial_{x}}\geq A_{0}^{-s}\geq 1}$ and
$0<A_{0}\leq A\leq A_{1}$, we have the two-sided bound

$A_{0}^{s}\left\|u\right\|_{Q^{s}_{\partial_{x}}}\lesssim_{s}\left\|u\right\|_{Q_{A\partial_{x}}^{s}}\lesssim_{s}A_{1}^{s}\left\|u\right\|_{Q^{s}_{\partial_{x}}}\,.$
(2.7)

  2. (2)

For $k\geq 1$ and

${c_{k}\geq\left(\nu^{\nicefrac{{k}}{{2}}}B_{k}\right)^{-s_{k}}}\geq 1\,,$
(2.8)

we have that

$\left(\nu^{\nicefrac{{k}}{{2}}}B_{k}\right)^{s_{k}}\left\|\mathbf{1}_{\mathcal{U}_{k}}u\right\|_{Q_{\partial_{x}}^{s_{k}}}\leq\left\|\mathbf{1}_{\mathcal{U}_{k}}u\right\|_{\mathbf{Z}_{k}}\leq\left(\nu^{\nicefrac{{k}}{{2}}}\left\|b^{(k)}\right\|_{L^{\infty}(D)}\right)^{s_{k}}\left\|u\right\|_{Q_{\partial_{x}}^{s_{k}}}\,.$
(2.9)

###### Proof.

Since time plays a passive role in this argument, we shall suppress the time
dependence. We first prove (2.6) following Hormander’s Lemma 4.1 [31]. After
making the identification $\widetilde{\sigma}=\sigma^{\nicefrac{{1}}{{s}}}$ in
(2.4), we may write that

$\displaystyle\widetilde{\sigma}^{-2s}$
$\displaystyle\|\exp(\widetilde{\sigma}A(y)\partial_{x})u-u\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}^{2}$
$\displaystyle=\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}}\|\exp(\widetilde{\sigma}A(y)\partial_{x})u-u\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}^{2}dh$
$\displaystyle\leq\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{\sigma}A(y),y)-u(x+hA_{1},y)|^{2}\,dx\,dy\,dt\,dh$
$\displaystyle\qquad+\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+hA_{1},y)-u(x,y)|^{2}\,dx\,dy\,dt\,dh$
$\displaystyle\leq\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{\sigma}A(y),y)-u(x+hA_{1},y)|^{2}\,dx\,dy\,dt\,dh+A_{1}^{2s}\|u\|_{Q_{\partial_{x}}^{s}}^{2}\,.$
(2.10)

In order to estimate the first integral, we define the new variables

$x^{\prime}=x+hA_{1},\qquad
h=\widetilde{h}+\widetilde{\sigma}\frac{A(y)}{A_{1}}\,.$ (2.11)

Using (2.11), we have that

$\displaystyle\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{\sigma}A(y),y)-u(x+hA_{1},y)|^{2}\,dx\,dy\,dt\,dh$
$\displaystyle\leq\widetilde{\sigma}^{-2s}\cdot
2\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{-\widetilde{\sigma}\nicefrac{{A}}{{A_{1}}}\leq\widetilde{h}\leq\widetilde{\sigma}}\int_{\Omega\times(0,c_{0}^{2})}|u(x^{\prime}-\widetilde{h}A_{1},y)-u(x^{\prime},y)|^{2}\,dx^{\prime}\,dy\,dt\,d\widetilde{h}$
$\displaystyle\leq\widetilde{\sigma}^{-2s}\cdot
2\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{-\widetilde{\sigma}\nicefrac{{A}}{{A_{1}}}\leq\widetilde{h}\leq\widetilde{\sigma}}A_{1}^{2s}\widetilde{h}^{2s}\left\|u\right\|_{Q_{\partial_{x}}^{s}}$
$\displaystyle\lesssim
A_{1}^{2s}\left\|u\right\|_{Q_{\partial_{x}}^{s}}^{2}\,.$ (2.12)

Combining (2) and (2) we have that

$\|u\|_{Q_{A\partial_{x}}^{s}}^{2}\lesssim
A_{1}^{2s}\|u\|_{Q_{\partial_{x}}^{s}}^{2}\,,$ (2.13)

which proves (2.6).

To prove the lower bound in (2.7), we use that one may assume that
$\sigma\in(0,1)$ in (2.5) and again make the identification
$\widetilde{\sigma}=\sigma^{\nicefrac{{1}}{{s}}}$ to write that

$\displaystyle A_{0}^{2s}\left\|u\right\|_{Q_{\partial_{x}}^{s}}^{2}$
$\displaystyle=A_{0}^{2s}\sup_{\widetilde{\sigma}\in(0,1)}\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}A_{0}^{-1}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{\sigma},y)-u(x,y)|^{2}\,dx\,dy\,dt\,dh\,.$

Adding and subtracting $u(x+hA(y),y)$ this time gives that

$\displaystyle\widetilde{\sigma}^{-2s}$
$\displaystyle\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}A_{0}^{-1}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{\sigma},y)-u(x,y)|^{2}\,dx\,dy\,dt\,dh$
$\displaystyle\lesssim\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}A_{0}^{-1}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{\sigma},y)-u(x+hA(y),y)|^{2}\,dx\,dy\,dt\,dh$
$\displaystyle\qquad+\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}A_{0}^{-1}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+hA(y),y)-u(x,y)|^{2}\,dx\,dy\,dt\,dh\,.$
(2.14)

To bound the second term, we make the change of variables
$h=\widetilde{h}^{\nicefrac{{1}}{{s}}}$ and obtain

$\displaystyle\widetilde{\sigma}^{-2s}$
$\displaystyle\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}A_{0}^{-1}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+hA(y),y)-u(x,y)|^{2}\,dx\,dy\,dt\,dh$
$\displaystyle=\widetilde{\sigma}^{-2s}\cdot(\widetilde{\sigma}A_{0}^{-1})^{-1}\cdot\frac{1}{s}\int_{0\leq\widetilde{h}\leq(\widetilde{\sigma}A_{0}^{-1})^{s}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{h}^{\nicefrac{{1}}{{s}}}A(y),y)-u(x,y)|^{2}\,dx\,dy\,dt\,\widetilde{h}^{\nicefrac{{1}}{{s}}-1}\,d\widetilde{h}$
$\displaystyle\leq\widetilde{\sigma}^{-2s-1}\cdot
A_{0}\cdot\frac{1}{s}\int_{0\leq\widetilde{h}\leq(\widetilde{\sigma}A_{0}^{-1})^{s}}\widetilde{h}^{\nicefrac{{1}}{{s}}+1}\left\|u\right\|_{Q_{A\partial_{x}}^{s}}^{2}\,d\widetilde{h}$
$\displaystyle\lesssim_{s}A_{0}^{-2s}\left\|u\right\|_{Q_{A\partial_{x}}^{s}}^{2}\,$

which gives the desired bound in (2.7). Notice that in order to appeal to the
$\left\|u\right\|_{Q_{A\partial_{x}}^{s}}$ norm, we have used that
$\widetilde{h}\leq(\widetilde{\sigma}A_{0}^{-1})^{s}\leq
A_{0}^{-s}\leq\sigma_{A\partial_{x}}$.

For the first term from (2.14), we use the change of variables

$x^{\prime}=x+A(y)h,\qquad h=h^{\prime}+\frac{\widetilde{\sigma}}{A(y)}\,,$
(2.15)

which is well defined by the assumption that $A(y)\geq A_{0}>0$. Then the
first term from (2.14) may be estimated as

$\displaystyle\widetilde{\sigma}^{-2s}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0\leq
h\leq\widetilde{\sigma}A_{0}^{-1}}\int_{\Omega\times(0,c_{0}^{2})}|u(x+\widetilde{\sigma},y)-u(x+hA(y),y)|^{2}\,dx\,dy\,dt\,dh$
$\displaystyle=\widetilde{\sigma}^{-2s}\int_{\Omega\times(0,c_{0}^{2})}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{{\widetilde{\sigma}}{A(y)}^{-1}\leq
h^{\prime}\leq\widetilde{\sigma}A_{0}^{-1}}|u(x^{\prime}-h^{\prime}A(y),y)-u(x^{\prime},y)|^{2}\,dh^{\prime}\,dx^{\prime}\,dy\,dt$
$\displaystyle\leq
2\widetilde{\sigma}^{-2s-1}A_{0}\cdot\frac{1}{s}\int_{\Omega\times(0,c_{0}^{2})}\int_{|\widetilde{h}|\leq
A_{0}^{-s}}|u(x^{\prime}-\widetilde{h}^{\nicefrac{{1}}{{s}}}A(y),y)-u(x^{\prime},y)|^{2}\widetilde{h}^{\nicefrac{{1}}{{s}}-1}\,d\widetilde{h}\,dx^{\prime}\,dy\,dt$
$\displaystyle\leq
2\widetilde{\sigma}^{-2s-1}A_{0}\cdot\frac{1}{s}\int_{|\widetilde{h}|\leq
A_{0}^{-s}}\widetilde{h}^{\nicefrac{{1}}{{s}}+1}\left\|u\right\|_{Q_{A\partial_{x}}^{s}}^{2}\,d\widetilde{h}$
$\displaystyle\lesssim_{s}A_{0}^{-2s}\left\|u\right\|_{Q_{A\partial_{x}}^{s}}^{2}$
(2.16)

Note that after making the change of variables
$\widetilde{h}^{\nicefrac{{1}}{{s}}}=-h^{\prime}$ above, we have again used
the assumption that $|\widetilde{h}|\leq A_{0}^{-s}\leq\sigma_{A\partial_{x}}$
to appeal to the $\left\|u\right\|_{Q_{A\partial_{x}}^{s}}$ norm.

Finally, the proof of (2.9) follows from (2.7) after using the definition of
$\mathcal{U}_{k}$, $B_{k}$, and $Z_{k}$. ∎

We shall use the following fractional Poincaré and interpolation inequalities.

######  Lemma 2.6 (Fractional Poincaré and Interpolation).

The following two estimates hold:

  1. (1)

If $u\in Q^{s}_{\partial_{x}}$ and $0<s^{\prime}<s<1$, then adopting the usual
notation for $\langle u\rangle(y,t)$, we have that999For a definition of
Banach-space valued Sobolev functions, one may for example refer to Amann [1].

