# mixing FOR generic rough shear flows

L. Galeati  Batiment 8, Office MA B2455, AMCV Group
EPFL Lausanne
Switzerland<EMAIL_ADDRESS>and  M. Gubinelli  Andrew Wiles Building
Mathematical Institute
University of Oxford<EMAIL_ADDRESS>
###### Abstract.

We study mixing and diffusion properties of passive scalars driven by generic
rough shear flows. Genericity is here understood in the sense of prevalence
and (ir)regularity is measured in the Besov–Nikolskii scale
$B^{\alpha}_{1,\infty}$, $\alpha\in(0,1)$. We provide upper and lower bounds,
showing that in general inviscid mixing in $H^{1/2}$ holds sharply with rate
$r(t)\sim t^{1/(2\alpha)}$, while enhanced dissipation holds with rate
$r(\nu)\sim\nu^{\alpha/(\alpha+2)}$. Our results in the inviscid mixing case
rely on the concept of $\rho$-irregularity, first introduced by Catellier and
Gubinelli (Stoc. Proc. Appl. 126, 2016) and provide some new insights compared
to the behavior predicted by Colombo, Coti Zelati and Widmayer (Ars Inven.
Anal., 2021).

MSC(2020): 35Q35, 37C20, 76F25, 76R50.

###### Key words and phrases:

Mixing, Enhanced Dissipation, Prevalence, $\rho$-irregularity, Rough Flows.

Note: This document has been written using GNU TEXmacs [28].

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Prevalence
    2. 2.2 A useful class of Gaussian transverse measures
  3. 3 Inviscid mixing
    1. 3.1 Lower bounds in terms of regularity
    2. 3.2 Upper bounds in terms of $\rho$-irregularity
    3. 3.3 Prevalence statements and proof of Theorem 1.4
  4. 4 Enhanced dissipation
    1. 4.1 Lower bounds in terms of regularity
    2. 4.2 Wei’s irregularity condition
    3. 4.3 Sufficient conditions for stochastic processes
    4. 4.4 Prevalence statements and proof of Theorems 1.5, 1.1
  5. 5 Further comments and future directions
  6. A Besov spaces
  7. B A simple extension of a result by Wei

##  1\. Introduction

We are interested in the long time behavior of solutions $f$ to

$\left\\{\begin{array}[]{l}\partial_{t}f+u\partial_{x}f=\nu\Delta f\\\
f|_{t=0}=f_{0},\quad\int_{\mathbb{T}}f_{0}(x,y)\mathrm{d}x=0\end{array}\right.$
(1.1)

on the $2$-dimensional flat torus $\mathbb{T}^{2}$. The PDE (1.1) is an
advection-diffusion equation associated to a shear flow
$u=u(y):\mathbb{T}\rightarrow\mathbb{R}$, $f:\mathbb{R}_{\geqslant
0}\times\mathbb{T}^{2}\rightarrow\mathbb{R}$ with initial condition $f_{0}\in
L^{2}(\mathbb{T}^{2})$ and where $\nu\in[0,1]$ is the diffusion coefficient.
Defining $\bar{u}:\mathbb{T}^{2}\rightarrow\mathbb{R}^{2}$ as
$\bar{u}(x,y):=(u(y),0)^{T}$, equation (1.1) may be written as

$\partial_{t}f+\bar{u}\cdot\nabla f=\nu\Delta f$ (1.2)

which is the equation for a passive scalar $f$ advected by the velocity field
$\bar{u}$. Note that $\bar{u}$ is a divergence-free vector field and a
stationary solution to $2$D Euler equations.

Exactly for this reason, shear flows have received a lot of attention in the
literature, in connection to the problem of understanding the interaction
between mixing and diffusion in fluid mechanics and the transfer of energy
from large to small scales for the scalar $f$. In particular, shear flows are
sufficiently simple to allow explicit calculations, while presenting a highly
non trivial behavior, as already observed by Kelvin in [32] in the case of the
Couette flow $u(y)=y$.

Observe that for continuous $u$, eq. (1.1) can be solved explicitly by
Feynman–Kac formula, giving

$f_{t}(x,y)=\mathbb{E}\left[f_{0}\left(x-\int_{0}^{t}u\left(y+\sqrt{2\nu}B^{2}_{s}\right)\mathrm{d}s+\sqrt{2\nu}B^{1}_{t},y+\sqrt{2\nu}B^{2}_{t}\right)\right]$
(1.3)

where $B=(B^{1},B^{2})$ is a standard 2D Brownian motion (Bm). In the case
$\nu=0$ we obtain

$f_{t}(x,y)=f_{0}(x-tu(y),y).$ (1.4)

Both formulas (1.3) and (1.4) can then be extended to the case $u\in
L^{1}(\mathbb{T})$, 111For $u\in L^{1}(\mathbb{T})$, the formal expression
$\int_{0}^{t}u\left(y+\sqrt{2\nu}B^{2}_{s}\right)\mathrm{d}s$ in (1.3) can be
made rigorous using the local time of $B^{2}$; alternatively, equation (1.1)
can be solved by applying the Fourier transform in the $x$-variable and
solving the family of equations for $f^{k}=P_{k}f$, see the beginning of
Appendix B for more details. in which case eq. (1.1) must be understood in the
weak sense, and generate continuous semigroups
$e^{t(-u\partial_{x}+\nu\Delta)}$ on $L^{2}(\mathbb{T}^{2})$. Yet, they do not
provide any immediate insight on the long time behavior of the solution $f$,
in particular on the decay in time of quantities like $\|f_{t}\|_{H^{-s}}$ and
$\|f_{t}\|_{L^{2}}$.

Following the line of research initiated in [43], [10], we consider rough
shear flows, in the sense of requiring $u\in
B^{\alpha}_{1,\infty}(\mathbb{T})$ for some $\alpha\in(0,1)$. Here
$B^{\alpha}_{1,\infty}(\mathbb{T})$ denote the Besov–Nikolskii spaces, see
Appendix A for their definition.

We are interested in understanding the behavior of generic $u\in
B^{\alpha}_{1,\infty}(\mathbb{T})$, a problem explicitly left open in [10].
For this purpose we adopt the measure-theoretic notion of genericity given by
the theory of prevalence, developed by Hunt, Sauer and Yorke [29] to provide
an analogous of “Lebesgue almost every” on infinite dimensional spaces, see
Section 2.1 for more details. In what follows the expression “for almost every
$\varphi\in E$”, where $E$ is a function space, is understood in the sense of
prevalence.

The next statement summarizes our main findings.

######  Theorem 1.1.

Let $\alpha\in(0,1)$. The following hold:

  1. i.

For almost every $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ we have inviscid
mixing in the scale $H^{1/2}(\mathbb{T}^{2})$, in the following sense: for any
$\tilde{\alpha}>\alpha$, there exists $C=C(\alpha,\tilde{\alpha},u)$ such
that, for any $f_{0}\in H^{1/2}(\mathbb{T})$ satisfying
$\int_{\mathbb{T}}f\left(x,\cdot\,\right)\mathrm{d}x\equiv 0$, it holds

$\|e^{-tu\partial_{x}}f_{0}\|_{H^{-1/2}}\leqslant
Ct^{-\frac{1}{2\tilde{\alpha}}}\|f_{0}\|_{H^{1/2}}\quad\forall\,t\geqslant 0.$

  2. ii.

For almost every $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ we have enhanced
dissipation in the following sense that: for any $\tilde{\alpha}>\alpha$ there
exist $C_{i}=C(\alpha,\tilde{\alpha},u)$ such that, for any $f_{0}\in
L^{2}(\mathbb{T})$ satisfying
$\int_{\mathbb{T}}f\left(x,\cdot\,\right)\mathrm{d}x\equiv 0$, it holds

$\|e^{t(-u\partial_{x}+\Delta)}f_{0}\|_{L^{2}}\leqslant
C_{1}\exp\left(-C_{2}t\nu^{\frac{\tilde{\alpha}}{\tilde{\alpha}+2}}\right)\|f_{0}\|_{L^{2}}\quad\forall\,t\geqslant
0,\nu\in[0,1].$

In the above statement, the condition
$\int_{\mathbb{T}}f\left(x,\cdot\,\right)\mathrm{d}x\equiv 0$ is necessary, as
it naturally ensures that $f$ witnesses the effect of the transport operator
$u\partial_{x}$; indeed $g_{t}(y):=\int_{\mathbb{T}}f_{t}(x,y)\mathrm{d}x$
must solve the standard heat equation $\partial_{t}g=\nu\partial_{y}^{2}g$ and
thus cannot exhibit any mixing/enhanced dissipation effect.

There is no obvious a priori reason to work with the spaces
$B^{\alpha}_{1,\infty}(\mathbb{T})$ (e.g. in [10] the authors deal with
$C^{\alpha}(\mathbb{T})=B^{\alpha}_{\infty,\infty}(\mathbb{T})$), rather they
arise naturally in our analysis. One of the main intuitions of the present
paper is the identification of such spaces as the correct one for studying
generic inviscid mixing and enhanced dissipation properties of shear flows. At
the same time, let us mention that the only truly relevant parameter is
$\alpha\in(0,1)$: indeed statements similar to those of Theorem 1.1 can be
given for the (smaller) spaces $B^{\alpha}_{p,q}(\mathbb{T})$ for any choice
of $p,q\in[1,\infty]$, see Remark 1.6 below.

Before moving further, let us heuristically motivate the connection between
Points i. and ii. of Theorem 1.1 and why it is natural to expect
$\nu^{\alpha/(\alpha+2)}$ to appear, given the decay
$\|f_{t}\|_{H^{-1/2}}\lesssim t^{-1/(2\alpha)}$. In fact, the argument can be
given in a much more general framework: let $f^{\nu}$ be a solution to (1.2)
with $\nu>0$, $\int_{\mathbb{T}^{d}}f_{0}(z)\mathrm{d}z=0$ and
$\bar{u}:\mathbb{T}^{d}\rightarrow\mathbb{R}^{d}$ be a divergence free vector
field; then $f^{\nu}$ satisfies the energy balance

$\frac{\mathrm{d}}{\mathrm{d}t}\|f^{\nu}_{t}\|_{L^{2}}^{2}=-2\nu\|\nabla
f^{\nu}_{t}\|_{L^{2}}^{2}.$

Now assume the solution $f$ to the transport equation
$\partial_{t}f+\bar{u}\cdot\nabla f=0$ to satisfy the decay
$\|f_{t}\|_{\dot{H}^{-s}}\lesssim t^{-s/\alpha}$ for suitable parameters
$\alpha>0,s\in(0,1]$ (for $s>1$, one may reduce to $s=1$ by Riesz–Thorin
interpolation theorem). For $\nu\ll 1$ and sufficiently short times, we expect
$f^{\nu}$ and $f$ to stay close and therefore $f^{\nu}$ to exhibit the same
decay as $f$. By the interpolation inequality

$\displaystyle\|f\|_{L^{2}}\lesssim$
$\displaystyle\|f\|_{\dot{H}^{-s}}^{\frac{1}{1+s}}\|\nabla
f\|_{L^{2}}^{\frac{s}{s+1}},$

we deduce that

$\frac{\mathrm{d}}{\mathrm{d}t}\|f^{\nu}_{t}\|_{L^{2}}^{-\frac{2}{s}}\sim\nu\|f^{\nu}_{t}\|_{L^{2}}^{-2\left(\frac{s+1}{s}\right)}\|\nabla
f^{\nu}_{t}\|_{L^{2}}^{2}\gtrsim\nu\|f^{\nu}_{t}\|_{\dot{H}^{-s}}^{-\frac{2}{s}}\gtrsim\nu
t^{\frac{2}{\alpha}}.$ (1.5)

Assume for simplicity $\|f_{0}\|_{L^{2}}=1$ and define $\tau>0$ to be the
first time such that $\|f^{\nu}_{t}\|_{L^{2}}=1/2$. Integrating (1.5) over
$[0,\tau]$ we obtain

$\displaystyle 1\sim 2^{\frac{2}{s}}-1$
$\displaystyle\gtrsim\,\nu\int_{0}^{\tau}t^{\frac{2}{\alpha}}\mathrm{d}t\sim\nu\tau^{1+\frac{2}{\alpha}}=\left(\nu^{\frac{\alpha}{\alpha+2}}\tau\right)^{\frac{\alpha}{\alpha+2}}.$

Namely, in order for the energy $\|f^{\nu}_{t}\|_{L^{2}}$ to be reduced by
half by the dynamics, we need to wait for at most
$\tau\lesssim\nu^{-\alpha/(\alpha+2)}$. Iterating the argument on intervals
$[n\tau,n(\tau+1)]$ would then produce an asymptotic decay at least of the
form $\exp(-Ct\nu^{\frac{\alpha}{\alpha+2}})$.

While the argument is clearly heuristic, it predicts the correct exponent
$\frac{\alpha}{\alpha+2}$ and works for any choice of the parameter $s>0$ (in
particular for $s=1/2$ as in Theorem 1.1) and not only for $s=1$, which is the
case receiving the most attention in the literature.

Unfortunately, there are only few rigorous quantitative results connecting
explicitly inviscid mixing and enhanced dissipation properties (see [13] and
the references therein) and they appear not to be optimal. For instance for
$s\in(0,1]$, an application of Corollary 2.3 from [13] would only predict a
decay

$\|f_{t}\|_{L^{2}}\leqslant\exp(-C\nu^{q_{s}}t)\|f_{0}\|_{L^{2}},\quad
q_{s}:=\frac{\alpha(1+s)}{\alpha+s+\alpha s};$

in particular $q_{1}=\frac{2\alpha}{2\alpha+1}$ while
$q_{1/2}=\frac{3\alpha}{3\alpha+1}$.

Relation with existing literature. Understanding the interaction between
mixing and diffusion is one of the most fundamental problems in fluid
mechanics, dating back to the works of Kelvin [32] and Reynolds [39].

In the pioneering work [11], such relation has been formalized mathematically
by introducing the concept of relaxation enhancing flows; the result has been
recently revisited in a more quantitative fashion in the works [13, 18]. The
use of weak norms $H^{-s}$ in order to quantify mixing of passive scalars was
first introduced in [34].

Shear flows and circular flows in particular have been recently studied by
several authors, employing a variety of technique, including stationary phase
methods and hypocoercivity schemes [2, 12, 14], spectral methods [43] and
stochastic analysis [15]. Roughly speaking, the main known results for (1.1)
are the following:

  * •

If $u\in C^{n+1}$ has a finite number of critical points with maximal order
$n$, then enhanced dissipation holds with
$r(\nu)\sim\nu^{\frac{n}{n+2}}(1+\log\nu^{-1})^{-1}$, see Theorem 1.1 in [2].

  * •

There exist $u\in C^{\alpha}$, $\alpha\in(0,1)$, for which enhanced
dissipation holds with $r(\nu)\sim\nu^{\frac{\alpha}{\alpha+2}}$, see Theorem
5.1 from [43].

  * •

The above results are sharp, up to logarithmic corrections, in the sense that
for $u\in C^{n+1}$ (resp. $u\in C^{\alpha}$) the best possible rate is
$r(\nu)\sim\nu^{\frac{n}{n+2}}$ (resp.
$r(\nu)\sim\nu^{\frac{\alpha}{\alpha+2}}$), see Theorem 4 in [15]; the proof
is based on the Lagrangian Fluctuation Dissipation relation introduced in
[16], [17].

Let us also mention the remarkable stable mixing estimate obtained in [12] for
$u$ satisfying Assumption (H) therein. Motivated by the above results, the
authors of [10] explore the mixing and enhanced dissipation properties of
rough shear flows, namely $u$ sharply $\alpha$–Hölder for $\alpha\in(0,1)$. In
particular, they construct a Weierstrass-type flow $u$ such that the following
hold (see Theorem 1.1 in [10]):

  1. 1.

enhanced dissipation holds with rate
$r(\nu)\sim\nu^{\frac{\alpha}{\alpha+2}}$, confirming the results from [43];

  2. 2.

along suitable sequences $t_{n}\rightarrow\infty$, inviscid mixing holds on
$H^{1}$ with rate $r(t)\sim t^{1/\alpha}$:

$\|e^{-t_{n}u\partial_{x}}f_{0}\|_{H^{-1}}\lesssim
t_{n}^{-\frac{1}{\alpha}}\|f_{0}\|_{H^{1}}.$

  3. 3.

however, to the authors’ surprise, there exist other sequences
$\tilde{t}_{n}\rightarrow\infty$ on which inviscid mixing only holds with rate
$r(t)\sim t$, in the sense that

$\|e^{-\tilde{t}_{n}u\partial_{x}}f_{0}\|_{H^{-1}}\gtrsim\tilde{t}_{n}^{-1}\|f_{0}\|_{H^{1}}.$

In particular, the inviscid mixing rate $r(t)\sim t$ is the same attained by
suitable Lipschitz functions; the authors wonder whether such a discrepancy
between Points 2. and 3. is to be expected for generic flows $u\in
C^{\alpha}$, see the paragraph “Perspectives”, p.3 in [10].

The main aim of the present work is to give a negative answer to the above
question, while letting a more natural picture emerge in the context of
generic rough shear flows. Theorem 1.1 shows that for generic $u\in
B^{\alpha}_{1,\infty}$ (similarly for $u\in C^{\alpha}$, see Remark 1.6)
inviscid mixing holds on $H^{1/2}$ with rate $r(t)\sim t^{1/2\alpha}$,
uniformly over all $t\geqslant 0$. Such a decay is also the best possible, see
Theorem 1.4 below. On the other hand, Theorem 1.1 confirms the enhanced
dissipation rate $r(\nu)\sim\nu^{\alpha/(\alpha+2)}$, already identified in
[43, 10], as a property of generic shear flows.

We believe that the use of less standard spaces $B^{\alpha}_{1,\infty}$ and
mixing norms $H^{-s}$ with $s\neq 1$ to be some of the main contributions of
this work, compared to previous literature; they arise naturally in
computations, rather than being a mathematical artifact. A complete picture is
however still missing; for instance, the question whether generic $u\in
B^{\alpha}_{1,\infty}$ satisfy inviscid mixing on $H^{1}$ with rate $r(t)\sim
t^{1/\alpha}$ is still open and goes beyond our current methods.

Structure of the proof. As done frequently in the literature, in order to
prove Theorem 1.1 for the PDE (1.1), we will pass to study its hypoelliptic
counterpart

$\partial_{t}f+u\partial_{x}f=\nu\partial_{y}^{2}f$ (1.6)

again under the assumption $\int_{\mathbb{T}}f_{0}(x,y)\mathrm{d}x=0$ for all
$y\in\mathbb{T}$.

For $k\in\mathbb{Z}_{0}:=\mathbb{Z}\setminus\\{0\\}$, define the Fourier
transform in the $x$-variable as

$(P_{k}f)(y):=\int_{\mathbb{T}}f(x,y)e^{-ikx}\mathrm{d}x$

so that any $f:\mathbb{T}^{2}\rightarrow\mathbb{R}$ has a decomposition
$f(x,y)=\sum_{k}(P_{k}f)(y)e^{ikx}$. If $f$ solves (1.6), then for each
$k\in\mathbb{Z}_{0}$ the function $f^{k}_{t}:=P_{k}f_{t}$ solves the one
dimensional complex valued PDE (harmonic oscillator)

$\partial_{t}f^{k}+ikuf^{k}=\nu\partial_{y}^{2}f^{k}.$ (1.7)

For $k\in\mathbb{Z}_{0}$, $\nu\geqslant 0$ and $u\in L^{1}(\mathbb{T})$, the
PDE (1.7) has an associated semigroup on $L^{2}(\mathbb{T};\mathbb{C})$, which
we denote by $e^{t(-iku+\nu\partial_{y}^{2})}$; observe that the parameter
$k$, up to its sign, may be removed by the rescaling $\tilde{t}=t|k|$,
$\tilde{\nu}=\nu/|k|$. In this way the study of asymptotic behavior of $f^{k}$
may be reduced to that of $f^{\pm 1}$, which motivates the following
definitions.

Note that whenever we refer to a rate $r:\mathbb{R}_{\geqslant
0}\rightarrow\mathbb{R}_{\geqslant 0}$, we always assume it to be a
continuous, increasing function.

######  Definition 1.2.

A velocity field $u\in L^{1}(\mathbb{T})$ is said to be mixing on the scale
$H^{s}(\mathbb{T};\mathbb{C})$, $s\geqslant 0$, with rate $r_{s\text{-mix}}$,
if there exist a constant $C>0$ such that

$\|e^{-itku}\|_{H^{s}\rightarrow
H^{-s}}\leqslant\frac{C}{r_{s\text{-mix}}(t|k|)}\quad\forall\>k\in\mathbb{Z}_{0},\;t\geqslant
1.$ (1.8)

######  Definition 1.3.

A velocity field $u\in L^{1}(\mathbb{T})$ is said to be diffusion enhancing on
$L^{2}(\mathbb{T};\mathbb{C})$ with rate $r_{\text{dif}}$ if there exists a
constant $C>0$ such that

$\|e^{t(-iku+\nu\partial_{y}^{2})}\|_{L^{2}\rightarrow L^{2}}\leqslant
C\exp\left(-r_{\text{dif}}\left(\frac{\nu}{|k|}\right)|k|t\right)\quad\forall\,k\in\mathbb{Z}_{0},\;\nu\in(0,1],\;t\geqslant
1.$ (1.9)

The following theorems, which are the main results of the paper, provide sharp
inviscid mixing and enhanced diffusion statements for generic shear flows. In
particular, they describe precisely the behavior of solutions to (1.1) at each
Fourier level $P_{k}$.

######  Theorem 1.4 (Inviscid case $\nu=0$).

Let $\alpha\in(0,1)$.

  1. a)

Lower bound. Suppose that $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ is mixing
on the scale $H^{1/2}(\mathbb{T};\mathbb{C})$ with rate $r_{1/2\text{-mix}}$,
in the sense of Definition 1.2; then necessarily
$r_{1/2\text{-mix}}(t)\lesssim t^{\frac{1}{2\alpha}}$.

  2. b)

Upper bound. Almost every $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ satisfies
the following property: for any $\tilde{\alpha}>\alpha$, $u$ is mixing on the
scale $H^{1/2}(\mathbb{T};\mathbb{C})$ with rate $r_{1/2\text{-mix}}(t)\gtrsim
t^{\frac{1}{2\tilde{\alpha}}}$.

######  Theorem 1.5 (Dissipative case $\nu>0$).

Let $\alpha\in(0,1)$.

  1. a)

Lower bound. Suppose that $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ is
diffusion enhancing with rate $r_{\text{dif}}$, in the sense of Definition
1.3; then necessarily
$r_{\text{dif}}(\nu)\lesssim\nu^{\frac{\alpha}{\alpha+2}}$.

  2. b)

Upper bound. Almost every $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ satisfies
the following property: for any $\tilde{\alpha}>\alpha$, $u$ is diffusion
enhancing with rate
$r_{\text{dif}}(\nu)\gtrsim\nu^{\tilde{\alpha}/(\tilde{\alpha}+2)}$.

Theorems 1.4 and 1.5 will be proven respectively in Sections 3 and 4, which
are structured in a very similar way. Roughly speaking, the strategy we adopt
in proving upper and lower bounds may be summarized in three main steps:

  1. 1.

In both cases, the lower bound follows from estimates which explicitly employ
the regularity assumption $u\in B^{\alpha}_{1,\infty}$; in the case $\nu>0$,
we need to preliminary establish a Lagrangian Fluctuation-Dissipation relation
for the PDE (1.7) (see Proposition 4.2) similarly in spirit to what was done
in [15].

  2. 2.

The upper bound is satisfied by any $u$ enjoying a suitable analytic property,
which encodes its irregularity. It turns out that the right properties are
given respectively by $\rho$-irregularity (see Definition 3.4) for $\nu=0$ and
by Wei’s irregularity condition (see Definition 4.4) for $\nu>0$. A shear flow
$u$ satisfying any of such properties necessarily enjoys only limited
regularity in the scales $B^{\alpha}_{1,\infty}$ (see Proposition 3.8 and
Lemma 4.8), confirming that these are the correct spaces to work with.

  3. 3.

Finally, we show that a.e. $u\in B^{\alpha}_{1,\infty}$ is $\rho$-irregular
(resp. satisfies Wei’s condition), see Section 3.3 (resp. Section 4.4). This
is achieved by probabilistic methods, using the law of fractional Brownian
motions (see Section 2.2 for details) to construct a measure witnessing the
prevalence of such properties.