$\left\|u-\langle
u\rangle(y,t)\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}+\left\|u\right\|_{L^{2}\left((0,c_{0}^{2});\dot{H}^{s^{\prime}}\left(\mathbb{T};L^{2}_{y}\right)\right)}\lesssim_{s,s^{\prime}}\left\|u\right\|_{Q^{s}_{\partial_{x}}}\,.$

  2. (2)

If $0\leq s_{1}<s_{2}\leq 1$, $0<\theta<1$, and $s=\theta s_{1}+\theta s_{2}$,
then

$\displaystyle\left\|u\right\|_{Q^{s}_{\partial_{x}}}$
$\displaystyle\lesssim\left\|u\right\|_{Q^{s_{1}}_{\partial_{x}}}^{\theta}\left\|u\right\|_{Q^{s_{2}}_{\partial_{x}}}^{1-\theta}$
$\displaystyle\left\|u\right\|_{\dot{H}^{s}\left(\mathbb{T};L^{2}_{y}\right)}$
$\displaystyle\lesssim\left\|u\right\|_{\dot{H}^{s_{1}}\left(\mathbb{T};L^{2}_{y}\right)}^{\theta}\left\|u\right\|_{\dot{H}^{s_{2}}\left(\mathbb{T};L^{2}_{y}\right)}^{1-\theta}\,.$

###### Proof.

To prove the first inequality, we appeal to Jensen’s inequality, the lower
bound on $|x-\widetilde{x}|^{-1-s}$ for $x,\widetilde{x}\in\mathbb{T}$, and
the bound on $u$ in $Q^{s}_{\partial_{x}}$ to write that

$\displaystyle\left\|u(x,y,t)-\langle
u\rangle(y,t)\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}^{2}$
$\displaystyle=\int\kern-3.39996pt\int\kern-3.39996pt\int_{\mathbb{T}\times(0,1)\times(0,c_{0}^{2})}\left|u(x,y,t)-\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}}u(\widetilde{x},y,t)\,d\widetilde{x}\right|^{2}\,dy\,dx\,dt$
$\displaystyle\lesssim\int\kern-3.39996pt\int\kern-3.39996pt\int_{\mathbb{T}\times(0,1)\times(0,c_{0}^{2})}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathbb{T}}\frac{|u(x,y,t)-u(\widetilde{x},y,t)|^{2}}{|x-\widetilde{x}|^{1+2s^{\prime}}}\,d\widetilde{x}\,dy\,dx\,dt$
$\displaystyle\lesssim\int\kern-3.39996pt\int_{\mathbb{T}\times(0,c_{0}^{2})}\int_{-\widetilde{x}}^{1-\widetilde{x}}\frac{\left\|u(\widetilde{x}+\sigma,y,t)-u(\widetilde{x},y,t)\right\|_{L^{2}_{y}}^{2}}{|\sigma|^{1+2s^{\prime}}}\,d\sigma\,d\widetilde{x}\,dt$
$\displaystyle\lesssim\int_{0}^{c_{0}^{2}}\int_{-1}^{1}\frac{\left\|u(\widetilde{x}+\sigma,y,t)-u(\widetilde{x},y,t)\right\|_{L^{2}_{y,\widetilde{x}}}^{2}}{|\sigma|^{1+2s^{\prime}}}\,d\sigma\,dt$
$\displaystyle\lesssim\|u\|_{Q^{s}_{\partial_{x}}}^{2}\int_{-1}^{1}\frac{|\sigma|^{2s}}{|\sigma|^{1+2s^{\prime}}}\,d\sigma$
$\displaystyle\lesssim\|u\|_{Q^{s}_{\partial_{x}}}^{2}\,.$ (2.17)

The implicit constant depends only on $s,s^{\prime}\in(0,1)$ and in particular
is independent of $u$. Notice that the second line is precisely the norm of
$u$ in
$L^{2}\left((0,1);H^{s^{\prime}}\left(\mathbb{T};L^{2}_{y}\right)\right)$,
concluding the proof.

When $s_{1}>0$, the third inequality follows from writing

$\displaystyle\frac{\left|u(x,y)-u(\widetilde{x},y)\right|^{2}}{|x-\widetilde{x}|^{1+2s}}=\frac{\left|u(x,y)-u(\widetilde{x},y)\right|^{2\theta}}{|x-\widetilde{x}|^{\theta(1+2s_{1})}}\frac{\left|u(x,y)-u(\widetilde{x},y)\right|^{2(1-\theta)}}{|x-\widetilde{x}|^{(1-\theta)(1+2s_{2})}},$
(2.18)

where $\widetilde{x}=x+\sigma$, and then applying Hölder’s inequality with
$L^{\frac{1}{\theta}}$ and $L^{\frac{1}{1-\theta}}$. The case $s_{1}=0$ and
the second inequality are similar, and we omit further details. ∎

##  3\. Bracket inequalities

The arguments in this section are based on the following equality, which may
be checked directly. To streamline calculations, we adopt the convention,
locally in Section 3, that $Z_{0}=X_{0}$, and
$\left\|u\right\|_{\mathbf{Z}_{0}}=\left\|u\right\|_{\mathbf{X}}$. Then for
$k\geq 0$,

$\displaystyle\exp(-\varepsilon\sigma^{m_{k}}Z_{k})\exp(-\varepsilon^{-1}\sigma
Y)\exp(\varepsilon\sigma^{m_{k}}Z_{k})\exp(\varepsilon^{-1}\sigma
Y)u(x,y,t)-u(x,y,t)$ (3.1)
$\displaystyle\quad=u\left(x+\varepsilon\sigma^{m_{k}}\nu^{\nicefrac{{k}}{{2}}}\left[b^{(k)}(y+\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}})-b^{(k)}(y)\right],y,t\right)-u(x,y,t)\,.$

Using the heuristic that

$\varepsilon\sigma^{m_{k}}\nu^{\nicefrac{{k}}{{2}}}\left[b^{(k)}(y+\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}})-b^{(k)}(y)\right]\approx\sigma^{m_{k+1}}\nu^{\frac{k+1}{2}}b^{(k+1)}(y)\,$

and ignoring terms from the Taylor expansion which have been hidden inside the
$\approx$ symbol, we would expect to control
$\left\|u\right\|_{{\mathbf{Z}_{k+1}}}$ using $\left\|u\right\|_{\mathbf{Y}}$
and $\left\|u\right\|_{\mathbf{Z}_{k}}$. The leftover terms in the Taylor
expansion will in fact be controlled by “higher brackets,” meaning
$\left\|u\right\|_{\mathbf{Z}_{j}}$ for $j\geq k+2$. The $\varepsilon$
accounts for the fact that the $\mathbf{Z}_{k}$ and $\mathbf{Y}$ norms allow
for finite differences of variable magnitude and helps us rebalance
inequalities to absorb remainder terms.

######  Lemma 3.1 (Bracket inequality).

If $0\leq k\leq N$ and $\varepsilon_{k+1}$ satisfies the technical condition
(3.7), then for an implicit constant independent of the choice of
$\varepsilon_{k+1}$, we have the estimate

$\displaystyle\left\|u\right\|_{\mathbf{Z}_{k+1}}$
$\displaystyle\lesssim\varepsilon_{k+1}^{s_{k}}\left\|u\right\|_{\mathbf{Z}_{k}}+\varepsilon_{k+1}^{-1}\left\|u\right\|_{\mathbf{Y}}+\sum_{j=k+2}^{N+1}\varepsilon_{k+1}^{s_{j}(k-j+1)}\left\|u\right\|_{\mathbf{Z}_{j}}$
$\displaystyle\qquad+\left(\varepsilon_{k+1}^{k-N-1}\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\,.$
(3.2)

######  Remark 3.2.

We check in Corollary 3.3 that such a choice of $\varepsilon_{k+1}$ is
possible, and then we fix values for $\varepsilon_{k+1}$ and also $c_{k+1}$.

Notice that we assume control over the norm $\left\|u\right\|_{\mathbf{X}}$,
which we show in Section 4 is controlled by the _a priori_ estimates enjoyed
by solutions to (1.5).

###### Proof of Lemma 3.1.

To ensure that the finite differences stay inside the domain, which has non-
trivial boundary in $t$ as well as in $y$ for the Dirichlet and Neumann cases,
we subdivide $\Omega$ into regions $\mathbb{T}\times I\times J$, where
$I\in\\{(0,\nicefrac{{1}}{{{2}}}),(\nicefrac{{1}}{{2}},1)\\}$ and
$J\in\\{(0,\nicefrac{{c_{0}^{2}}}{{{{2}}}}),(\nicefrac{{c_{0}^{2}}}{{{2}}},c_{0}^{2})\\}$.
We only write the case when $I=(0,\nicefrac{{1}}{{2}})$ and
$J=(0,\nicefrac{{c_{0}^{2}}}{{2}})$. The cases with
$I=(\nicefrac{{1}}{{2}},1)$ follow upon replacing $Y$ by $-Y$, and the cases
with $J=(\nicefrac{{c_{0}^{2}}}{{2}},c_{0}^{2})$ follow upon replacing $X$ by
$-X$. We divide the remainder of the proof into steps. In the first step, we
write out the Taylor expansion corresponding to (3.1). In the second step, we
use the admissible ranges of $\sigma$ to derive the constraints (3.4), (3.5),
and (3.6) on $\varepsilon_{k+1}$ in terms of $c_{k}$. Finally, the last step
bounds each term using the appropriate Besov norm. The parameter $k$ is fixed
throughout the rest of the proof, and we simply write $\varepsilon$ rather
than $\varepsilon_{k+1}$.