######  Remark 1.6.

Let us stress that points a) of Theorems 1.4-1.5 hold for all $u\in
B^{\alpha}_{1,\infty}$, not only generic elements. Since $\mathbb{T}$ is
finite, we have the embeddings $B^{\alpha}_{p,q}\hookrightarrow
B^{\alpha}_{1,\infty}$ for any $p,q\in[1,\infty]$, thus the lower bound is
true for all $u\in B^{\alpha}_{p,q}$ as well. On the other hand, the proofs of
points b) of Theorems 1.4-1.5 can be easily readapted to provide the same
statements for almost every $u\in B^{\alpha}_{p,q}$, for any choice of
$p,q\in[1,\infty]$.

In particular, one could always work with the spaces
$C^{\alpha}=B^{\alpha}_{\infty,\infty}$ if desired. There are however several
reasons for working with $B^{\alpha}_{1,\infty}$ or more generally
$B^{\alpha}_{p,q}$ instead of $C^{\alpha}$.

Mathematically, such spaces include genuinely discontinuous functions, as well
as (possibly continuous) functions of finite $p$-variation for any
$p\in[1,\infty]$: it holds

$B^{1/p}_{p,1}\hookrightarrow V^{p}_{c}\hookrightarrow V^{p}\hookrightarrow
B^{1/p}_{p,\infty},$

see Proposition 4.3 from [35], Proposition 2.3 from [23] for more details.

Physically, a simple way to explain singularities in fully developed
turbulence is by means of structure functions (see e.g. [21]), which are
closely related to the finite difference characterization of Besov spaces
$B^{\alpha}_{p,\infty}$. Turbulence is also believed to be closely connected
to multifractality (again we refer to the appendix of [21]), a feature which
is absent from generic $u\in C^{\alpha}$ (which are monofractal) but instead
manifested by almost every $u\in B^{\alpha}_{p,q}$, see [31, 20, 19].

Our results show that the only relevant parameter in understanding mixing and
enhanced dissipation rates for a.e. $u\in B^{\alpha}_{p,q}$ is
$\alpha\in(0,1)$, regardless of the values of $p,q$; thus there is no apparent
connection between mixing and multifractal features of $u$, at least in the
setting of shear flows.

Structure of the paper. In Section 2 we shortly recall some of the main tools
we will be working with, specifically the theory of prevalence and a relevant
class of Gaussian processes, which includes fractional Brownian motion.

Sections 3 and 4 contain the proofs of Theorems 1.4 and 1.5 and are designed
in a similar manner: in both cases we will first prove the lower bound, then
introduce the concept of $\rho$-irregularity (resp. Wei’s condition) and
explain its connection to the upper bound, as well as to the irregularity of
$u$; finally, we show by probabilistic means that a.e. $u\in
B^{\alpha}_{1,\infty}$ satisfies such property. The end of Section 4 also
contains the proof of Theorem 1.1.

In Appendix A we collect some well known results on Besov spaces, while
Appendix B contains a technical extension of the results from [43] needed to
work in our setting.

Acknowledgments. The authors were supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the Hausdorff
Center for Mathematics under Germany’s Excellence Strategy – EXC-2047/1 –
390685813 and through CRC 1060 - projekt number 211504053.

Notations and conventions. We will use the notation $a\lesssim b$ to mean that
there exists a constant $c>0$ such that $a\leqslant cb$; $a\lesssim_{x}b$
highlights the dependence $c=c(x)$. The notation $a\sim b$ stands for
$a\lesssim b$ and $b\lesssim a$, similarly for $a\sim_{x}b$.

Whenever needed, we will identify the $d$-dimensional torus $\mathbb{T}^{d}$
with either $[0,2\pi]^{d}$ or $[-\pi,\pi]^{d}$ with periodic boundary
condition, and functions $\varphi:\mathbb{T}^{d}\rightarrow\mathbb{R}$ with
$2\pi$-periodic functions defined on $\mathbb{R}^{d}$. We will use
$d_{\mathbb{T}^{d}}(x,y)$ to denote the canonical distance on the flat torus
$\mathbb{T}^{d}$, namely
$d_{\mathbb{T}^{d}}(x,y)=\inf_{k\in\mathbb{Z}^{d}}|x+2\pi k-y|$, where
$|\cdot|$ denotes the Euclidean distance on $\mathbb{R}^{d}$. With a slight
abuse, we will keep writing $|x|$ for $x\in\mathbb{T}^{d}$ to denote
$d_{\mathbb{T}^{d}}(x,0)$.

$L^{p}(\mathbb{T}^{d})$ denotes classical Lebesgue spaces,
$C^{\alpha}(\mathbb{T}^{d})$ Hölder spaces and
$H^{s}(\mathbb{T}^{d})=W^{s,2}(\mathbb{T}^{d})$ fractional Sobolev spaces.
$B^{\alpha}_{p,q}(\mathbb{T}^{d})$ denotes Besov spaces on $\mathbb{T}^{d}$;
we refer to Appendix A for a detailed discussion of their definition and main
properties. Here let us shortly recall, that for $\alpha\in(0,1)$ and
$p\in[1,\infty)$, $f\in B^{\alpha}_{p,q}(\mathbb{T}^{d})$ if and only if $f\in
L^{p}(\mathbb{T}^{d})$ and it has finite Gagliardo-Niremberd type seminorm

$\llbracket f\rrbracket_{B^{\alpha}_{p,\infty}(\mathbb{T}^{d})}:=\sup_{x\neq
y\in\mathbb{T}^{d}}\frac{\left\|f\left(\cdot\,+x\right)-f\left(\cdot\,+y\right)\right\|_{L^{p}}}{d_{\mathbb{T}^{d}}(x,y)^{s}};$
(1.10)

see equations (A.1)-(A.2) for more details. Similarly,
$B^{\alpha}_{p,q}(0,\pi)$ denotes Besov spaces on $[0,\pi]$.

Given $p\in[1,\infty)$ and a compact interval $I\subset\mathbb{R}$, we denote
by $V^{p}=V^{p}(I)$ the Banach space of functions $f:I\rightarrow\mathbb{R}$
of finite $p$-variation, with norm

$\|f\|_{V^{p}}=|f(0)|+\sup_{\pi\in\Pi(I)}\left(\sum_{[t_{i},t_{i+1}]\in\pi}|f(t_{i+1})-f(t_{i})|^{p}\right)^{\frac{1}{p}}$

where the supremum is taken over the set $\Pi(I)$ of all finite partition of
$I$, identified with sequences $\\{t_{i}\\}_{i=0}^{n}$ such that $\min
I=t_{0}<t_{1}<\cdots<t_{n}=\max I$. $V^{p}_{c}$ stands for the closed subspace
of $V^{p}$ of continuous functions. $V^{p}(\mathbb{T})$ is defined by
identifying $\mathbb{T}$ with the interval $[-\pi,\pi]$.

Whenever a stochastic process $X=(X_{t})_{t\geqslant 0}$ is considered, if not
specified we tacitly assume the existence of an abstract underlying filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geqslant
0},\mathbb{P})$, such that the $\sigma$-algebra $\mathcal{F}$ and the
filtration $(\mathcal{F}_{t})_{t\geqslant 0}$ satisfy the usual assumptions
and $(X_{t})_{t\geqslant 0}$ is adapted to $(\mathcal{F}_{t})_{t\geqslant 0}$.
Whenever we say that $(\mathcal{F}_{t})_{t\geqslant 0}$ is the natural
filtration generated by $X$, then it is tacitly implied that it is actually
its right continuous, normal augmentation wrt. $\mathbb{P}$. We denote by
$\mathbb{E}$ integration (equiv. expectation) wrt. the probability
$\mathbb{P}$.

##  2\. Preliminaries

###  2.1. Prevalence

The theory of prevalence has been developed by Hunt, Sauer and Yorke in [29]
in order to provide a measure theoretic notion of genericity in infinite
dimensional spaces. It is a natural generalization of the concept of “full
Lebesgue measure sets” from the finite dimensional setting. We follow here the
exposition given in [29], although for our purposes it will be enough to work
with Banach spaces $E$.

######  Definition 2.1.

Let $E$ be a complete metric vector space. A Borel set $A\subset E$ is said to
be shy if there exists a measure $\mu$ such that:

  1. i.

There exists a compact set $K\subset E$ such that $0<\mu(K)<\infty$.

  2. ii.

For every $v\in E$, $\mu(v+A)=0$.

In this case, the measure $\mu$ is said to be transverse to $A$. More
generally, a subset of $E$ is shy if it is contained in a shy Borel set. The
complement of a shy set is called a prevalent set.

Sometimes it is said informally that the measure $\mu$ “witnesses” the
prevalence of $A^{c}$.

It follows immediately from Point i. of Definition 2.1 that, if such a measure
$\mu$ exists, then it can be assumed to be a compactly supported probability
measure on $E$. On the other hand, in order to exhibit the existence of $\mu$
satisfying Points. i.-ii., it suffices to find another tight probability
measure $\tilde{\mu}$ only satisfying requirement ii. If $E$ is separable,
then any probability measure on $E$ is tight and therefore Point i. is
automatically satisfied.

The following properties hold for prevalence (all proofs can be found in
[29]):

  1. 1.

If $E$ is finite dimensional, then a set $A$ is shy if and only if it has zero
Lebesgue measure.

  2. 2.

If $A$ is shy, then so is $v+A$ for any $v\in E$.

  3. 3.

Prevalent sets are dense.

  4. 4.

If $\dim(E)=+\infty$, then compact subsets of $E$ are shy.

  5. 5.

Countable union of shy sets is shy; conversely, countable intersection of
prevalent sets is prevalent.

From now, whenever we say that a statement holds for a.e. $v\in E$, we mean
that the set of elements of $E$ for which the statement holds is a prevalent
set. Property 1. states that this convention is consistent with the finite
dimensional case.

In the context of a function space $E$, it is natural to consider as
probability measure the law induced by an $E$-valued random variable. Namely,
given stochastic process $W$ defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ taking values in a separable Banach space
$E$, in order to show that a property $\mathcal{P}$ holds for a.e. $f\in E$,
it suffices to show that

$\mathbb{P}\left(\text{$f+W$ satisfies property
$\mathcal{P}$}\right)=1,\qquad\forall\,f\in E.$ (2.1)

Clearly, we are assuming that the set $A=\left\\{w\in E:\text{$w$ satisfies
property $\mathcal{P}$}\right\\}$ is Borel measurable; if $E$ is not
separable, we need to additionally require that the law of $W$ is tight, so as
to satisfy Point i. of Definition 2.1.

As a consequence of properties 4. and 5., the set of all possible realizations
of a probability measure $\mu$ on a separable infinite dimensional Banach
space is a shy set, as it is contained in a countable union of compact sets
(this is true more in general for any tight measure on a Banach space). This
fact highlights the difference between a statement of the form “Property
$\mathcal{P}$ holds for a.e. $f$ (in the sense of prevalence)” and “Property
$\mathcal{P}$ holds for $\mu$-a.e. $f$”; indeed, the second statement doesn’t
provide any information regarding whether the property might be prevalent or
not. Intuitively, the elements satisfying a prevalence statement are “many
more” than just the realizations of a given measure $\mu$.

###  2.2. A useful class of Gaussian transverse measures

From now on, given an interval $[0,T]$ and a probability measure $\mu$ on
$C([0,T])$, we will denote by $(X_{t})_{t\in[0,T]}$ the associated canonical
process, which is given by $X_{t}(\omega)=\omega(t)$ for $\omega\in C([0,T])$,
and by $\mathcal{F}_{t}=\sigma(\\{X_{s},s\leqslant t\\})$ the associated
natural filtration.

A key point of the present work is to verify that suitable properties
$\mathcal{P}$ are satisfied by a.e. $f\in E$ for suitable
$E=B^{\alpha}_{1,\infty}$. The discussion from Section 2.1, in particular
equation (2.1), suggests to look for classes of processes which are stable
under deterministic additive perturbations and in [24] we identified the local
nondeterministic (LND) Gaussian processes as a useful class in the study of
prevalence in function spaces. We recall in the next definition that a real
valued process $X$ is Gaussian if for any $n\in\mathbb{N}$ and
$t_{1},\ldots,t_{n}\in[0,T]$, $(X_{t_{1}},\ldots,X_{t_{n}})$ is a
$\mathbb{R}^{n}$-valued Gaussian variable.

######  Definition 2.2.

Given $\beta>0$, a real valued Gaussian process $X$ is strongly locally
nondeterministic with parameter $\beta$, $\beta$-SLND for short, if there
exists a constant $C_{X}$ such that

$\operatorname{Var}(X_{t}|\mathcal{F}_{s})\geqslant C_{X}|t-s|^{2\beta}$ (2.2)

uniformly over $s,t\in[0,T]$ with $s<t$.