Step 1.  From Taylor’s theorem, we have that

$\displaystyle b^{(k)}(y+\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}})$
$\displaystyle=b^{(k)}(y)+\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}}b^{(k+1)}(y)$
$\displaystyle\qquad+\sum_{j=k+2}^{N+1}\frac{(\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}})^{j-k}}{(j-k)!}b^{(j)}(y)\,+\,O\left((\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}})^{N-k+2}\right)\,,$

where the prefactor on
$O\left((\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}})^{N-k+2}\right)$ is
bounded from above by $\left\|b^{(N+2)}\right\|_{L^{\infty}}$. Multiplying
through by $\varepsilon\sigma^{k+2}\nu^{\nicefrac{{k}}{{2}}}$ and rearranging,
it follows that

$\displaystyle\sigma^{k+3}\nu^{\frac{k+1}{2}}b^{(k+1)}(y)$
$\displaystyle=\varepsilon\sigma^{k+2}\nu^{\nicefrac{{k}}{{2}}}\left(b^{(k)}(y+\varepsilon^{-1}\sigma\nu^{\nicefrac{{1}}{{2}}})-b^{(k)}(y)\right)$
$\displaystyle\qquad-\sum_{j=k+2}^{N+1}\frac{\sigma^{j+2}\nu^{\nicefrac{{j}}{{2}}}}{(j-k)!}\varepsilon^{k-j+1}b^{(j)}(y)\,+\,O\left(\sigma^{N+4}\right)\,,$

where now the prefactor on $O(\sigma^{N+4})$ is bounded from above by
$\varepsilon^{k-N-1}\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}$.
From (2.2) and (3.1), it follows that (recall that when $k=0$, $Z_{0}=X_{0}$)

$\displaystyle\exp(\sigma^{m_{k+1}}Z_{k+1})u-u=\exp(-\varepsilon\sigma^{m_{k}}Z_{k})\exp(-\varepsilon^{-1}\sigma
Y)\exp(\varepsilon\sigma^{m_{k}}Z_{k})\exp(\varepsilon^{-1}\sigma Y)Ru-u\,,$
(3.3)

where the remainder operator $R$ is defined by

$\displaystyle
Ru=\left(\prod_{j=k+2}^{N+1}\exp\left(-\frac{\varepsilon^{k-j+1}}{(j-k)!}{\sigma^{m_{j}}Z_{j}}\right)\right)\exp(H\partial_{x})u\,,$

and $H(y,\sigma)=O(\sigma^{m_{N+2}})$ with prefactor bounded above by
$\varepsilon^{k-N-1}\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}$.
Notice that each of the above transformations is volume preserving, but the
transformations induced by $Z_{j}$, $j\geq 1$, and $H$ preserve the domain
$\mathbb{T}\times I\times J$, while the transformations induced by $X_{0}$ and
$Y$ do not and will be estimated separately.

Step 2.  The left-hand side of (3.3) must be measured in $L^{2}$ for
$\sigma\in(0,c_{k+1})$, which necessitates that the same range of $\sigma$ is
admissible in the right-hand side. Since all admissible values of $\sigma$ are
positive, we may track the largest value of $\sigma$, which for $Z_{k+1}$ is
$c_{k+1}$. In order for $\exp(\varepsilon^{-1}\sigma Y)$ to be well defined,
we need

$\varepsilon^{-1}c_{k+1}\leq\frac{\nu^{-\frac{1}{2}}}{2}\,.$ (3.4)

In order for $\exp(\varepsilon\sigma^{m_{k}}Z_{k})$ to be well defined, we
need

$\varepsilon c_{k+1}^{m_{k}}\leq c_{k}^{m_{k}}\,.$ (3.5)

We require that the term with $\exp(H)u$ must be well defined for
$\sigma\in(0,c_{k+1})$. However, this term will be bounded using the
$Q_{\partial_{x}}^{s_{N+2}}$ norm, for which any values of $\sigma$ are
admissible. The other terms in the remainder for $k+2\leq j\leq N+1$ require
that

$\frac{\varepsilon^{k-j+1}}{(j-k)!}c_{k+1}^{m_{j}}\leq c_{j}^{m_{j}}\,.$ (3.6)

Since the first and third conditions provide lower bounds on $\varepsilon$ and
the second provides an upper bound, we consolidate the three conditions by
imposing that

$\max\left(2c_{k+1}\nu^{\nicefrac{{1}}{{2}}},\max_{k+2\leq j\leq
N+1}\left(\left(\frac{c_{k+1}}{c_{j}}\right)^{m_{j}}\frac{1}{(j-k)!}\right)^{\frac{1}{j-1-k}}\right)\leq\varepsilon\leq\left(\frac{c_{k}}{c_{k+1}}\right)^{m_{k}}\,.$
(3.7)

Notice that the right hand side involves an upper bound in terms of $c_{k}$
and $c_{k+1}$, while the left hand side involves lower bounds with powers of
$c_{j}$ in the denominator where $j\geq k+2$, and so we shall pick our
sequence of ‘$c_{k}$’s in increasing order with respect to $k$.

Step 3.  To estimate the left-hand side of (3.3) in $L^{2}(\mathbb{T}\times
I\times J)$, we introduce a telescoping sum on the right-hand side and apply
the triangle inequality:

$\displaystyle\big{\|}\exp(\sigma^{m_{k+1}}$ $\displaystyle
Z_{k+1})u-u\big{\|}_{L^{2}}$
$\displaystyle\leq\big{\|}\exp(-\varepsilon\sigma^{m_{k}}Z_{k})\exp(-\varepsilon^{-1}\sigma
Y)\exp(\varepsilon\sigma^{m_{k}}Z_{k})\exp(\varepsilon^{-1}\sigma Y)Ru$
$\displaystyle\qquad\qquad\qquad\qquad-\exp(-\varepsilon^{-1}\sigma
Y)\exp(\varepsilon\sigma^{m_{k}}Z_{k})\exp(\varepsilon^{-1}\sigma
Y)Ru\big{\|}_{L^{2}}$ $\displaystyle\quad+\left\|\exp(-\varepsilon^{-1}\sigma
Y)\exp(\varepsilon\sigma^{m_{k}}Z_{k})\exp(\varepsilon^{-1}\sigma
Y)Ru-\exp(\varepsilon\sigma^{m_{k}}Z_{k})\exp(\varepsilon^{-1}\sigma
Y)Ru\right\|_{L^{2}}$
$\displaystyle\quad+\left\|\exp(\varepsilon\sigma^{m_{k}}Z_{k})\exp(\varepsilon^{-1}\sigma
Y)Ru-\exp(\varepsilon^{-1}\sigma Y)Ru\right\|_{L^{2}}$
$\displaystyle\quad+\left\|\exp(\varepsilon^{-1}\sigma Y)Ru-
Ru\right\|_{L^{2}}+\left\|Ru-u\right\|_{L^{2}}\,.$

Let $\iota=\mathbf{1}_{k=0}$. Changing variables in the integrals on the
right-hand side of the previous inequality yields

$\displaystyle\left\|\exp(\sigma^{m_{k+1}}Z_{k+1})u-u\right\|_{L^{2}(\mathbb{T}\times
I\times J)}$
$\displaystyle\quad\leq\left\|\exp(-\varepsilon\sigma^{m_{k}}Z_{k})u-u\right\|_{L^{2}(\mathbb{T}\times
I\times(\varepsilon\iota\sigma^{m_{k}}+J))}+\left\|\exp(-\varepsilon^{-1}\sigma
Y)u-u\right\|_{L^{2}\left(\mathbb{T}\times(\varepsilon^{-1}\nu^{\nicefrac{{1}}{{2}}}\sigma+I)\times(\varepsilon\iota\sigma^{m_{k}}+J)\right)}$
$\displaystyle\qquad+\left\|\exp(\varepsilon\sigma^{m_{k}}Z_{k})u-u\right\|_{L^{2}\left(\mathbb{T}\times(\varepsilon^{-1}\nu^{\nicefrac{{1}}{{2}}}\sigma+I)\times
J\right)}+\left\|\exp(\varepsilon^{-1}\sigma
Y)u-u\right\|_{L^{2}(\mathbb{T}\times I\times J)}$
$\displaystyle\qquad\quad+\left\|Ru-u\right\|_{L^{2}(\mathbb{T}\times I\times
J)}\,.$

Hence,

$\left\|\exp(\sigma^{m_{k+1}}Z_{k+1})u-u\right\|_{L^{2}(\mathbb{T}\times
I\times J)}\lesssim\left\|Ru-u\right\|_{L^{2}(\mathbb{T}\times I\times
J)}+\sigma\varepsilon^{s_{k}}\left\|u\right\|_{\mathbf{Z}_{k}}+\sigma\varepsilon^{-1}\left\|u\right\|_{\mathbf{Y}}\,.$
(3.8)

To estimate $\left\|Ru-u\right\|_{L^{2}(\mathbb{T}\times I\times J)}$, we
again introduce a telescoping sum and change variables:

$\displaystyle\left\|Ru-u\right\|_{L^{2}(\mathbb{T}\times I\times J)}$
$\displaystyle\lesssim\sum_{j=k+2}^{N+1}\left\|\exp\left(-\frac{\varepsilon^{k-j+1}}{(j-k)!}\sigma^{m_{j}}Z_{j}\right)-u\right\|_{L^{2}(\mathbb{T}\times
I\times J)}+\left\|\exp(H\partial_{x})u-u\right\|_{L^{2}(\mathbb{T}\times
I\times J)}$ (3.9)
$\displaystyle\lesssim\sum_{j=k+2}^{N+1}\sigma\varepsilon^{s_{j}(k-j+1)}\left\|u\right\|_{\mathbf{Z}_{j}}+\sigma\left(\varepsilon^{k-N-1}\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}.$

Here, we have applied estimate (2.6) from Lemma 2.5 with $s=s_{N+2}$,
$A(y)=h(y)\sigma^{m_{N+2}}$ for some unknown function $h(y)$ depending on the
exact form of the remainder $H(y,\sigma)$, and
$A_{1}=\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}$ to estimate
the term containing $H\partial_{x}$. Combining (3.8) and (3.9) completes the
proof. ∎

######  Corollary 3.3 (Improved bracket inequality, choice of $c_{k}$, and
conditions on $\nu_{0}$).

For all $0\leq k\leq N+1$, $\varepsilon>0$, and $\nu\ll_{b}1$,

$\left\|u\right\|_{\mathbf{Z}_{k}}\leq\varepsilon\left\|u\right\|_{\mathbf{X}}+C(\varepsilon,b)\left\|u\right\|_{\mathbf{Y}}+C(\varepsilon,b)\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\,.$
(3.10)

###### Proof.

The proof proceeds in steps according to the value of a dummy parameter
$\ell\in\\{1,...,N+1\\}$, so that when $\ell$ reaches the terminal value
$N+1$, we derive several conditions on $\nu_{0}$ and conclude the proof. We
begin by choosing $c_{0}=\nu^{-\frac{N+1}{2(N+3)}}$.