In (2.2) above, $\operatorname{Var}\left(\cdot\,|\mathcal{F}_{s}\right)$
denotes the conditional variance; equivalently, Definition 2.2 amount to the
condition that, for any $s<t$, there is a decomposition
$X_{t}=X^{(1)}_{s,t}+X^{(2)}_{s,t}$ where $X^{(1)}_{s,t}$ is Gaussian and
adapted to $\mathcal{F}_{s}$ while $X^{(2)}_{s,t}$ is Gaussian, independent of
$\mathcal{F}_{s}$, with variance $\operatorname{Var}(X^{(2)}_{s,t})\geqslant
C_{X}|t-s|^{2\beta}$. The increments of the process $X$ are therefore
“intrinsically chaotic” in a way that can be quantified precisely by the
parameter $\beta$. Let us shortly mention that Definition 2.2 is not the only
notion of LND in the literature and there are several non-equivalent ones; see
[44] for a review.

The importance of the $\beta$-SLND property comes from the following
elementary fact, which can be readily checked from the definition (see also
Remark 26 from [24]); in the statement, $f:[0,T]\rightarrow\mathbb{R}$ can be
naturally unbounded.

######  Lemma 2.3.

Let $\\{X_{t}\\}_{t\in[0,T]}$ be a $\beta$-SLND Gaussian process and
$f:[0,T]\rightarrow\mathbb{R}$ be a measurable function; then $X+f$ is also a
$\beta$-SLND Gaussian process.

Lemma 2.3 will be our main leverage to establish prevalence statements, as it
reduces the difficulty to that of verifying that any $\beta$-SLND Gaussian
process satisfies $\mu$-a.s. the property $\mathcal{P}$ of interest; this will
indeed be the strategy implemented in Sections 3.3 and 4.4 respectively.

In this sense, we could work with any possible Gaussian law $\mu$ whose
associated canonical process is $\beta$-SLND, without further specification.
To keep things less abstract, we will however use a well-known one-parameter
family from this class, which are the laws
$\left\\{\mu^{H},\,H\in(0,1)\right\\}$ of fractional Brownian motion (fBm) of
parameter $H\in(0,1)$. The material recalled next is mostly classical and can
be found in the monograph [37].

The law of fBm of Hurst parameter $H\in(0,1)$ is defined as the unique
Gaussian measure $\mu^{H}$ on $\Omega=C([0,T])$ such that

$\int_{\Omega}X_{t}(\omega)\mu^{H}(\mathrm{d}\omega)=0,\quad\int_{\Omega}X_{t}(\omega)X_{s}(\omega)\mu^{H}(\mathrm{d}\omega)=\frac{1}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}).$

For $H=1/2$, the law of fBm corresponds to the classical Wiener measure;
instead for $H\neq 1/2$, the associated canonical process $X$ is not a
semimartingale nor a Markov process.

The support of $\mu^{H}$ in terms of Besov spaces is well understood, with
sharp results going back to [9] (see also [42] for a modern proof which
extends to the vector valued case): it holds

$\mu^{H}(C^{H-\varepsilon})=1\quad\forall\,\varepsilon>0,\quad\mu^{H}(B^{H}_{p,\infty})=1\quad\forall\,p\in[1,\infty),$

while

$\mu^{H}(C^{H})=0,\quad\mu^{H}(B^{H}_{p,q})=0\quad\forall\,p,q\in[1,\infty).$

In particular fBm trajectories are sharply not $H$-Hölder continuous, but by
Ascoli–Arzelà $\mu^{H}$ is a tight probability measure on
$B^{H-\varepsilon}_{p,\infty}$ for any $\varepsilon>0$ and any
$p\in[1,\infty]$. As promised, this class of Gaussian measures does satisfy
the LND property.

######  Lemma 2.4.

Let $X$ be the canonical process associated to $\mu^{H}$, $H\in(0,1)$. Then
$X$ is $H$-SLND; moreover, the Gaussian process
$Y_{t}:=\int_{0}^{t}X_{s}\mathrm{d}s$ is $(1+H)$-SLND.

###### Proof.

The first claim is classical and can be found in the review [44] and the
references therein; alternative, a self-contained proof, based on the
Mandelbrot–Van Ness representation of fBm, is given in Section 2.4 from [24];
the same representation can be used to establish the second half of the claim
involving the process $Y$, see Example iv. from Section 4.2 in [24]. ∎

Among the reasons for using $\mu^{H}$, instead of just any Gaussian measure
satisfying a suitable LND condition, let us finally mention that this process
can be simulated numerically in a very efficient way.

##  3\. Inviscid mixing

This section contains the proof of Theorem 1.4, which we split in several
steps.

Recall the setting: in order to study the transport equation
$\partial_{t}f+u\partial_{x}f=0$, we pass to Fourier modes
$f^{k}_{t}(y)=(P_{k}f_{t})(y)$, solving $\partial_{t}f^{k}+ikuf^{k}=0$; namely
$f^{k}_{t}(y)=e^{-iktu(y)}f^{k}_{0}(y)$.

It is then natural to take a slightly more general perspective and study maps
of the form $y\mapsto e^{i\xi u(y)}g(y)$ with $\xi\in\mathbb{R}$, $g\in
H^{s}(\mathbb{T})$.

###  3.1. Lower bounds in terms of regularity

We show here that the regularity of $u$, measured in the Besov–Nikolskii scale
$B^{\alpha}_{1,\infty}$, necessarily implies a lower bound on the decay of
solutions in the $H^{-1/2}$-norm. The proof is partly inspired by that of
Proposition 3.2 from [10].

######  Lemma 3.1.

Let $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ for some $\alpha\in(0,1)$. Then
for any $g\in H^{1}(\mathbb{T})$ there exists a constant $C=C(\alpha,g)$ such
that

$\|e^{i\xi u}g\|_{H^{-1/2}}\geqslant
C(1+\|u\|_{B^{\alpha}_{1,\infty}})^{-\frac{1}{2\alpha}}|\xi|^{-\frac{1}{2\alpha}}\quad\forall\,|\xi|\geqslant
1.$ (3.1)

###### Proof.

Fix $\xi$ with $|\xi|\geqslant 1$ and set $\bar{g}:=e^{i\xi u}g$; we claim
that $\bar{g}\in B^{\alpha/2}_{2,\infty}$. By Sobolev and Besov embeddings,
$g\in L^{\infty}\cap B^{\alpha/2}_{2,\infty}$; $e^{i\xi u}\in L^{\infty}$, so
it’s enough to show that $e^{i\xi u}\in B^{\alpha/2}_{2,\infty}$. By the basic
estimate $|e^{ia}-e^{ib}|\leqslant\sqrt{2}|a-b|^{1/2}$, it holds

$\displaystyle\left\|e^{i\xi u\left(\cdot\,+y\right)}-e^{i\xi
u(\cdot+\tilde{y})}\right\|_{L^{2}}$
$\displaystyle\lesssim|\xi|^{1/2}\left\|u\left(\cdot\,+y\right)-u\left(\cdot\,+\tilde{y}\right)\right\|_{L^{1}}^{1/2}$
$\displaystyle\lesssim|\xi|^{1/2}\|u\|^{1/2}_{B^{\alpha}_{1,\infty}}d_{\mathbb{T}}(y,\tilde{y})^{\alpha/2}.$

By the equivalent characterization of Besov–Nikolskii spaces, this implies

$\|e^{i\xi u}\|_{B^{\alpha/2}_{2,\infty}}\lesssim
1+|\xi|^{1/2}\|u\|^{1/2}_{B^{\alpha}_{1,\infty}}\lesssim(1+\|u\|_{B^{\alpha}_{1,\infty}})^{1/2}|\xi|^{1/2}$

and so by Proposition A.4 in Appendix A we conclude that $\bar{g}\in
B^{\alpha/2}_{2,\infty}$ with

$\|\bar{g}\|_{B^{\alpha/2}_{2,\infty}}\lesssim\|g\|_{H^{1}}(1+\|u\|_{B^{\alpha}_{1,\infty}})^{1/2}|\xi|^{1/2}.$
(3.2)

Clearly $\|\bar{g}\|_{L^{2}}=\|g\|_{L^{2}}$. Using the interpolation
inequality from Corollary A.6 in Appendix A (for the choice $s_{1}=1/2$,
$s_{2}=\alpha/2$) we obtain

$\|g\|_{L^{2}}=\|\bar{g}\|_{L^{2}}\lesssim\|\bar{g}\|_{H^{-1/2}}^{\frac{\alpha}{1+\alpha}}\|\bar{g}\|_{B^{\alpha/2}_{2,\infty}}^{\frac{1}{1+\alpha}}.$
(3.3)

Rearranging now the terms in (3.3) and applying the estimate (3.2) we find

$\|\bar{g}\|_{H^{-1/2}}\gtrsim\|\bar{g}\|_{B^{\alpha/2}_{2,\infty}}^{-\frac{1}{\alpha}}\|g\|_{L^{2}}^{1+\frac{1}{\alpha}}\gtrsim\|g\|_{L^{2}}^{1+\frac{1}{\alpha}}\|g\|_{H^{1}}^{-\frac{1}{\alpha}}(1+\|u\|_{B^{\alpha}_{1,\infty}})^{-\frac{1}{2\alpha}}|\xi|^{-\frac{1}{2\alpha}}$
(3.4)

where the hidden constant in (3.4) only depends on $\alpha$. Using the
definition of $\bar{g}$ and relabelling the constant to include the
$g$-dependent terms yields the conclusion. ∎

######  Corollary 3.2.

Let $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ be mixing on
$H^{1/2}(\mathbb{T})$ with rate $r_{1/2\text{-mix}}$, in the sense of
Definition 1.2. Then there exists a constant $C=C(\alpha,u)$ such that

$r_{1/2\text{-mix}}(t)\leqslant Ct^{\frac{1}{2\alpha}}.$

###### Proof.

Consider $g(y)=e^{iy}$, so that $\|g\|_{H^{1/2}}\sim\|g\|_{H^{1}}\sim 1$; then
by Definition 1.2 applied for the choice $k=1$ and Lemma 3.1 for $\xi=-t$, it
holds

$\displaystyle\frac{1}{r(t)}\gtrsim$
$\displaystyle\|e^{-itu}\|_{H^{1/2}\rightarrow
H^{-1/2}}\geqslant\|e^{-itu}g\|_{H^{-1/2}}\gtrsim_{\alpha}(1+\|u\|_{B^{\alpha}_{1,\infty}})^{-\frac{1}{2\alpha}}t^{-\frac{1}{2\alpha}};$

up to relabelling constants, this yields the conclusion. ∎

######  Remark 3.3.

In fact, the statement of Lemma 3.1 can be generalized as follows. For
$\alpha\in(0,1)$, $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$, $g\in
H^{1}(\mathbb{T})$ and any $s>0$ there exists a constant $C(\alpha,g,s)$ such
that

$\|e^{i\xi u}g\|_{H^{-s}}\geqslant
C(1+\|u\|_{B^{\alpha}_{1,\infty}})^{-\frac{s}{\alpha}}|\xi|^{-\frac{s}{\alpha}}\quad\forall\,|\xi|\geqslant
1.$

Then arguing as in Corollary 3.2 by choosing $g(y)=e^{iy}$, one can conclude
that the best possible rate for inviscid mixing on the scale
$H^{s}(\mathbb{T})$ is $r_{s\text{-mix}}(t)\sim t^{s/\alpha}$. Taking $s=1$
provides the rate $t^{1/\alpha}$, which is in line with Proposition 3.2 from
[10].

###  3.2. Upper bounds in terms of $\rho$-irregularity

The concept of $\rho$-irregularity was first introduced in [6] in the study of
regularization by noise phenomena. Its applications to PDEs have been
subsequently explored in [7, 8, 24, 5].

######  Definition 3.4.

Let $\gamma\in[0,1)$, $\rho>0$; a measurable map
$u:[0,\pi]\rightarrow\mathbb{R}$ is said to be $(\gamma,\rho)$-irregular if
there exists a constant $C>0$ such that

$\left|\int_{I}e^{i\xi u(z)}\mathrm{d}z\right|\leqslant
C|I|^{\gamma}|\xi|^{-\rho}\quad\forall\,\xi\in\mathbb{R},\,I\subset[0,\pi]$
(3.5)

where $I$ stands for a subinterval of $[0,\pi]$ and $|I|$ denotes its length.
A similar definition holds for $u:\mathbb{R}\rightarrow\mathbb{R}$; a map
$u:\mathbb{T}\rightarrow\mathbb{R}$ is said to be $(\gamma,\rho)$-irregular if
its $2\pi$-periodic extension $u:\mathbb{R}\rightarrow\mathbb{R}$ has this
property. We say that $u$ is $\rho$-irregular for short if there exists
$\gamma>1/2$ such that it is $(\gamma,\rho)$-irregular.

In all of the cases covered by Definition 3.4, following the original
definition from [6], we denote the optimal constant $C$ in (3.5) by
$\|\Phi^{u}\|_{\gamma,\rho}$. This is due to the notation
$\Phi^{u}_{t}(\xi):=\int_{0}^{t}e^{i\xi u(z)}\mathrm{d}z$ and the fact that,
for $u:[0,\pi]\rightarrow\mathbb{R}$, by (3.5) it holds

$\|\Phi^{u}\|_{\gamma,\rho}=\sup_{\xi\in\mathbb{R},0\leqslant
s<t\leqslant\pi}\frac{|\Phi^{u}_{t}(\xi)-\Phi^{u}_{s}(\xi)|}{|t-s|^{\gamma}|\xi|^{-\rho}}.$

The property of $\rho$-irregularity may be rephrased in the following form,
more suited for our purposes.

######  Lemma 3.5.

Let $u:\mathbb{T}\rightarrow\mathbb{R}$ be $(\gamma,\rho)$-irregular, then

$\|e^{i\xi
u}\|_{B^{\gamma-1}_{\infty,\infty}}\lesssim\|\Phi^{u}\|_{\gamma,\rho}|\xi|^{-\rho}\quad\forall\,\xi\in\mathbb{R}.$

###### Proof.

For $\bar{y}\in[-\pi,\pi]$ and $\xi\in\mathbb{R}$, define the function

$\displaystyle v^{\xi}(\bar{y})$ $\displaystyle=\int_{-\pi}^{\bar{y}}e^{i\xi
u(y)}\mathrm{d}y-\left(\frac{\bar{y}+\pi}{2\pi}\right)\int_{-\pi}^{\pi}e^{i\xi
u(y)}\mathrm{d}y;$

by periodicity it can be identified with a function on $\mathbb{T}$. Then by
definition of $(\gamma,\rho)$-irregularity it holds
$\|v^{\xi}\|_{C^{\gamma}}\lesssim\|\Phi^{u}\|_{\gamma,\rho}|\xi|^{-\rho}$ and
so by Proposition A.2 we deduce that

$\begin{split}\|e^{i\xi
u}\|_{B^{\gamma-1}_{\infty\infty}}&=\left\|(v^{\xi})^{\prime}+\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i\xi
u(y)}\mathrm{d}y\right\|_{B^{\gamma-1}_{\infty,\infty}}\\\
&\lesssim\|v^{\xi}\|_{C^{\gamma}}+\frac{1}{2\pi}\left|\int_{-\pi}^{\pi}e^{i\xi
u(y)}\mathrm{d}y\right|\\\
&\lesssim\|\Phi^{u}\|_{\gamma,\rho}|\xi|^{-\rho}.\qed\end{split}$

The relation between $\rho$-irregularity and inviscid mixing comes from the
next result.

######  Lemma 3.6.

Let $u:\mathbb{T}\rightarrow\mathbb{R}$ be $(\gamma,\rho)$-irregular for some
$\gamma>1/2$. Then there exists a constant $C=C(\gamma)$ such that

$\|e^{i\xi u}g\|_{H^{-1/2}}\leqslant
C\|\Phi^{u}\|_{\gamma,\rho}|\xi|^{-\rho}\|g\|_{H^{1/2}}\quad\forall\,\xi\neq
0,\,g\in H^{1/2}.$ (3.6)

As a consequence, $u$ is mixing on the scale $H^{1/2}$ with rate
$r_{1/2\text{-mix}}(t)=t^{\rho}$, in the sense of Definition 1.2.

###### Proof.

The proof of the estimate (3.6) relies on several properties of Besov spaces,
for which we refer the reader to Appendix A. By assumption $\gamma+1/2>1$,
thus we can apply Proposition A.3 (for the choice $s_{1}=\gamma-1$,
$s_{2}=1/2$, $p_{1}=q=\infty$, $p_{2}=p=2$) and Lemma 3.5 to obtain

$\displaystyle\|e^{i\xi u}g\|_{B^{\gamma-1}_{2,\infty}}$
$\displaystyle\lesssim\|e^{i\xi
u}\|_{B^{\gamma-1}_{\infty,\infty}}\|g\|_{B^{1/2}_{2,\infty}}$
$\displaystyle\lesssim\|\Phi^{u}\|_{\gamma,\rho}|\xi|^{-\rho}\|g\|_{B^{1/2}_{2,2}}$
$\displaystyle=\|\Phi^{u}\|_{\gamma,\rho}|\xi|^{-\rho}\|g\|_{H^{1/2}}.$

Again by the hypothesis $\gamma-1>-1/2$ and so by Besov embeddings
$B^{\gamma-1}_{2,\infty}\hookrightarrow H^{-1/2}$, yielding the first claim.
Applying estimate (3.6) for $k\in\mathbb{Z}_{0}$, $\xi=-tk$ gives

$\|e^{-itku}\|_{H^{1/2}\rightarrow
H^{-1/2}}\leqslant\frac{C\|\Phi^{u}\|_{\gamma,\rho}}{(t|k|)^{\rho}}$

and thus the conclusion. ∎

The property of $\rho$-irregularity implies roughness of $u$, as the name
suggests. To quantify this precisely, we recall the concept of Hölder
roughness, as presented in [22].

######  Definition 3.7.

A measurable map $u:\mathbb{T}\rightarrow\mathbb{R}$ is said to be
$\alpha$-Hölder rough if there exists $L=L_{\alpha}(u)$ such that: for any
$\delta>0$ and any $\bar{y}\in\mathbb{T}$, there exists $z\in\mathbb{T}$
satisfying

$d_{\mathbb{T}}(\bar{y},z)\leqslant\delta\quad\text{ and
}\quad|u(\bar{y})-u(z)|\geqslant L_{\alpha}(u)\delta^{\alpha}.$

The optimal constant $L_{\alpha}(u)$ is called the modulus of $\alpha$-Hölder
roughness of $u$.

Definition 3.7 is equivalent to requiring

$L_{\alpha}(u)=\inf_{\bar{y}\in\mathbb{T},\delta>0}\sup_{z\in
B_{\delta}(\bar{y})}\frac{|u(z)-u(\bar{y})|}{\delta^{\alpha}}>0.$ (3.7)

A detailed study of analytic properties of $\rho$-irregular paths was carried
out in Section 5 of [24]; in particular, there exists a critical prameter
$\alpha^{\ast}$, associated to the pair $(\gamma,\rho)$, linked to the
(ir)regularity of $u$ in Hölder and Besov–Nikolskii scales.

######  Proposition 3.8.

Let $u:\mathbb{T}\rightarrow\mathbb{R}$ be $(\gamma,\rho)$-irregular and
define $\alpha^{\ast}:=(1-\gamma)/\rho$. Then:

  1. a)

$u$ is $\alpha$-Hölder rough for any $\alpha>\alpha^{\ast}$ with
$L_{\alpha}(u)=+\infty$.

  2. b)

$u$ has infinite $p$-variation on any subinterval $I\subset\mathbb{T}$ and for
any $p>1/\alpha^{\ast}$.

  3. c)

$u$ does not belong to $B^{\alpha}_{1,\infty}$ for any $\alpha>\alpha^{\ast}$.

###### Proof.

For functions $u:[0,T]\rightarrow\mathbb{R}$, points a) and b) are proved in
[24], cf. Corollary 65 and Corollary 68 therein; we recall here shortly the
idea of proof.

Going through the proof of Theorem 63 from [24], one can establish the (much
stronger) fact that, if $u$ is $(\gamma,\rho)$-irregular, then for any
$\tilde{\alpha}>\alpha^{\ast}$ it holds

$\lim_{\varepsilon\rightarrow
0^{+}}\inf_{y\in(0,T)}\varepsilon^{-1}\mathcal{L}(h\in(0,\varepsilon):|u(y+h)-u(y)|\geqslant\varepsilon^{\tilde{\alpha}})=1,$
(3.8)

where $\mathcal{L}$ denotes the Lebesgue measure on $\mathbb{R}$. In
particular, there exists $\varepsilon_{0}>0$ such that, for all
$0<\varepsilon<\varepsilon_{0}$, it must hold

$\mathcal{L}(h\in(0,\varepsilon):|u(y+h)-u(y)|\geqslant\varepsilon^{\tilde{\alpha}})\geqslant\varepsilon/2>0\quad\forall\,y\in(0,T);$

therefore for any $y\in(0,T)$ we can find infinitely many, arbitrarily small
$h$ such that $|u(y+h)-u(y)|\geqslant h^{\tilde{\alpha}}$; playing with the
arbitrariness of $\tilde{\alpha}$, one can then easily establish both
properties of Hölder roughness and infinite $p$-variation.

Up to identifying $u:\mathbb{T}\rightarrow\mathbb{R}$ with a $2\pi$-periodic
function, it’s easy to check that property (3.8) carries over to this setting
as well, as it is only related to the local behaviour or $u$ around any fixed
$y$; same goes for the proofs of points a) and b).

We now focus on establishing claim c), which is instead an original
contribution of this work. Fix $\alpha>\alpha^{\ast}$ and choose
$\tilde{\alpha}\in(\alpha^{\ast},\alpha)$; by estimate (3.8) (with the infimum
taken over $y\in\mathbb{T}$ instead of $(0,T)$), for all $\varepsilon>0$
sufficiently small, it must hold

$\displaystyle\pi$
$\displaystyle\leqslant\int_{\mathbb{T}}\varepsilon^{-1}\mathcal{L}(h\in(0,\varepsilon):|u(y+h)-u(y)|\geqslant\varepsilon^{\tilde{\alpha}})\mathrm{d}y$
$\displaystyle\leqslant\int_{\mathbb{T}}\varepsilon^{-1-\tilde{\alpha}}\int_{0}^{\varepsilon}|u(y+h)-u(y)|\mathrm{d}h\mathrm{d}y$
$\displaystyle=\varepsilon^{-1-\tilde{\alpha}}\int_{0}^{\varepsilon}\left\|u\left(\cdot\,+h\right)-u(\cdot)\right\|_{L^{1}}\mathrm{d}h$
$\displaystyle\leqslant\varepsilon^{\alpha-\tilde{\alpha}}\llbracket
u\rrbracket_{B^{\alpha}_{1,\infty}},$

where in the second passage we used Markov’s inequality. Since
$\alpha>\tilde{\alpha}$, letting $\varepsilon\rightarrow 0^{+}$ we can
conclude that $\llbracket u\rrbracket_{B^{\alpha}_{1,\infty}}=+\infty$. ∎

######  Remark 3.9.

If $u$ is $\rho$-irregular, then Proposition 3.8-c) implies that $u$ does not
belong to $B^{\alpha}_{1,\infty}$ for any $\alpha>(2\rho)^{-1}$. Conversely,
if $u\in B^{\alpha}_{1,\infty}$, then it can only be $\rho$-irregular for
parameters $\rho$ satisfying $\rho\leqslant(2\alpha)^{-1}$.

###  3.3. Prevalence statements and proof of Theorem 1.4

Given the results of Sections 3.1–3.2, it is natural to wonder whether generic
elements of $B^{\alpha}_{1,\infty}$ are “almost as irregular as possible”, in
the sense of being $\rho$-irregular for any $\rho<(2\alpha)^{-1}$; we provide
here a positive answer.

In order to do so, we will first prove the statement for elements of
$B^{\alpha}_{1,\infty}(0,\pi)$, see Theorem 3.11, and only later deduce the
same property for $B^{\alpha}_{1,\infty}(\mathbb{T})$ by a “deperiodization”
procedure (cf. Corollary 3.13 below).

Differently from Section 2.2, whenever dealing with a measure $\mu$ supported
on $C([0,\pi])$, it will be useful to denote by $u=\\{u_{y}\\}_{y\in[0,\pi]}$
the associated canonical process; we will instead employ the letter $\varphi$
to denote deterministic functions, either defined on $[0,\pi]$ or on
$\mathbb{T}$.