Step 1.  Choose $c_{1}$ so that (2.8) is satisfied for $k=1$. Now choose
$\varepsilon_{1}$ in (3.2) (level $k=0$) so that

$\displaystyle\left\|u\right\|_{\mathbf{Z}_{1}}$
$\displaystyle\leq\nicefrac{{\varepsilon}}{{2}}\left\|u\right\|_{\mathbf{X}}+C(\varepsilon,b)\bigg{[}\left\|u\right\|_{\mathbf{Y}}+\sum_{j=2}^{N+1}\left\|u\right\|_{\mathbf{Z}_{j}}+\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\bigg{]}\,,$

and so that (3.5) is satisfied at level $k=0$. This choice of
$\varepsilon_{1}$ may be made _independently of $\nu$_ as long as we have
chosen $c_{1}$ to be a $\nu$-independent multiple of $c_{0}$. As a result of
the choices of $c_{0}$, $c_{1}$, and $\varepsilon_{1}$, we have additional new
lower bounds on all $c_{m}$ for $m\geq 2$ from (3.6). Note that our choices
and the resulting additional lower bounds may depend on $b$ and $\varepsilon$,
but are independent of $\nu$.

Step 2.  Choose $c_{2}$ so that the new lower bound from (3.6) is satisfied
for $j=2$ and $k=0$, and so that (2.8) is satisfied for $k=2$. Now choose
$\varepsilon_{2}$ in (3.2) so that for $k=1,2$,

$\displaystyle\left\|u\right\|_{\mathbf{Z}_{k}}$
$\displaystyle\leq\nicefrac{{3\varepsilon}}{{4}}\left\|u\right\|_{\mathbf{X}}+C(\varepsilon,b)\bigg{[}\left\|u\right\|_{\mathbf{Y}}+\sum_{j=3}^{N+1}\left\|u\right\|_{\mathbf{Z}_{j}}+\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\bigg{]}\,,$

and so that (3.5) is satisfied at level $k=1$. Notice that the sum crucially
begins at $j=3$ in the above inequality. As a result of the choices of
$\varepsilon_{2}$ and $c_{2}$, we have additional new lower bounds on all
$c_{m}$ for $m\geq 3$ from (3.6) which are $\nu$-independent, provided that
$c_{2}$ is a $\nu$-independent multiple of $c_{1}$.

Step $k$.  Assume that Steps $1,...,k-1$ have been completed successfully,
thus fixing choices of $c_{0},...,c_{k-1}$ and
$\varepsilon_{1},...,\varepsilon_{k-1}$. Choose $c_{k}$ so that all lower
bounds resulting from choices of $c_{k^{\prime}}$ for $k^{\prime}<k$ from
(3.6) are satisfied, and so that (2.8) is satisfied at level $k$. Now choose
$\varepsilon_{k}$ in (3.2) so that for $\widetilde{k}=1,2,...,k$,

$\displaystyle\left\|u\right\|_{\mathbf{Z}_{\widetilde{k}}}$
$\displaystyle\leq\frac{(2^{\widetilde{k}}-1)\varepsilon}{2^{\widetilde{k}}}\left\|u\right\|_{\mathbf{X}}+C(\varepsilon,b)\bigg{[}\left\|u\right\|_{\mathbf{Y}}+\sum_{j=k+1}^{N+1}\left\|u\right\|_{\mathbf{Z}_{j}}+\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\bigg{]}\,,$

and so that (3.5) is true at level $k-1$. The sum in the above inequality now
begins at $k+1$, and we have additional new lower bounds an all $c_{m}$ for
$m\geq k+1$ from (3.6).

Conclusion.  Upon reaching $k=N+1$, the sum in the above inequality is empty,
and we thus conclude the proof after ensuring that $\nu$, and by extension
$\nu_{0}$, is small enough so that (3.4) is satisfied for all $k$. Note that
this is possible since $c_{0}=\nu^{-\frac{N+1}{2(N+3)}}\ll\nu^{-\frac{1}{2}}$,
and each subsequent $c_{k}$ was chosen so that (2.8), (3.5), and (3.6) were
satisfied. The latter two conditions are $\nu$-independent and the first
depends on $\nu^{-\frac{ks_{k}}{2}}$, which is increasing in $k$ and bounded
above by $\nu^{-\frac{N+1}{2(N+3)}}$. ∎

######  Proposition 3.4 (Subelliptic estimate, Poincaré inequality, and choice
of $\nu_{0}$).

For all $\nu\leq\nu_{0}\ll_{b}1$ and $\varepsilon>0$, we have that

$\nu^{\frac{N+1}{2(N+3)}}\left(\left\|u-\langle
u\rangle\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}+\left\|u\right\|_{Q^{s_{N+1}}_{\partial_{x}}}\right)\lesssim_{b}\varepsilon\left\|u\right\|_{\mathbf{X}}+C(\varepsilon)\left\|u\right\|_{\mathbf{Y}}\,,$
(3.11)

where the implicit constant is independent of $\nu$ and $\varepsilon$ but
depends on $b$.

###### Proof.

From Lemma 2.5 and Lemma 2.6, we have that for $k\geq 1$,

$\left\|\mathbf{1}_{\mathcal{U}_{k}}u\right\|_{Q^{s_{k}}_{\partial_{x}}}\lesssim\left(\nu^{\nicefrac{{k}}{{2}}}B_{k}\right)^{-s_{k}}\left\|\mathbf{1}_{\mathcal{U}_{k}}u\right\|_{\mathbf{Z}_{k}}\,.$

Since
$\nu^{-\nicefrac{{ks_{k}}}{{2}}}\leq\nu^{-\nicefrac{{(k+1)s_{k+1}}}{{2}}}$ for
all $k\geq 1$, we obtain using Corollary 3.3 that

$\displaystyle\left\|u\right\|_{Q^{s_{N+1}}_{\partial_{x}}}$
$\displaystyle\lesssim_{b}\nu^{-\frac{N+1}{2(N+3)}}\sum_{k=1}^{N+1}\left\|u\right\|_{\mathbf{Z}_{k}}$
(3.12)
$\displaystyle\lesssim_{b}\varepsilon\left\|u\right\|_{\mathbf{X}}+C(\varepsilon,b)\left\|u\right\|_{\mathbf{Y}}+C(\varepsilon,b)\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\,.$

We then apply the fractional Poincaré inequality from Lemma 2.6 to $u-\langle
u\rangle$ to deduce that

$\left\|u-\langle
u\rangle\right\|_{L^{2}}\lesssim_{b}\varepsilon\left\|u\right\|_{\mathbf{X}}+C(\varepsilon,b)\left\|u\right\|_{\mathbf{Y}}+C(\varepsilon,b)\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\,.$
(3.13)

Applying the fractional interpolation inequality from Lemma 2.6 to the
$\left\|u\right\|_{Q_{\partial_{x}}^{s_{N+2}}}$ term on the right-hand side of
(3.12) and (3.13) and using that

$C(\varepsilon,b)\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\ll_{\nu}1$

if $\nu\leq\nu_{0}\ll_{b}1$ concludes the proof. Note that we have chosen
$\nu_{0}$ so that the above inequality is satisfied and the conditions on
$\nu_{0}$ from Corollary 3.3 are satisfied.

∎

##  4\. Interpolation inequality

The estimates in the previous section relied on an iteration written in terms
of norms with positive indices of differentiability. However, the _a priori_
estimate for $\left(\partial_{t}+b(y)\partial_{x}\right)u$ from (1.5) is in
the space $\mathbf{Y}^{*}$, which in the Dirichlet-in-$y$ case one may
identify with $L^{2}_{t,x}H^{-1}_{y}$ up to a $\nu$-dependent factor. Combined
with the estimate for $u$ in $L^{2}_{t,x}H^{1}_{y}$, one may heuristically
expect, by interpolation, that
“$\left(\partial_{t}+b(y)\partial_{x}\right)^{\nicefrac{{1}}{{2}}}u\in
L^{2}_{t,x}L^{2}_{y}$. ” We will show that such an interpolation inequality is
possible by following Hörmander’s arguments in Section 5 of [31]. Note that
Corollary 3.3 and Proposition 3.4 control a number of norms of $u$ by a large
constant times $\left\|u\right\|_{\mathbf{Y}}$ and a small constant times
$\left\|u\right\|_{\mathbf{X}}$, and so it will be acceptable to use these
quantities when estimating $\left\|u\right\|_{\mathbf{X}}$.

######  Definition 4.1 (Mollification).

For $1\leq k\leq N+1$, let $Z_{k}$ be defined as in (2.1d) and (2.1e), and
$s_{k}$ and $m_{k}$ be defined as in (2.2).

  1. (1)

Let $\phi\in C_{0}^{\infty}(-1,1)$ be a smooth, even, non-negative function
with unit $L^{1}$ norm. We set

$\displaystyle\phi_{\eta,k}u(x,y,t)$
$\displaystyle=\int_{\mathbb{R}}\left(\exp\left(\eta^{m_{k}}\sigma
Z_{k}\right)u\right)(x,y,t)\phi(\sigma)\,d\sigma\,$
$\displaystyle=\int_{\mathbb{R}}u(x+\eta^{m_{k}}\sigma\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y),y,t)\phi(\sigma)\,d\sigma\,.$
(4.1)

Note that since $\phi$ is even, $\phi_{\eta,k}$ is self-adjoint on
$L^{2}(\Omega\times(0,c_{0}^{2}))$.

  2. (2)

Let $\Phi\in C_{0}^{\infty}(-1,1)$ be a smooth, even, non-negative function
with unit $L^{1}$ norm. We set

$\displaystyle\Phi_{\eta}u(x,y,t)$
$\displaystyle=\int_{\mathbb{R}}(\exp(\eta^{m_{N+2}}x^{\prime}\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}}\partial_{x}))u(x,y,t)\Phi(x^{\prime})\,dx^{\prime}$
$\displaystyle=\int_{\mathbb{R}}u\left(x+\eta^{m_{N+2}}x^{\prime}\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}},y,t\right)\Phi(x^{\prime})\,dx^{\prime}\,$
(4.2)

and note that $\Phi_{\eta}$ is self-adjoint on
$L^{2}(\Omega\times(0,c_{0}^{2}))$ as well.

  3. (3)

For $\eta>0$ and $1\leq j\leq N+1$, define $\phi_{\eta,\geq j}u$ by

$\phi_{\eta,\geq j}u=\prod_{k=j}^{N+1}\phi_{\eta,k}\Phi_{\eta}u\,.$ (4.3)

When $j=1$, we simply write $\phi_{\eta}u$, and when $j=N+2$, we define
$\phi_{\eta,\geq j}u=\Phi_{\eta}u$.

######  Remark 4.2.

Since the smoothing operators are mollifications-in-$x$ for each fixed $t$ and
$y$, they actually commute. Moreover, we have

$[X_{0},\phi_{\eta,k}]u=\left[\exp(\tau X_{0}),\phi_{\eta,k}\right]u=0\,,$
(4.4)

with the same equalities holding for $\phi_{\eta,\geq j}$ and $\phi_{\eta}$.

######  Lemma 4.3 (Commutator Identity).

Let $Z$, $Z_{0}$ be smooth vector fields, and recall from (2.1a) that
$(\textnormal{ad}Z_{0})Z$ denotes the vector field $[Z_{0},Z]=Z_{0}Z-ZZ_{0}$.
Let $M\in\mathbb{N}_{0}$. Define

$\exp(-\tau\,\textnormal{ad}Z_{0})Zu=\exp(-\tau Z_{0})Z\exp(\tau
Z_{0})u(x,y,t)$ (4.5)

Then we have the identity

$\displaystyle\exp(-\tau\,\textnormal{ad}Z_{0})Zu=\sum_{k=0}^{M}\frac{(-\tau)^{k}}{k!}\left(\textnormal{ad}Z_{0}\right)^{k}Zu(x,y,t)+\int_{0}^{\tau}(\textnormal{ad}Z_{0})^{M+1}Zu(x,y,t)(\tau-s)^{M}\,ds\,.$
(4.6)

###### Proof.

Consider the formal identity

$\exp(-\tau Z_{0})Z\exp(\tau
Z_{0})u=\sum_{k=0}^{\infty}\frac{(-\tau)^{k}}{k!}(\textnormal{ad}Z_{0})^{k}Zu,$
(4.7)

which can be ‘checked’ by differentiating the left-hand side $k$ times with
respect to $\tau$, evaluating at $\tau=0$, and comparing with the Taylor
series on the right-hand side. Since the series may not converge, we instead
truncate the Taylor series to order $M$, which, after using the integral form
of the remainder term, leads directly to (4.6). ∎

######  Lemma 4.4 (Mollification Estimates).

With $\phi_{\eta}$ defined as in Definition 4.1, the following estimates hold.

  1. (1)

Let $\eta^{2}\leq\nicefrac{{c_{0}^{2}}}{{2}}$ be given. Then

$\left\|\phi_{\eta}u-u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\lesssim\eta\left(\sum_{k=1}^{N+1}\left\|u\right\|_{\mathbf{Z}_{k}}+\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\right)\,.$
(4.8)

  2. (2)

For $0\leq\tau\leq\eta^{2}\leq\nicefrac{{c_{0}^{2}}}{{{2}}}$,

$\displaystyle\left\|\exp(\tau
X_{0})\exp(-\tau\textnormal{ad}X_{0})Y\phi_{\eta}\phi_{\eta}u\right\|_{L^{2}(\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}}))}$
$\displaystyle\qquad\qquad\lesssim\left\|u\right\|_{\mathbf{Y}}+\sum_{k=1}^{N+1}\left\|u\right\|_{\mathbf{Z}_{k}}+\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{{Q}^{s_{N+2}}_{\partial_{x}}}\,,$
(4.9)

with a similar estimate holding if $\tau$ is replaced with $-\tau$ and the
time interval $(0,\nicefrac{{c_{0}^{2}}}{{2}})$ is replaced with
$(\nicefrac{{c_{0}^{2}}}{{2}},c_{0}^{2})$.

The estimate (4.8) is used to control the terms (4.16) and (4.18) in the proof
of Proposition 4.5 (Interpolation inequality), whereas (4.9) is used to
control the more involved term (4.17).

###### Proof of (4.8).

We may write $\phi_{\eta}u-u$ in a telescoping sum as

$\displaystyle\phi_{\eta}u-u$
$\displaystyle=\prod_{k=1}^{N+1}\phi_{\eta,k}\Phi_{\eta}u-u$
$\displaystyle=\sum_{m=1}^{N+1}\left(\prod_{k=1}^{m}\phi_{\eta,k}u\
-\prod_{k=1}^{m-1}\phi_{\eta,k}u\right)+\prod_{k=1}^{N+1}\phi_{\eta,k}\Phi_{\eta}u-\prod_{k=1}^{N+1}\phi_{\eta,k}u\,.$

Using again that convolution is a bounded operator from
$L^{2}(\Omega\times(0,c_{0}^{2}))$ to itself, it suffices to estimate in
$L^{2}(\Omega\times(0,c_{0}^{2}))$ the quantities

$\phi_{\eta,k}u-u,\qquad\Phi_{\eta}u-u$

for $1\leq k\leq N+1$. This is achieved by writing

$\displaystyle\left\|\phi_{\eta,k}u(x,y,t)-u(x,y,t)\right\|^{2}_{L^{2}(\Omega\times(0,c_{0}^{2}))}$
$\displaystyle\qquad\leq\int_{0}^{c_{0}^{2}}\int_{0}^{1}\int_{\mathbb{T}}\int_{\mathbb{R}}\left(\exp(\sigma\eta^{m_{k}}Z_{k})u(x,y,t)-u(x,y,t)\right)^{2}\phi(\sigma)\,d\sigma\,dx\,dy\,dt\,$
$\displaystyle\qquad=\int_{0}^{c_{0}^{2}}\int_{0}^{1}\int_{\mathbb{T}}\int_{\mathbb{R}}\left(\exp((\widetilde{\sigma}\eta)^{m_{k}}Z_{k})u(x,y,t)-u(x,y,t)\right)^{2}\phi(\widetilde{\sigma}^{m_{k}})m_{k}\widetilde{\sigma}^{m_{k}-1}\,d\widetilde{\sigma}\,dx\,dy\,dt\,$
$\displaystyle\qquad\lesssim\eta^{2}\left\|u\right\|^{2}_{\mathbf{Z}_{k}}\,,$

with a similar estimate for $\Phi_{\eta}$. Note that since $\phi$ is compactly
supported in $(-1,1)$, we may assume $\widetilde{\sigma}=\sigma^{s_{k}}$ is
contained in the interval $(0,1)$, and so
$\exp((\widetilde{\sigma}\eta)^{m_{k}}Z_{k})u$ is well defined since
$\widetilde{\sigma}\eta\leq\eta\leq\nicefrac{{c_{0}}}{{\sqrt{2}}}\leq c_{k}$
for all $k\geq 1$ by (2.3). ∎

###### Proof of (4.9).

To make our situation more concrete, we exploit the identity

$\displaystyle\exp(-\tau\textnormal{ad}X_{0})Y=Y+\tau
Z_{1}=\nu^{\nicefrac{{1}}{{2}}}\partial_{y}+\tau\nu^{\nicefrac{{1}}{{2}}}b^{\prime}(y)\partial_{x}\,,$

which does not include any translations in time. Since $\exp(\tau X_{0})$
consists of a volume-preserving diffeomorphism in the $x$ variable and a shift
in the time variable, it will suffice to estimate

$\left\|\exp(-\tau\textnormal{ad}X_{0})Y\phi_{\eta}\phi_{\eta}u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}=\left\|\left(Y+\tau
Z_{1}\right)\phi_{\eta}\phi_{\eta}u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\,.$

Step 1. Estimating $Y\phi_{\eta}\phi_{\eta}u$. To commute $Y$ past the
mollifications, we will demonstrate

$\left\|[Y,\phi_{\eta,\geq j}\phi_{\eta,\geq
j}]u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\lesssim\left\|u\right\|_{\mathbf{Y}}+\sum_{k=1}^{N+1}\left\|u\right\|_{\mathbf{Z}_{k}}+\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}$
(4.10)

using decreasing induction on $j$ and beginning with the case $j=N+2$. Since
$Yu$ is clearly bounded in $L^{2}$ by the right-hand side of (4.10), this will
then produce an estimate for $Y\phi_{\eta}\phi_{\eta}u$ which matches (4.9).

Beginning with the case $j=N+2$, recall from (4.3) that $\phi_{\eta,\geq
N+2}=\Phi_{\eta}$. Then since $Y$ differentiates in $y$ and $\Phi_{\eta}$
mollifies in $x$, $Y$ in fact commutes with both operators, demonstrating the
base case of the induction. Therefore, we may assume that $1\leq j\leq N+1$
and that (4.10) holds for $j^{\prime}>j$. We calculate that

$\displaystyle\left[Y,\phi_{\eta,\geq j}\phi_{\eta,\geq j}\right]u$
$\displaystyle=\left[Y,\phi_{\eta,j}\phi_{\eta,j}\phi_{\eta,\geq
j+1}\phi_{\eta,\geq j+1}\right]u$
$\displaystyle=\phi_{\eta,j}\phi_{\eta,j}[Y,\phi_{\eta,\geq
j+1}\phi_{\eta,\geq
j+1}]u+\left[Y,\phi_{\eta,j}\phi_{\eta,j}\right]\phi_{\eta,\geq
j+1}\phi_{\eta,\geq j+1}u\,.$

Due to the induction hypothesis, we may immediately bound the first term, so
we focus on the second term. We may expand this term using a modification of
Lemma 4.3 with $Z_{0}=Z_{j}$ and $Z=Y$ in which we have applied
$\exp(\eta^{m_{j}}(\sigma-\sigma^{*})Z_{j})$ to both sides:

$\displaystyle
Y\int\kern-3.39996pt\int_{\mathbb{R}^{2}}\exp\left(\eta^{m_{j}}\left(\sigma-\sigma^{*}\right)Z_{j}\right)\phi_{\eta,\geq
j+1}\phi_{\eta,\geq j+1}u(x,y,t)\,\phi(\sigma)\phi(\sigma^{*})\,d\sigma
d\sigma^{*}$
$\displaystyle\qquad\qquad\qquad-\int\kern-3.39996pt\int_{\mathbb{R}^{2}}\exp\left(\eta^{m_{j}}\left(\sigma-\sigma^{*}\right)Z_{j}\right)Y\phi_{\eta,\geq
j+1}\phi_{\eta,\geq j+1}u(x,y,t)\,\phi(\sigma)\phi(\sigma^{*})\,d\sigma
d\sigma^{*}$
$\displaystyle\overset{\eqref{eq:adjointrepdef}}{=}\int\kern-3.39996pt\int_{\mathbb{R}^{2}}\bigg{[}\exp\left(\eta^{m_{j}}\left(\sigma-\sigma^{*}\right)Z_{j}\right)\exp\left(-\eta^{m_{j}}\left(\sigma-\sigma^{*}\right)\textnormal{ad}Z_{j}\right)Y\phi_{\eta,\geq
j+1}\phi_{\eta,\geq j+1}u(x,y,t)$
$\displaystyle\qquad\qquad\qquad-\exp\left(\eta^{m_{j}}\left(\sigma-\sigma^{*}\right)Z_{j}\right)Y\phi_{\eta,\geq
j+1}\phi_{\eta,\geq j+1}u(x,y,t)\bigg{]}\phi(\sigma)\phi(\sigma^{*})\,d\sigma
d\sigma^{*}$
$\displaystyle\overset{\eqref{eq:barf:identity:1}}{=}\int\kern-3.39996pt\int_{\mathbb{R}^{2}}\exp\left(\eta^{m_{j}}(\sigma-\sigma^{*})Z_{j}\right)$
$\displaystyle\qquad\qquad\times\sum_{k=1}^{\infty}\frac{(-\eta)^{km_{j}}(\sigma-\sigma^{*})^{k}}{k!}\left(\textnormal{ad}Z_{j}\right)^{k}Y\phi_{\eta,\geq
j+1}\phi_{\eta,\geq j+1}u(x,y,t)\,\phi(\sigma)\phi(\sigma^{*})\,d\sigma
d\sigma^{*}\,.$ (4.11)