Before proceeding further, we need to recall the following key result
established in [24], cf. Theorem 29 therein.

######  Proposition 3.10.

Let $\mu$ be a Gaussian measure on $C([0,T])$ whose canonical process $u$ is
$\beta$-SLND for some $\beta>0$. Then for any $\rho<(2\beta)^{-1}$ it holds

$\mu^{H}\left(u\text{ is }\rho\text{-irregular}\right)=1.$

We can combine Proposition 3.10 with the invariance of the $\beta$-SLND
property from Lemma 2.3 to deduce a first prevalence statement.

######  Theorem 3.11.

Let $\alpha\in(0,1)$; then a.e. $\varphi\in B^{\alpha}_{1,\infty}(0,\pi)$ is
$\rho$-irregular for every $\rho<(2\alpha)^{-1}$.

###### Proof.

Given $\rho>0$, define the set

$\mathcal{A}_{\rho}=\left\\{\varphi\in
B^{\alpha}_{1,\infty}(0,\pi):\varphi\text{ is $\rho$-irregular}\right\\};$

it holds

$\mathcal{A}_{\rho}=\bigcup_{n,m=3}^{\infty}\mathcal{A}_{\rho,n,m},$

with

$\mathcal{A}_{\rho,n,m}:=\left\\{\varphi\in
B^{\alpha}_{1,\infty}(0,\pi):\varphi\text{ is $(\gamma,\rho)$-irr. for
}\gamma=\frac{1}{2}+\frac{1}{n},\|\Phi^{\varphi}\|_{\gamma,\rho}\leqslant
m\right\\}.$

The sets $\mathcal{A}_{\rho,n,m}$ are closed in the topology of
$B^{\alpha}_{1,\infty}(0,\pi)$ (the map
$\varphi\mapsto\|\Phi^{\varphi}\|_{\gamma,\rho}$ is lower semicontinuous in
the topology of $L^{1}(0,\pi)$), thus $\mathcal{A}_{\rho}$ is Borel
measurable. If we show that $\mathcal{A}_{\rho}$ is prevalent in
$B^{\alpha}_{1,\infty}(0,\pi)$ for any $\rho<(2\alpha)^{-1}$, then the same
holds for

$\mathcal{A}=\left\\{\varphi\in B^{\alpha}_{1,\infty}(0,\pi):\varphi\text{ is
$\rho$-irregular for every
}\rho<\frac{1}{2\alpha}\right\\}=\bigcap_{n=1}^{\infty}\mathcal{A}_{\frac{1}{2\alpha}-\frac{1}{n}}$

providing the conclusion.

Now fix $\rho<(2\alpha)^{-1}$ and choose $H\in(0,1)$ such that $H>\alpha$,
$\rho<(2H)^{-1}$; denote by $\mu^{H}$ the law of fractional Brownian motion on
$C([0,\pi])$ and by $u=\\{u_{y},y\in[0,\pi]\\}$ the associated canonical
process. Since $\mu^{H}$ is supported on $C^{H-\varepsilon}([0,\pi])$ for any
$\varepsilon>0$ and $H>\alpha$, it is also a tight probability measure on
$B^{\alpha}_{1,\infty}(0,\pi)$; thus we only need to verify Property ii. from
Definition 2.1, equivalently property (2.1) for $E=B^{\alpha}_{1,\infty}$.

Fix $\varphi\in B^{\alpha}_{1,\infty}(0,\pi)$; by Proposition 2.4, $u$ is a
$H$-SLND process and so by Lemma 2.3 the same holds for $u+\varphi$. In turn,
by our choise of the parameters and Proposition 3.10, this implies that
$\varphi+u$ is $\mu^{H}$-a.s. $\rho$-irregular; as the argument holds for any
$\varphi\in B^{\alpha}_{1,\infty}(0,\pi)$, we have shown that

$\mu^{H}(\varphi+\mathcal{A}_{\rho})=1\quad\forall\,\varphi\in
B^{\alpha}_{1,\infty}(0,\pi),$

namely that $\mu^{H}$ witnesses the prevalence of $\mathcal{A}_{\rho}$ in
$B^{\alpha}_{1,\infty}(0,\pi)$. ∎

We pass to show how to exploit Theorem 3.11 to establish similar statement for
functions defined on the torus.

We identify the torus $\mathbb{T}$ with the interval $[-\pi,\pi]$, up to
$-\pi\sim\pi$; thus any measurable function
$\varphi:\mathbb{T}\rightarrow\mathbb{R}$ can be identified with
$\varphi:[-\pi,\pi]\rightarrow\mathbb{R}$ such that
$\varphi(-\pi)=\varphi(\pi)$. Any such $\varphi$ is in a 1-1 correspondence
with a pair $(\varphi_{1},\varphi_{2})$ of measurable functions defined on
$[0,\pi]$, given by $\varphi_{1}(y):=\varphi(y)$,
$\varphi_{2}(y):=\varphi(-y)$; they satisfy the constraint
$\varphi_{1}(\pi)=\varphi_{2}(\pi)$. The $\rho$-irregularity property of the
periodic function $\varphi$ is actually equivalent to that of the aperiodic
functions $\varphi_{i}$.

######  Lemma 3.12.

A measurable function $\varphi:\mathbb{T}\rightarrow\mathbb{R}$ is
$(\gamma,\rho)$-irregular if and only if the functions
$\varphi_{1},\varphi_{2}:[0,\pi]\rightarrow\mathbb{R}$ are so.

###### Proof.

The proof is elementary. Given $I\subset[-\pi,\pi]$, setting
$I_{1}=I\cap[0,\pi]$, $I_{2}=I\cap[-\pi,0]$ it holds
$\max\\{|I_{1}|,|I_{2}|\\}\leqslant|I|\leqslant 2\max\\{|I_{1}|,|I_{2}|\\}$,
so that

$\max\\{\|\Phi^{\varphi_{1}}\|_{\gamma,\rho},\|\Phi^{\varphi_{2}}\|_{\gamma,\rho}\\}\leqslant\|\Phi^{\varphi}\|_{\gamma,\rho}\leqslant
2\max\\{\|\Phi^{\varphi_{1}}\|_{\gamma,\rho},\|\Phi^{\varphi_{2}}\|_{\gamma,\rho}\\}.\qed$

Conversely, given a measurable $\tilde{\varphi}:[0,\pi]\rightarrow\mathbb{R}$,
we can associate it another function
$\varphi=T\tilde{\varphi}:\mathbb{T}\rightarrow\mathbb{R}$ by setting
$T\tilde{\varphi}(y)=\tilde{\varphi}(|y|)$, which corresponds to
$(T\tilde{\varphi})_{1}=(T\tilde{\varphi})_{2}=\tilde{\varphi}$. It
immediately follows from Lemma 3.12 that $T\tilde{\varphi}$ is
$(\gamma,\rho)$-irregular if and only if $\tilde{\varphi}$ is so; it is also
easy to check that, if $\tilde{\varphi}\in B^{\alpha}_{1,\infty}(0,\pi)\cap
L^{\infty}(0,\pi)$, then $T\tilde{\varphi}\in
B^{\alpha}_{1,\infty}(\mathbb{T})$.

We are finally ready to prove a prevalence statement in
$B^{\alpha}_{1,\infty}(\mathbb{T})$.

######  Corollary 3.13.

Let $\alpha\in(0,1)$, then a.e. $\varphi\in B^{\alpha}_{1,\infty}(\mathbb{T})$
is $\rho$-irregular for any $\rho<(2\alpha)^{-1}$.

###### Proof.

The proof that the set

$\mathcal{A}:=\left\\{\varphi\in
B^{\alpha}_{1,\infty}(\mathbb{T}):\varphi\text{ is }\rho\text{-irregular for
any }\rho<\frac{1}{2\alpha}\right\\}$

is Borel in the topology of $B^{\alpha}_{1,\infty}(\mathbb{T})$ is identical
to that of Theorem 3.11 and thus omitted; as therein, we can introduce the
sets $\mathcal{A}_{\rho}$ and reduce the task to establish the prevalence of
the set $\mathcal{A}_{\rho}$ for any fixed $\rho<(2\alpha)^{-1}$.

Choose $H\in(0,1)$ such that $H>\alpha$, $\rho<(2H)^{-1}$ and denote by
$\mu^{H}$ the associated law of fBm; since it is supported on
$B^{\alpha}_{1,\infty}(0,\pi)\cap L^{\infty}(0,\pi)$, we can define a new
measure on $B^{\alpha}_{1,\infty}(\mathbb{T})$ by
$\nu^{H}:=T_{\sharp}\mu^{H}$, where
$(T\tilde{\varphi})(y)=\tilde{\varphi}(|y|)$ for $y\in[0,\pi]$ and
$T_{\sharp}$ denotes the pushforward measure.

Recall the notation $\varphi_{1},\varphi_{2}$ from Lemma 3.12; for any
$\varphi\in B^{\alpha}_{1,\infty}(\mathbb{T})$ it holds

$\displaystyle\nu^{H}(\varphi+\mathcal{A})$
$\displaystyle=\mu^{H}\left(\left\\{u\in
B^{\alpha}_{1,\infty}(0,\pi):\,Tu+\varphi\text{ is
}\rho\text{-irregular}\right\\}\right)$
$\displaystyle=\mu^{H}\left(\bigcap_{i=1}^{2}\left\\{u\in
B^{\alpha}_{1,\infty}(0,\pi):\,u+\varphi_{i}\text{ is
}\rho\text{-irregular}\right\\}\right)=1;$

in the last passage we used the already established properties of the measure
$\mu^{H}$ from the proof of Theorem 3.11, as well as the fact that the
intersection of sets of full measure is still of full measure. Overall, this
shows that $\nu^{H}$ witnesses the prevalence of the set $\mathcal{A}_{\rho}$;
the conclusion follows using the fact that countable intersection of prevalent
sets is prevalent. ∎

We are now ready to complete the

###### Proof of Theorem 1.4.

The lower bound comes from Corollary 3.2, while the upper bound from a
combination of Lemma 3.6 and Corollary 3.13. ∎

##  4\. Enhanced dissipation

This section contains the proof of Theorem 1.5 split in several steps.

Recall the setting: we want to study the asymptotic behavior of the family of
complex-valued PDEs (1.7), equivalently obtain upper and lower bounds on

$\|e^{tL_{k,\nu}}\|_{L^{2}(\mathbb{T};\mathbb{C})\rightarrow
L^{2}(\mathbb{T};\mathbb{C})}\quad\text{as }t\rightarrow\infty,$

where $L_{k,\nu}:=-iku+\nu\partial_{y}^{2}.$

###  4.1. Lower bounds in terms of regularity

We show here that if $u$ has regularity of degree $\alpha\in(0,1)$, as
measured in a suitable Besov–Nikolskii scale, then the its best possible
diffusion enhancing rate is $r_{\text{dif}}(\nu)\sim\nu^{\alpha/(2+\alpha)}$.
The precise statement goes as follows.

######  Proposition 4.1.

Let $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ be diffusion enhancing with rate
$r_{\text{dif}}$, in the sense of Definition 1.3; then there exists a constant
$C>0$ such that

$r_{\text{dif}}(\nu)\leqslant C\nu^{\frac{\alpha}{\alpha+2}}$

for all $\nu\in(0,1]$.

In order to provide estimates for $e^{tL_{k,\nu}}$ it is convenient to study
more generally the properties of solutions $g:\mathbb{T}\rightarrow\mathbb{C}$
to

$\partial_{t}g+i\xi ug=\nu\partial_{y}^{2}g$ (4.1)

in function of the parameters $\xi\in\mathbb{R}$, $\nu\in(0,1)$ and the shear
flow $u$.

The proof of Proposition 4.1 follows a similar strategy to [15] and is based
on deriving a Lagrangian Fluctuation-Dissipation relation (FDR) for the PDE
(4.1), which is a result of independent interest.