Notice that the infinite series representation is well defined since

$\displaystyle-\left(\textnormal{ad}Z_{j}\right)Y=Z_{j+1},\qquad\left(\textnormal{ad}Z_{j}\right)^{2}Y=-\left(\textnormal{ad}Z_{j}\right)Z_{j+1}=-\left[\nu^{\frac{j}{2}}b^{(j)}(y)\partial_{x},\nu^{\frac{j+1}{2}}b^{(j+1)}(y)\partial_{x}\right]=0\,,$

and so the only non-zero term in (4.11) is the term in the series for $k=1$.

It will suffice to obtain an $L^{2}$ bound for

$\displaystyle Z_{j+1}\phi_{\eta,\geq j+1}\phi_{\eta,\geq j+1}u(x,y,t)\,$
(4.12)

for $1\leq j\leq N+1$. In fact, we will estimate a version of this term for
$j\geq 0$, as the $j=0$ case will be required shortly. We begin by relabeling
$j$ with $0\leq j\leq N+1$ as $j$ with $1\leq j\leq N+2$ in (4.12). Then we
analyze how the derivatives ‘hit’ the mollifiers: For $1\leq j<N+2$, we
have101010The notation $Z_{j}\cdot u(\cdots)$ below emphasizes that the
differential operator $Z_{j}$ is applied to the object $u(\cdots)$, in
contrast to $(Z_{j}u)(\cdots)$.

$\displaystyle Z_{j}\phi_{\eta,\geq j}\phi_{\eta,\geq j}u(x,y,t)$
$\displaystyle=Z_{j}\prod_{k=j}^{N+1}\phi_{\eta,k}\Phi_{\eta}\prod_{k=j}^{N+1}\phi_{\eta,k}\Phi_{\eta}\,u(x,y,t)$
$\displaystyle=\int_{\mathbb{R}^{2(N+3-j)}}\Phi(x^{\prime})\Phi(x^{*})\prod_{k=j}^{N+1}\phi(\sigma_{k})\phi(\sigma_{k}^{*})$
$\displaystyle\quad\times Z_{j}\cdot
u\left(x+\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}}\eta^{{m_{N+2}}}(x^{\prime}-x^{*})+\sum_{k=j}^{N+1}\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y)\eta^{m_{k}}(\sigma_{k}-\sigma_{k}^{*}),y,t\right)\,dx^{\prime}\,dx^{*}\,\prod_{k=j}^{N+1}d\sigma_{k}\,d\sigma_{k}^{*}$
$\displaystyle=\int_{\mathbb{R}^{2(N+3-j)}}\Phi(x^{\prime})\Phi(x^{*})\left(\prod_{k=j}^{N+1}\phi(\sigma_{k})\phi(\sigma_{k}^{*})\right)\nu^{\nicefrac{{j}}{{2}}}b^{(j)}(y)$
$\displaystyle\quad\times\partial_{x}u\left(x+\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}}\eta^{{m_{N+2}}}(x^{\prime}-x^{*})+\sum_{k=j}^{N+1}\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y)\eta^{m_{k}}\left(\sigma_{k}-\sigma_{k}^{*}\right),y,t\right)\,dx^{\prime}\,dx^{*}\,\prod_{k=j}^{N+1}d\sigma_{k}\,d\sigma_{k}^{*}\,.$

We need to multiply this expression by $\eta^{m_{j-1}}$ and then bound the
resulting object by the right hand side of (4.9). Towards this end, we first
convert the $\nu^{\nicefrac{{j}}{{2}}}b^{(j)}(y)\partial_{x}$ to
$\eta^{-m_{j}}\left(\partial_{\sigma_{j}}-\partial_{\sigma_{j}^{*}}\right)$,
yielding

$\displaystyle\eta^{m_{j-1}-m_{j}}\int_{\mathbb{R}^{2(N+3-j)}}\left(\partial_{\sigma_{j}}-\partial_{\sigma_{j}^{*}}\right)\cdot
u\left(x+\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}}\eta^{{m_{N+2}}}(x^{\prime}-x^{*})+\sum_{k=j}^{N+1}\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y)\eta^{m_{k}}\left(\sigma_{k}-\sigma_{k}^{*}\right),y,t\right)$
$\displaystyle\qquad\qquad\qquad\times\Phi(x^{\prime})\Phi(x^{*})\left(\prod_{k=j}^{N+1}\phi(\sigma_{k})\phi(\sigma_{k}^{*})\right)\,dx^{\prime}\,dx^{*}\,\prod_{k=j}^{N+1}d\sigma_{k}\,d\sigma_{k}^{*}$
$\displaystyle\overset{\footnotesize\rm
IBP}{=}-\eta^{-1}\int_{\mathbb{R}^{2(N+3-j)}}u\left(x+\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}}\eta^{{m_{N+2}}}(x^{\prime}-x^{*})+\sum_{k=j}^{N+1}\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y)\eta^{m_{k}}\left(\sigma_{k}-\sigma_{k}^{*}\right),y,t\right)$
$\displaystyle\qquad\qquad\qquad\times\Phi(x^{\prime})\Phi(x^{*})\left(\partial_{\sigma_{j}}-\partial_{\sigma_{j}^{*}}\right)\left(\prod_{k=j}^{N+1}\phi(\sigma_{k})\phi(\sigma_{k}^{*})\right)\,dx^{\prime}\,dx^{*}\,\prod_{k=j}^{N+1}d\sigma_{k}\,d\sigma_{k}^{*}$
$\displaystyle=-\eta^{-1}\int_{\mathbb{R}^{2(N+3-j)}}\bigg{[}u\left(x+\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}}\eta^{{m_{N+2}}}(x^{\prime}-x^{*})+\sum_{k=j}^{N+1}\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y)\eta^{m_{k}}\left(\sigma_{k}-\sigma_{k}^{*}\right),y,t\right)$
$\displaystyle\qquad\qquad\qquad\qquad-u\left(x+\nu^{\frac{N+2}{2}}\|b^{(N+2)}\|_{L^{\infty}}\eta^{{m_{N+2}}}(x^{\prime}-x^{*})+\sum_{k=j+1}^{N+1}\nu^{\nicefrac{{k}}{{2}}}b^{(k)}(y)\eta^{m_{k}}\left(\sigma_{k}-\sigma_{k}^{*}\right),y,t\right)\bigg{]}$
$\displaystyle\qquad\qquad\qquad\times\Phi(x^{\prime})\Phi(x^{*})\left(\partial_{\sigma_{j}}-\partial_{\sigma_{j}^{*}}\right)\left(\prod_{k=j}^{N+1}\phi(\sigma_{k})\phi(\sigma_{k}^{*})\right)\,dx^{\prime}\,dx^{*}\,\prod_{k=j}^{N+1}d\sigma_{k}\,d\sigma_{k}^{*}\,.$
(4.13)

Notice that the second-to-last term, which may be subtracted for free due to
the fact that we have differentiated with respect to
$\sigma_{j}-\sigma_{j}^{*}$, crucially gives a finite difference along $Z_{j}$
in the $x$ variable. Now since $\sigma_{k}-\sigma_{k}^{*}$ is at most of order
one, we have a finite difference of order $\eta^{m_{j}}$ multiplied by
$\eta^{-1}$, and so we can then apply Jensen’s inequality and integrate in
$L^{2}(\Omega\times(0,c_{0}^{2}))$ to control the $L^{2}$ norm of the right-
hand side of (4.13) by $\left\|u\right\|_{\mathbf{Z}_{j}}$.

It remains to treat the case $j=N+2$, in which we must bound
$Z_{N+2}\Phi_{\eta}\Phi_{\eta}u$. The argument is similar, and we omit further
details.

Step 2. Estimating $\tau Z_{1}\phi_{\eta}\phi_{\eta}u$. Notice that
$Z_{1}\phi_{\eta}\phi_{\eta}$ is precisely what we calculated in the previous
step for $j=1$. The only difference in the calculation is that instead of
multiplying by a power of $\eta$, we multiply by $\tau$. But by the assumption
that $\tau\leq\eta^{2}$, we pick up a prefactor of $\eta^{-1}$ exactly as in
(4.13). We omit further details.

∎

######  Proposition 4.5 (Interpolation inequality).

For a constant independent of $u$, we have the estimate

$\left\|u\right\|_{\mathbf{X}}\lesssim\left\|u\right\|_{\mathbf{Y}}+\left\|X_{0}u\right\|_{\mathbf{Y}^{*}}+\sum_{k=1}^{N+1}\left\|u\right\|_{\mathbf{Z}_{k}}+\left(\nu^{\frac{N+2}{2}}\left\|b^{(N+2)}\right\|_{L^{\infty}}\right)^{s_{N+2}}\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}\,.$
(4.14)

###### Proof of Proposition 4.5.

For $\eta\leq\nicefrac{{c_{0}}}{{\sqrt{2}}}$, we wish to demonstrate that

$\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}(\exp(\eta^{2}X_{0})u-u)^{2}\,dx\,dy\,dt+\int_{\Omega\times(\nicefrac{{c_{0}^{2}}}{{2}},c_{0}^{2})}(\exp(-\eta^{2}X_{0})u-u)^{2}\,dx\,dy\,dt\lesssim\eta^{2}M[u]^{2}\,,$
(4.15)

where we have used $M[u]$ to denote the right hand side of (4.15). We focus on
estimating the first term on the left-hand side of (4.15), since the second
term is similar. We split up the integrand as

$\displaystyle\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left(\exp(\eta^{2}X_{0})u-u\right)^{2}\,dx\,dy\,dt$
$\displaystyle\quad\lesssim\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left(\exp(\eta^{2}X_{0})u-\exp(\eta^{2}X_{0})\phi_{\eta}u\right)^{2}\,dx\,dy\,dt$
(4.16)
$\displaystyle\quad\quad+\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left(\exp(\eta^{2}X_{0})\phi_{\eta}u-\phi_{\eta}u\right)^{2}\,dx\,dy\,dt$
(4.17)
$\displaystyle\quad\quad+\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left(\phi_{\eta}u-u\right)^{2}\,dx\,dy\,dt\,.$
(4.18)

The first and third terms on the right-hand side may be bounded by appealing
to (4.8). Focusing then on the second term and substituting in the variable
$\tau$ for values $\tau\leq\eta^{2}$, it will suffice to show that

$\frac{d}{d\tau}\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left(\exp(\tau
X_{0})\phi_{\eta}u-\phi_{\eta}u\right)^{2}\,dx\,dy\,dt\lesssim M[u]^{2}\,.$

Calculating the left-hand side, we write that

$\displaystyle\frac{d}{d\tau}\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left(\exp(\tau
X_{0})\phi_{\eta}u-\phi_{\eta}u\right)^{2}\,dx\,dy\,dt$
$\displaystyle\quad=2\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left[\phi_{\eta}u(t+\tau,x+\tau
b(y),y)-\phi_{\eta}u(t,x,y)\right]$
$\displaystyle\qquad\qquad\times\left(\partial_{t}+b(y)\partial_{x}\right)\phi_{\eta}u(t+\tau,x+\tau
b(y),y)\,dx\,dy\,dt$
$\displaystyle\quad=2\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left[\phi_{\eta}u(t,x,y)-\phi_{\eta}u(t-\tau,x-\tau
b(y),y)\right]\left(\partial_{t}+b(y)\partial_{x}\right)\phi_{\eta}u(t,x,y)\,dx\,dy\,dt$
$\displaystyle\quad=2\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left[\phi_{\eta}u(t,x,y)-\phi_{\eta}u(t-\tau,x-\tau
b(y),y)\right]\phi_{\eta}\left(X_{0}u\right)(t,x,y)\,dx\,dy\,dt$
$\displaystyle\qquad+2\int_{\Omega\times(0,\nicefrac{{c_{0}^{2}}}{{2}})}\left[\phi_{\eta}u(t,x,y)-\phi_{\eta}u(t-\tau,x-\tau
b(y),y)\right]\left[X_{0},\phi_{\eta}\right]u(t,x,y)\,dx\,dy\,dt\,.$

The second term in the last equality vanishes after appealing to (4.4). To
estimate the first term, first note that from (1), we may convert the
convolution of $\left(\partial_{t}+b\partial_{x}\right)u$ with $\phi_{\eta}$
into convolution with $\phi_{\eta}$ on the other term in the integrand. We
then appeal to (4.4) to commute $\phi_{\eta}$ past the $\exp(-\tau X_{0})$.
Using the bound on $\partial_{t}u+b\partial_{x}u$ in $\mathbf{Y}^{*}$, we see
that the proof will be complete if we can estimate

$Y\left(\phi_{\eta}\phi_{\eta}u-\exp\left(-\tau
X_{0}\right)\phi_{\eta}\phi_{\eta}u\right)\,.$

Appealing to Lemma 4.3 to rewrite this expression and then to (4.9) concludes
the proof. ∎

Using Proposition 4.5, we may immediately obtain an improved subelliptic
estimate. Indeed, after employing the higher bracket inequality to absorb the
$\left\|u\right\|_{\mathbf{Z}_{k}}$ terms as in the proof of Corollary 3.3 and
the subelliptic estimate to absorb the
$\left\|u\right\|_{Q^{s_{N+2}}_{\partial_{x}}}$ term as in the proof of
(3.11), we obtain the following.

######  Theorem 4.6 (Improved subelliptic estimate).

For all $\nu\leq\nu_{0}\ll_{b}1$, we have that

$\nu^{\frac{N+1}{2(N+3)}}\left(\left\|u-\langle
u\rangle\right\|_{L^{2}}+\left\|u\right\|_{Q^{s_{N+1}}_{\partial_{x}}}\right)\lesssim_{b}\left\|X_{0}u\right\|_{\mathbf{Y}^{*}}+\left\|u\right\|_{\mathbf{Y}}\,$
(4.19)

for an implicit constant which is independent of $\nu$ but depends on $b$.

##  5\. Proofs of Theorems 1.1 and 1.2

###### Proof of Theorem 1.1.

For $t\geq 0$, we define the time-translated seminorm

$\|u\|_{\mathbf{Y}(t)}:=\nu^{\nicefrac{{1}}{{2}}}\|\partial_{y}u\|_{L^{2}(\Omega\times(t,t+c_{0}^{2}))}\,,$

with an analogous definition for $\mathbf{Y}^{*}(t)$. For solutions of (1.5),
we have the identity

$\|f(\cdot,t_{2})\|_{L^{2}(\Omega)}^{2}-\|f(\cdot,t_{1})\|_{L^{2}(\Omega)}^{2}=-2\int_{t_{1}}^{t_{2}}\|Yf(\cdot,s)\|_{L^{2}(\Omega)}^{2}ds\,.$
(5.1)

For $k\in\mathbb{N}$, setting $t_{1}=0$ and $t_{2}=kc_{0}^{2}$ we can deduce
from (5.1) that

$\sum_{i=0}^{k-1}\left\|f\right\|_{\mathbf{Y}(ic_{0}^{2})}^{2}\leq\left\|f_{\rm
in}\right\|_{L^{2}(\Omega)}^{2}\,.$

We now have the crucial estimate

$\displaystyle\|X_{0}f\|_{\mathbf{Y}^{*}}\overset{\eqref{eq:fpde}}{=}\|Y^{2}f\|_{\mathbf{Y}^{*}}$
$\displaystyle=\sup_{u}\left|\int_{\Omega\times(0,c_{0}^{2})}Y^{2}f\cdot
u\,dx\,dy\,dt\right|$ (5.2) $\displaystyle\overset{\footnotesize\rm
IBP}{=}\sup_{u}\left|\int_{\Omega\times(0,c_{0}^{2})}Yf\cdot
Yu\,dx\,dydt\right|\leq\|Yf\|_{L^{2}(\Omega\times(0,c_{0}^{2})}$

where the supremum is over $u\in
L^{2}_{t,x}H^{1}_{y}(\Omega\times(0,c_{0}^{2}))$ ($(H^{1}_{0})_{y}$ in the
Dirichlet case) satisfying $\|u\|_{\mathbf{Y}}=1$, see Definition 2.4. Notice
that the boundary conditions satisfied by $f$ and the test function $u$ allow
the above integration by parts. Applying (5.2) and Theorem 4.6 (Improved
subelliptic estimate) to time translates of the solution, we obtain

$\displaystyle\nu^{\frac{N+1}{N+3}}\|f\|_{L^{2}(\Omega\times(0,kc_{0}^{2}))}^{2}$
$\displaystyle=\nu^{\frac{N+1}{N+3}}\sum_{i=0}^{k-1}\|f\|_{L^{2}(\Omega\times(ic_{0}^{2},(i+1)c_{0}^{2}))}^{2}$
$\displaystyle\leq
2C_{b}\sum_{i=0}^{k-1}\left\|f\right\|_{\mathbf{Y}(ic_{0}^{2})}^{2}$
$\displaystyle\leq C_{b}\left\|f_{\rm in}\right\|_{L^{2}(\Omega)}^{2}\,.$
(5.3)

We estimate the left-hand side from below using the monotonicity of the
$L^{2}$ norm for solutions of (1.5) and the choice
$c_{0}^{2}=\nu^{-\frac{N+1}{N+3}}$ from Corollary 3.3:

$k\|f(\cdot,kc_{0}^{2})\|_{L^{2}(\Omega)}^{2}\leq
kc_{0}^{2}\nu^{\frac{N+1}{N+3}}\inf_{t^{\prime}\in(0,kc_{0}^{2})}\|f(\cdot,t^{\prime})\|_{L^{2}(\Omega)}^{2}\leq\nu^{\frac{N+1}{N+3}}\|f\|_{L^{2}(\Omega\times(0,kc_{0}^{2}))}^{2}\,.$
(5.4)

Combining (5.4) with (5.3), we have

$k\|f(\cdot,kc_{0}^{2})\|_{L^{2}(\Omega)}^{2}\leq C_{b}\|f_{\rm
in}\|_{L^{2}(\Omega)}^{2}\,.$

Taking $k_{0}$ sufficiently large yields

$\|f(\cdot,k_{0}c_{0}^{2})\|_{L^{2}(\Omega)}^{2}\leq\frac{1}{e}\left\|f_{\rm
in}\right\|_{L^{2}(\Omega)}^{2}\,.$

We iterate this estimate to obtain, for an integer multiple $j\geq 0$ of
$k_{0}c_{0}^{2}$,

$\|f(\cdot,jk_{0}c_{0}^{2})\|_{L^{2}(\Omega)}^{2}\leq e^{-j}\|f_{\rm
in}\|_{L^{2}(\Omega)}^{2}\,.$

The monotonicity of the $L^{2}$ norm for solutions of (1.5) gives

$\|f(\cdot,t)\|_{L^{2}(\Omega)}^{2}\leq
e\exp\left(-tk_{0}^{-1}\nu^{\frac{N+1}{N+3}}\right)\|f_{\rm
in}\|_{L^{2}(\Omega)}^{2}\,,$

for any $t\geq 0$, which concludes the proof of the theorem.∎

###### Proof of Theorem 1.2.

. Let $q\colon[0,+\infty)\to[0,+\infty)$ be a decreasing function satisfying

$\sum_{j\in\mathbb{N}}q^{2}(2^{j})<+\infty\,.$

For example, we may set $q(j)=(\log(2+|j|))^{-\frac{1}{2}-\varepsilon}$. We
require the following characterization of the Besov space
$Q^{s}_{\partial_{x}}$.111111For discussion of real-valued Besov spaces and
equivalences, one may refer to Bahouri, Chemin, and Danchin [4]. For vector-
valued and Banach space-valued analogues, one may refer to Amann [1], which
contains (5.5) in Section 5. Let $(\dot{\Delta}_{j}^{x})_{j\geq 0}$ be a
suitable sequence of Littlewood-Paley projections onto dyadic annuli in the
$x$-direction in frequency space satisfying $\sum_{j\geq
0}\dot{\Delta}_{j}u\equiv u$ for functions
$u\colon\Omega\times(0,c_{0}^{2})\to\mathbb{R}$ with $\langle
u\rangle(y,t)\equiv 0$. Then