######  Proposition 4.2.

Let $u\in L^{1}(\mathbb{T})$, $g$ be a solution to (4.1) with initial data
$g_{0}\in L^{2}(\mathbb{T};\mathbb{C})$; for any
$(t,y)\in\mathbb{R}_{\geqslant 0}\times\mathbb{T}$, define the complex random
variable

$Z^{y}_{t}=\exp\left(-i\xi\int_{0}^{t}u\left(y+\sqrt{2\nu}B_{s}\right)\mathrm{d}s\right)g_{0}\left(y+\sqrt{2\nu}B_{t}\right)$

where $B$ is a standard real-valued BM. Then we have the following Lagrangian
FDR:

$\|g_{0}\|_{L^{2}}^{2}-\|g_{t}\|_{L^{2}}^{2}=\int_{\mathbb{T}}\operatorname{Var}(Z^{y}_{t})\mathrm{d}y.$
(4.2)

###### Proof.

Without loss of generality, we can assume $u$ and $g_{0}$ to be smooth, as
identity (4.2) in the general case will follow from an approximation argument
(the definition of $Z^{y}_{t}$ is meaningful for any $u\in L^{1}(\mathbb{T})$,
thanks to the properties of the local time of a Brownian motion). Let us
however first show that the r.h.s. of (4.2) is a well-defined quantity, which
can be estimated independently of the smoothness of $u$, $g_{0}$. Indeed, for
any $t\geq 0$ it holds

$\displaystyle\int_{\mathbb{T}}\mathbb{E}[|Z^{y}_{t}|^{2}]\mathrm{d}y$
$\displaystyle=\int_{\mathbb{T}}\mathbb{E}\big{[}\,|g_{0}|^{2}(y+\sqrt{2\nu}B_{t})\big{]}\mathrm{d}y=\mathbb{E}\left[\int_{\mathbb{T}}|g_{0}|^{2}(y+\sqrt{2\nu}B_{t})\mathrm{d}y\right]=\|g_{0}\|_{L^{2}}^{2},$

where in the last step we used the invariance of the $L^{2}$-norm of $g_{0}$
under (random) translations; the pointwise bound
$\operatorname{Var}(Z^{y}_{t})\leq\mathbb{E}[|Z^{y}_{t}|^{2}]$ then readily
yields an estimate for the r.h.s. of (4.2).

Now, by the Feynman–Kac formula, the solution $g$ to (4.1) is given by
$g(t,y)=\mathbb{E}[Z^{y}_{t}]$. Moreover since $u$ is real valued, we have the
energy balance

$\|g_{0}\|_{L^{2}}^{2}-\|g_{t}\|_{L^{2}}^{2}=2\nu\int_{0}^{t}\|\partial_{y}g_{s}\|_{L^{2}}^{2}\mathrm{d}s;$

and more generally, the map $(t,y)\mapsto|g|^{2}(t,y)$ satisfies

$\partial_{t}|g|^{2}=\nu\partial_{y}^{2}|g|^{2}-2\nu|\partial_{y}g|^{2}.$

Now let $h$ to be a solution of $\partial_{t}h=\nu\partial_{y}^{2}h$ with
initial data $h_{0}=|g_{0}|^{2}$. It holds

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{T}}[|g|^{2}-h]\mathrm{d}y=-2\nu\|\partial_{y}g\|_{L^{2}}^{2},$

which implies that

$\|g_{0}\|_{L^{2}}^{2}-\|g_{t}\|_{L^{2}}^{2}=2\nu\int_{0}^{t}\|\partial_{y}g\|_{L^{2}}^{2}=\int_{\mathbb{T}}[h_{t}(y)-|g_{t}(y)|^{2}]\mathrm{d}y.$

Finally, since by Feynman–Kac,
$h(t,y)=\mathbb{E}\left[|g_{0}|^{2}(y+\sqrt{2\nu}B_{t})\right]$, we obtain

$\displaystyle\|g_{0}\|_{L^{2}}^{2}-\|g_{t}\|_{L^{2}}^{2}$
$\displaystyle=\int_{\mathbb{T}}\left(\mathbb{E}\left[|g_{0}|^{2}(y+\sqrt{2\nu}B_{t})\right]-|\mathbb{E}[Z^{y}_{t}]|^{2}\right)\mathrm{d}y$
$\displaystyle=\int_{\mathbb{T}}(\mathbb{E}[|Z^{y}_{t}|^{2}]-|\mathbb{E}[Z^{y}_{t}]|^{2})\mathrm{d}y$

which gives the conclusion. ∎

######  Lemma 4.3.

Let $g_{0}\in H^{1}(\mathbb{T};\mathbb{C})$, $u\in
B^{\alpha}_{1,\infty}(\mathbb{T})$ for some $\alpha\in(0,1)$ and
$\xi\in\mathbb{R}$. Then there exists $C=C(\alpha)>0$ such that the solution
$g$ to (4.1) satisfies

$\|g_{0}\|_{L^{2}}^{2}-\|g_{t}\|_{L^{2}}^{2}\leqslant
C\|g_{0}\|^{2}_{H^{1}}\left(\nu t+\llbracket
u\rrbracket_{B^{\alpha}_{1,\infty}}|\xi|\nu^{\frac{\alpha}{2}}t^{1+\frac{\alpha}{2}}\right)\quad\forall\,t,\nu>0.$

###### Proof.

Recall the elementary identity
$2\operatorname{Var}(X)=\mathbb{E}[|X-\tilde{X}|^{2}]$ for $\tilde{X}$ being
an i.i.d. copy of $X$. In our setting, we take

$\tilde{Z}^{y}_{t}=\exp\left(-i\xi\int_{0}^{t}u\left(y+\sqrt{\nu}\tilde{B}_{s}\right)\mathrm{d}s\right)g_{0}\left(y+\sqrt{\nu}\tilde{B}_{t}\right)$

where $\tilde{B}$ is another BM independent of $B$. Therefore

$\displaystyle\|g_{0}\|_{L^{2}}^{2}-\|g_{t}\|_{L^{2}}^{2}$
$\displaystyle=\,\frac{1}{2}\int_{\mathbb{T}}\mathbb{E}[|Z^{y}_{t}-\tilde{Z}^{y}_{t}|^{2}]\mathrm{d}y$
$\displaystyle\leqslant\,\mathbb{E}\left[\int_{\mathbb{T}}\left|g_{0}\left(y+\sqrt{\nu}B_{t}\right)-g_{0}\left(y+\sqrt{\nu}\tilde{B}_{t}\right)\right|^{2}\mathrm{d}y\right]$
$\displaystyle+\|g_{0}\|^{2}_{L^{\infty}}\mathbb{E}\left[\int_{\mathbb{T}}\left|e^{-i\xi\int_{0}^{t}u\left(y+\sqrt{\nu}B_{s}\right)\mathrm{d}s}-e^{-i\xi\int_{0}^{t}u\left(y+\sqrt{\nu}\tilde{B}_{s}\right)\mathrm{d}s}\right|^{2}\mathrm{d}y\right].$

Using the inequality $|e^{i\xi a}-e^{i\xi
b}|\leqslant\sqrt{2}|\xi|^{1/2}|b-a|^{1/2}$ and the characterization of Besov
spaces in terms of finite differences (see Appendix A), we deduce

$\displaystyle\|g_{0}\|_{L^{2}}^{2}-\|g_{t}\|_{L^{2}}^{2}$
$\displaystyle\lesssim\,\mathbb{E}\left[\left\|g_{0}\left(\cdot\,+\sqrt{\nu}B_{t}\right)-g_{0}\left(\cdot\,+\sqrt{\nu}B_{t}\right)\right\|_{L^{2}}^{2}\right]$
$\displaystyle\quad+\|g_{0}\|_{L^{\infty}}^{2}|\xi|\mathbb{E}\left[\int_{\mathbb{T}}\int_{0}^{t}\left|u\left(y+\sqrt{\nu}B_{s}\right)-u\left(y+\sqrt{\nu}\tilde{B}_{s}\right)\right|\mathrm{d}s\mathrm{d}y\right]$
$\displaystyle\lesssim\|g_{0}\|^{2}_{H^{1}}\left(\nu\mathbb{E}[|B_{t}-\tilde{B}_{t}|^{2}]+|\xi|\int_{0}^{t}\mathbb{E}\left[\left\|u\left(\cdot\,+\sqrt{\nu}B_{s}\right)-u\left(\cdot\,+\sqrt{\nu}\tilde{B}_{s}\right)\right\|_{L^{1}}\right]\mathrm{d}s\right)$
$\displaystyle\lesssim\|g_{0}\|^{2}_{H^{1}}\left(\nu t+\llbracket
u\rrbracket_{B^{\alpha}_{1,\infty}}|\xi|\nu^{\frac{\alpha}{2}}\int_{0}^{t}\mathbb{E}[|B_{s}-\tilde{B}_{s}|^{\alpha}]\mathrm{d}s\right);$

computing the last expectation yields the conclusion. ∎

We are now ready to complete the

###### Proof of Proposition 4.1.

The proof goes along the same lines as Lemma 2 from [15]. We argue by
contradiction. Assume there exists no such constant $C$, then it must hold

$\liminf_{\nu\rightarrow
0^{+}}\nu^{-\frac{\alpha}{\alpha+2}}r_{\text{dif}}(\nu)=+\infty.$ (4.3)

Now take $g_{0}(y)=(2\pi)^{-1/2}e^{iy}$, so that
$\|g_{0}\|_{L^{2}}=1\sim\|g_{0}\|_{H^{1}}$; by Definition 1.3 and Lemma 4.3
applied to $\xi=1$ we deduce that there exist constants $C_{1},C_{2}>0$ such
that, for any $\nu\leqslant 1$ and $t\geqslant 1$, it holds

$\displaystyle 1-C_{1}e^{-r_{\text{dif}}(\nu)t}$ $\displaystyle\leqslant
1-\|e^{tL_{1,\nu}}\|_{L^{2}}^{2}\leqslant 1-\|g_{t}\|_{L^{2}}^{2}$
$\displaystyle\leqslant C_{2}\|g_{0}\|^{2}_{H^{1}}\left(\nu t+\llbracket
u\rrbracket_{B^{\alpha}_{1,\infty}}\nu^{\frac{\alpha}{2}}t^{1+\frac{\alpha}{2}}\right)$
$\displaystyle\lesssim C_{2}(1+\llbracket
u\rrbracket_{B^{\alpha}_{1,\infty}})\nu^{\frac{\alpha}{2}}t^{1+\frac{\alpha}{2}}.$

Let $\nu_{n}\downarrow 0$ be a sequence realizing the liminf in (4.3) and
choose

$t_{n}=\left(r_{\text{dif}}(\nu_{n})\nu_{n}^{\alpha/(\alpha+2)}\right)^{-1/2};$

then we obtain

$1-C_{1}\exp\left(-\left(\nu_{n}^{-\frac{\alpha}{\alpha+2}}r_{\text{dif}}(\nu)\right)^{1/2}\right)\lesssim_{u}\left(\nu_{n}^{-\frac{\alpha}{\alpha+2}}r_{\text{dif}}(\nu)\right)^{-\frac{\alpha+2}{4}}.$

Taking the limit as $n\rightarrow\infty$ on both sides we find $1\leqslant 0$
which is absurd. ∎

###  4.2. Wei’s irregularity condition

A major role in the analysis of dissipation enhancement by rough shear flows
is played by the following condition, first introduced in [43].

######  Definition 4.4.

We say that $u\in L^{1}(0,T)$ satisfies Wei’s condition with parameter
$\alpha>0$ if, setting $\psi(y)=\int_{0}^{y}u(z)\mathrm{d}z$, it holds

$\Gamma_{\alpha}(u):=\left[\inf_{\delta\in(0,1),\bar{y}\in[0,T-\delta]}\delta^{-2\alpha-3}\inf_{c_{1},c_{2}\in\mathbb{R}}\int_{\bar{y}}^{\bar{y}+\delta}|\psi(y)-c_{1}-c_{2}y|^{2}\mathrm{d}y\right]^{1/2}>0.$
(4.4)

A similar definition holds for $u\in L^{1}_{\operatorname{loc}}(\mathbb{R})$;
$u\in L^{1}(\mathbb{T})$ is said to satisfy Wei’s condition once it is
identified with a $2\pi$-periodic map on $\mathbb{R}$.

######  Remark 4.5.

Denoting by $\mathcal{P}_{1}$ the set of all polynomials of degree at most
one, for $u\in L^{1}_{\operatorname{loc}}(\mathbb{R})$ the definition is
equivalent to

$\Gamma_{\alpha}(u)=\left(\inf_{I\subset\mathbb{R},|I|<1}|I|^{-2\alpha-3}\inf_{P\in\mathcal{P}_{1}}\int_{I}|\psi(y)-P(y)|^{2}\mathrm{d}y\right)^{1/2}>0;$

this highlights its “complementarity” to the seminorm
$\llbracket\psi\rrbracket_{\mathcal{L}^{2,2\alpha+3}_{1}}$ associated to the
higher order Campanato space $\mathcal{L}_{1}^{2,2\alpha+3}$, as defined in
[4]. Observe that $\Gamma_{\alpha}$ is homogeneous, i.e.
$\Gamma_{\alpha}(\lambda u)=\lambda\Gamma_{\alpha}(u)$ for all
$\lambda\geqslant 0$.