$\sup_{j\geq
0}\,2^{js}\left\|\dot{\Delta}_{j}^{x}u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}\sim_{s}\left\|u\right\|_{Q^{s}_{\partial_{x}}(\Omega\times(0,c_{0}^{2}))}\,.$
(5.5)

Using this equivalence, we have that

$\displaystyle\left\||\partial_{x}|^{s}q(|\partial_{x}|)u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}^{2}$
$\displaystyle=\sum_{k\in
2\pi\mathbb{Z}}|k|^{2s}q^{2}(|k|)\int\kern-3.39996pt\int_{(0,1)\times(0,c_{0}^{2})}\left|\hat{u}(k,y,t)\right|^{2}\,dy\,dt$
$\displaystyle\lesssim_{s}\sum_{j\geq
0}2^{2js}q^{2}(2^{j})\left\|\dot{\Delta}_{j}^{x}u\right\|_{L^{2}(\Omega\times(0,c_{0}^{2}))}^{2}$
$\displaystyle\lesssim_{s,q}\left\|u\right\|_{Q^{s}_{\partial_{x}}(\Omega\times(0,c_{0}^{2}))}^{2}\,.$

The subelliptic estimate (3.11) combined with the energy estimates gives

$\nu^{\frac{N+1}{N+3}}\left\|f\right\|_{Q^{s_{N+1}}_{\partial_{x}}(\Omega\times(0,c_{0}^{2}))}^{2}\lesssim_{b}\left\|f_{\rm
in}\right\|_{L^{2}}^{2}-\left\|f(\cdot,c_{0}^{2})\right\|_{L^{2}(\Omega)}^{2}\,.$

By the energy estimate, the above embeddings, and a sum over the intervals
$(nc_{0}^{2},(n+1)c_{0}^{2})$ for $n\in\mathbb{N}_{0}$, we have

$\nu^{\frac{N+1}{N+3}}\left\||\partial_{x}|^{\frac{1}{N+3}}q(|\partial_{x}|)f\right\|_{L^{2}(\Omega\times\mathbb{R}^{+})}^{2}\lesssim_{b}\left\|f_{\rm
in}\right\|_{L^{2}}^{2}\,.$

Let $t>0$. By the pigeonhole principle or Chebyshev’s inequality, by choosing
$C$ sufficiently large, we can ensure that there exists a measurable set
$A\subset(0,t)$ with $|A|\geq t/2$ which consists of times $\bar{t}\in
A\subset(0,t)$ satisfying

$\left\||\partial_{x}|^{\frac{1}{N+3}}q(|\partial_{x}|)f(\cdot,\bar{t})\right\|_{L^{2}(\Omega)}^{2}\leq
Ct^{-1}\nu^{-\frac{N+1}{N+3}}\left\|f_{\rm in}\right\|_{L^{2}}^{2}\,.$

Since $|\partial_{x}|^{\frac{1}{N+3}}q(|\partial_{x}|)f$ also satisfies (1.5),
we have that the left-hand side is monotone decreasing in time. This
guarantees the smoothing estimate

$\left\||\partial_{x}|^{\frac{1}{N+3}}q(|\partial_{x}|)f(\cdot,t)\right\|_{L^{2}(\Omega)}^{2}\leq
Ct^{-1}\nu^{-\frac{N+1}{N+3}}\left\|f_{\rm in}\right\|_{L^{2}}^{2}$

for any $t>0$. One may continue smoothing by applying the estimate again to
the function $|\partial_{x}|^{\frac{1}{N+3}}q(|\partial_{x}|)f(\cdot,t)$.
Specifically, we may obtain

$\left\||\partial_{x}|^{\frac{2}{N+3}}q^{2}(|\partial_{x}|)f(\cdot,t)\right\|_{L^{2}(\Omega)}\leq
C_{0}2^{-1}t^{-1}\nu^{-\frac{N+1}{N+3}}\left\|f_{\rm in}\right\|_{L^{2}}\,,$

which is more convenient for summing. For given $t>0$, we may iterate this
smoothing estimate $k$ times on intervals of length $t/k$. This gives

$C_{0}^{-k}t^{k}k^{-k}\nu^{\frac{N+1}{N+3}k}\left\|\left[|\partial_{x}|^{\frac{2}{N+3}}q^{2}(|\partial_{x}|)\right]^{k}f(\cdot,t)\right\|_{L^{2}(\Omega)}\leq
2^{-k}\left\|f_{\rm in}\right\|_{L^{2}}\,.$

for all $t>0$ and $k\geq 0$. By Stirling’s formula for $k^{-k}$, the Taylor
series of $\exp$, and the triangle inequality, we have

$\left\|\exp\left(\mu_{0}\nu^{\frac{N+1}{N+3}}|\partial_{x}|^{\frac{2}{N+3}}q^{2}(|\partial_{x}|)t\right)f(\cdot,t)\right\|_{L^{2}(\Omega)}\lesssim\left\|f_{\rm
in}\right\|_{L^{2}}\,,$

where $\mu_{0}\ll C_{0}^{-1}$. This gives the desired Gevrey smoothing. ∎

##  Appendix A Proof of sample interpolation inequality

###### Proof.

It suffices to prove the (apparently) weaker inequality

$\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}\lesssim\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}+\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}$
(A.1)

for all $f\in C^{\infty}_{0}$. After rescaling in $y$, we have

$\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}\lesssim\lambda\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}+\lambda^{-1}\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}.$

Optimizing $\lambda$ gives the full inequality. The optimization step is
possible because, by assumption, each term on the RHS of (A.1) is non-zero.

For $\ell>0$, consider the mollification-in-$x$ at scale $\ell^{3}$:

$S_{\ell}f(x,y):=\ell^{-3}\int_{\mathbb{R}}f(x-s,y)\varphi(\ell^{-3}s)\,ds.$

We will require the properties

$\|S_{\ell}f-f\|_{L^{2}}\lesssim\ell\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}$
(A.2)

and

$\|\partial_{x}S_{\ell}f\|_{L^{2}}\lesssim\ell^{-2}\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}.$
(A.3)

Let $h\in\mathbb{R}$. We expand

$\displaystyle\|e^{h^{2}y\partial_{x}}f-f\|_{L^{2}}$
$\displaystyle\leq\|e^{h^{2}y\partial_{x}}f-e^{h^{2}y\partial_{x}}S_{\ell}f\|_{L^{2}}$
$\displaystyle+\|e^{h^{2}y\partial_{x}}S_{\ell}f-S_{\ell}f\|_{L^{2}}$
$\displaystyle+\|S_{\ell}f-f\|_{L^{2}}\,.$

The first and third terms on the right-hand side are estimated in the same
way, namely, by the property (A.2) of the mollification operator. We focus on
the second term. We have

$\frac{d}{dh}\|e^{h^{2}y\partial_{x}}S_{\ell}f-S_{\ell}f\|_{L^{2}}^{2}=4h\langle
e^{h^{2}y\partial_{x}}S_{h}f,e^{h^{2}y\partial_{x}}y\partial_{x}S_{h}f\rangle=4h\langle
e^{h^{2}y\partial_{x}}S_{h}^{*}S_{h}f,y\partial_{x}f\rangle\,,$

which is estimated by $\dot{H}^{1}/\dot{H}^{-1}$ duality:

$h|\langle
e^{h^{2}y\partial_{x}}S_{h}^{*}S_{h}f,y\partial_{x}f\rangle|\lesssim
h\|e^{h^{2}y\partial_{x}}S_{h}^{*}S_{h}f\|_{L^{2}_{x}\dot{H}^{1}_{y}}\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}\,.$

It remains to estimate
$h\|\partial_{y}e^{h^{2}y\partial_{x}}S_{h}^{*}S_{h}f\|_{L^{2}}$. We have

$h\partial_{y}e^{h^{2}y\partial_{x}}S_{h}^{*}S_{h}f=h^{3}\partial_{x}e^{h^{2}y\partial_{x}}S_{h}^{*}S_{h}f+he^{h^{2}y\partial_{x}}\partial_{y}S^{*}_{h}S_{h}f-h\partial_{y}S^{*}_{h}S_{h}f\,.$

The second and third terms are estimated in $L^{2}$ by
$h\|\partial_{y}f\|_{L^{2}}$. For the 1st term, we allow $\partial_{x}$ to hit
the mollifier and use the property (A.3):

$h^{3}\|\partial_{x}e^{h^{2}y\partial_{x}}S_{h}^{*}S_{h}f\|_{L^{2}}\lesssim
h^{3}\ell^{-2}\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}\lesssim\ell\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}$

for $h\leq\ell$. This gives

$\left|\frac{d}{dh}\|e^{h^{2}y\partial_{x}}S_{\ell}f-S_{\ell}f\|_{L^{2}}^{2}\right|\lesssim(h\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}+\ell\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}})\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}\,.$

In particular, for $h=\ell$, we have

$\|e^{h^{2}y\partial_{x}}S_{\ell}f-S_{\ell}f\|_{L^{2}}^{2}\lesssim
h^{2}(\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}+\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}})\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}\,.$

Combining all estimates with $h=\ell$, we have

$\|e^{h^{2}y\partial_{x}}f-f\|_{L^{2}}^{2}\lesssim
h^{2}\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}^{2}+h^{2}(\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}+\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}})\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}$

or

$\displaystyle\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}^{2}$
$\displaystyle\lesssim\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}}^{2}+(\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}+\|f\|_{Q^{\nicefrac{{1}}{{3}}}_{\partial_{x}}})\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}$
$\displaystyle\lesssim\|f\|_{Q^{1}_{\partial_{y}}}^{\nicefrac{{2}}{{3}}}\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}^{\nicefrac{{4}}{{3}}}+\|f\|_{L^{2}_{x}\dot{H}^{1}_{y}}\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}+\|f\|_{Q^{1}_{\partial_{y}}}^{\nicefrac{{1}}{{3}}}\|f\|_{Q^{\nicefrac{{1}}{{2}}}_{y\partial_{x}}}^{\nicefrac{{2}}{{3}}}\|y\partial_{x}f\|_{L^{2}_{x}\dot{H}^{-1}_{y}}\,,$

Then Young’s inequality produces (A.1). ∎

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