The importance of condition (4.4) comes from the following result.

######  Theorem 4.6.

Let $u\in L^{1}(\mathbb{T})$ be such that $\Gamma_{\alpha}(u)>0$ for some
$\alpha>0$. Then there exist positive constants $C_{1},C_{2}$, depending on
$\alpha$ and $\Gamma_{\alpha}(u)$, such that

$\|e^{tL_{k,\nu}}\|_{L^{2}\rightarrow L^{2}}\leqslant
C_{1}\exp\left(-C_{2}\nu^{\frac{\alpha}{\alpha+2}}|k|^{\frac{2}{\alpha+2}}t\right)\quad\forall\,\nu\in(0,1),k\in\mathbb{Z}_{0},t\geqslant
0.$ (4.5)

Namely, $u$ is diffusion enhancing with rate $r_{\operatorname{dif}}(x)\sim
x^{\alpha/(\alpha+2)}$, in the sense of Definition 1.3.

The statement comes from Theorem 5.1 from [43]; therein $u$ is required to be
continuous, but this restriction is not necessary, see Appendix B for the
proof.

Following the same approach as in Section 3, we proceed to show that the
condition $\Gamma_{\alpha}(u)$ implies irregularity of $u$; we start by
relating it to the property of $\alpha$-Hölder roughness, in the sense of
Definition 3.7.

######  Lemma 4.7.

Let $u\in L^{1}(\mathbb{T})$ be such that $\Gamma_{\alpha}(u)>0$ for some
$\alpha>0$. Then $u$ is $\alpha$-Hölder rough and it holds
$L_{\alpha}(u)\geqslant\Gamma_{\alpha}(u)$.

###### Proof.

Fix $\delta>0$, $\bar{y}\in[-\pi,\pi]$; it holds

$\displaystyle\inf_{c_{1},c_{2}\in\mathbb{R}}\int_{\bar{y}}^{\bar{y}+\delta}|\psi(y)-c_{1}-c_{2}y|^{2}\mathrm{d}y$
$\displaystyle\leqslant\int_{\bar{y}}^{\bar{y}+\delta}|\psi(y)-\psi(\bar{y})-\psi^{\prime}(\bar{y})(y-\bar{y})|^{2}\mathrm{d}y$
$\displaystyle\leqslant\int_{\bar{y}}^{\bar{y}+\delta}\left(\int_{\bar{y}}^{y}|u(z)-u(\bar{y})|\mathrm{d}z\right)^{2}\mathrm{d}y$
$\displaystyle\leqslant\delta^{2\alpha+3}\left(\sup_{z\in
B_{\delta}(\bar{y})}\frac{|u(z)-u(\bar{y})|}{\delta^{\alpha}}\right)^{2}.$

As the inequality holds for all $\delta$ and $\bar{y}$, we obtain
$\Gamma_{\alpha}(u)^{2}\leqslant L_{\alpha}(u)^{2}$ and the conclusion. ∎

We can also relate Wei’s condition to regularity in the Besov–Nikolskii scales
$B^{\alpha}_{1,\infty}$.

######  Lemma 4.8.

Let $u\in L^{1}(\mathbb{T})$ be such that $\Gamma_{\alpha}(u)>0$ for some
$\alpha\in(0,1)$. Then $u$ does not belong to $B^{\tilde{\alpha}}_{1,\infty}$
for any $\tilde{\alpha}>\alpha$ and does not belong to $B^{\alpha}_{1,q}$ for
any $q<\infty$.

###### Proof.

For any $\bar{y}\in[-\pi,\pi]$ and $\delta>0$ it holds

$\displaystyle\delta^{2\alpha+3}\Gamma_{\alpha}(u)^{2}$
$\displaystyle\leqslant\int_{\bar{y}}^{\bar{y}+\delta}\left|\int_{\bar{y}}^{y}[u(z)-u(\bar{y})]\mathrm{d}z\right|^{2}\mathrm{d}y$
$\displaystyle\leqslant\int_{\bar{y}}^{\bar{y}+\delta}\left(\int_{\bar{y}}^{\bar{y}+\delta}|u(z)-u(\bar{y})|\mathrm{d}z\right)^{2}\mathrm{d}y$

thus implying that

$\inf_{\bar{y}\in\mathbb{T}}\int_{0}^{\delta}|u(\bar{y}+h)-u(\bar{y})|\mathrm{d}h\geqslant\delta^{1+\alpha}\Gamma_{\alpha}(u)\qquad\forall\,\delta\in(0,1).$
(4.6)

Now fix $\tilde{\alpha}>\alpha$; starting from (4.6) and arguing as in the
proof of Proposition 3.8 (with $\varepsilon$ replaced by $\delta$), one
obtains

$2\pi\Gamma_{\alpha}(u)\leqslant\delta^{\tilde{\alpha}-\alpha}\llbracket
u\rrbracket_{B^{\tilde{\alpha}}_{1,\infty}},$

which implies the first claim by letting $\delta\rightarrow 0^{+}$.
Integrating (4.6) over $\bar{y}\in\mathbb{T}$ yields

$\int_{0}^{\delta}\left\|u\left(\cdot\,+h\right)-u(\cdot)\right\|_{L^{1}}\mathrm{d}h\geqslant\delta^{1+\alpha}\Gamma_{\alpha}(u)\quad\forall\,\delta\in(0,1);$
(4.7)

now assume by contradiction that $u\in B^{\alpha}_{1,q}$ for some $q<\infty$,
then by its equivalent characterization (see Appendix A) and the uniform
integrability of $h\mapsto h^{-1-\alpha
q}\left\|u\left(\cdot\,+h\right)-u(\cdot)\right\|^{q}_{L^{1}}$ it must hold

$\lim_{\delta\rightarrow
0^{+}}\int_{0}^{\delta}\frac{\left\|u\left(\cdot\,+h\right)-u(\cdot)\right\|_{L^{1}}^{q}}{|h|^{1+\alpha
q}}\mathrm{d}h=0.$ (4.8)

On the other hand, by estimate (4.7) and Jensen’s inequality, it holds

$\displaystyle\int_{0}^{\delta}\frac{\left\|u\left(\cdot\,+h\right)-u(\cdot)\right\|_{L^{1}}^{q}}{|h|^{1+\alpha
q}}\mathrm{d}h$ $\displaystyle\geqslant\delta^{-1-\alpha
q}\int_{0}^{\delta}\left\|u\left(\cdot\,+h\right)-u(\cdot)\right\|_{L^{1}}^{q}\mathrm{d}h$
$\displaystyle\geqslant\delta^{-q(1+\alpha)}\left(\int_{0}^{\delta}\left\|u\left(\cdot\,+h\right)-u(\cdot)\right\|_{L^{1}}\mathrm{d}h\right)^{q}$
$\displaystyle\geqslant\Gamma_{\alpha}(u)^{q}>0$

uniformly in $\delta\in(0,1)$, contradicting (4.8). ∎

######  Remark 4.9.

It follows from Lemma 4.8 and the construction presented Section 2 from [10]
that, for any $\alpha\in\mathbb{Q}$ as in Lemma 2.1 therein, there exists a
Weierstrass-type function which belongs to $C^{\alpha}(\mathbb{T})$, satisfies
Wei’s condition with parameter $\alpha$ and does not belong to
$B^{\alpha}_{p,q}$ for any $p\in[1,\infty],q\in[1,\infty)$, nor to any
$B^{\tilde{\alpha}}_{p,q}$ with $\tilde{\alpha}>\alpha$.

In light of Theorem 4.6, in order to show that almost every shear flow $u$
enhances dissipation, it will suffice to show that almost every $u$ satisfies
Wei’s condition. We therefore need to find sufficient conditions in order for
$\Gamma_{\alpha}(u)>0$ to hold. We start with the following simple fact, whose
proof is almost identical to that of Lemma 3.12, which simplifies the problem
by allowing us to work with not necessarily periodic functions.

######  Lemma 4.10.

A map $u:\mathbb{T}\rightarrow\mathbb{R}$ satisfies $\Gamma_{\alpha}(u)>0$ if
and only if the maps $u_{i}:[0,\pi]\rightarrow\mathbb{R}$ defined by
$u_{1}(y)=u(y)$, $u_{2}(y)=u(-y)$ do so.

In this way, we can reduce the task to identifying sufficient conditions for
functions defined on a standard interval $[0,\pi]$. For any $\delta>0$, we
denote by $\Delta^{2}_{\delta}$ the discrete Laplacian operator
$\Delta^{2}_{\delta}f(y)=f(y+2\delta)-2f(y+\delta)+f(y)$.

######  Lemma 4.11.

For any $\alpha>0$ and any $(\bar{y},\delta)$ it holds

$\delta^{-2\alpha-3}\inf_{c_{1},c_{2}}\int_{\bar{y}}^{\bar{y}+3\delta}|\psi(y)-c_{1}-c_{2}y|^{2}\mathrm{d}y\geqslant\frac{1}{12}\left(\int_{\bar{y}}^{\bar{y}+\delta}|\Delta_{\delta}^{2}\psi(y)|^{-\frac{1}{1+\alpha}}\mathrm{d}y\right)^{-2(1+\alpha)}$
(4.9)

###### Proof.

First observe that $\Delta^{2}_{\delta}(c_{1}+c_{2}y)\equiv 0$ for any
$c_{1}$, $c_{2}$ and that for any $f$ it holds

$\int_{\bar{y}}^{\bar{y}+3\delta}|f(y)|^{2}\mathrm{d}y\geqslant\frac{1}{12}\int_{\bar{y}}^{\bar{y}+\delta}|\Delta^{2}_{\delta}f(y)|^{2}\mathrm{d}y.$

Next, applying Jensen inequality for $g(x)=x^{-\frac{1}{2(1+\alpha)}}$, which
is convex on $(0,\infty)$, it holds

$\left(\frac{1}{\delta}\int_{\bar{y}}^{\bar{y}+\delta}|\Delta^{2}_{\delta}f(y)|^{2}\mathrm{d}y\right)^{-\frac{1}{2(1+\alpha)}}\leqslant\frac{1}{\delta}\int_{\bar{y}}^{\bar{y}+\delta}|\Delta^{2}_{\delta}f(y)|^{-\frac{1}{1+\alpha}}\mathrm{d}y.$

Algebraic manipulations of the second inequality and the choice
$f(y)=\psi(y)-c_{1}-c_{2}y$ yield (4.9). ∎

In view of Lemma 4.11, given $\alpha>0$ and an integrable
$u:[0,\pi]\rightarrow\mathbb{R}$, we define

$G_{\alpha}(\bar{y},\delta):=\int_{\bar{y}}^{\bar{y}+\delta}|\Delta^{2}_{\delta}\psi(y)|^{-\frac{1}{1+\alpha}}\mathrm{d}y,$
(4.10)

where $\psi$ is defined as usual by $\psi(y)=\int_{0}^{y}u(z)\mathrm{d}z$.

######  Lemma 4.12.

For any $\alpha\in(0,1)$ and $\varepsilon>0$, define
$\beta:=\alpha+\varepsilon(1+\alpha)$ and

$K_{\alpha,\varepsilon}(u):=\sup_{n\in\mathbb{N},1\leqslant k\leqslant
2^{n}-1}2^{-n\varepsilon}G_{\alpha}(\pi k2^{-n},\pi 2^{-n-1}).$

Then there exists a constant $C=C(\alpha,\varepsilon)$ such that

$\Gamma_{\beta}(u)\geqslant C(K_{\alpha,\varepsilon}(u))^{-1-\alpha}.$

###### Proof.

First observe that, for any $\beta\in(0,1),$

$|\Gamma_{\beta}(u)|^{2}\sim_{\beta}\inf_{\delta\in(0,1/3),\bar{y}\in[0,1-3\delta]}\delta^{-2\beta-3}\inf_{c_{1},c_{2}\in\mathbb{R}}\int_{\bar{y}}^{\bar{y}+3\delta}|\psi(y)-c_{1}-c_{2}y|^{2}\mathrm{d}y$

so to conclude it suffices to provide a lower bound on the latter for our
choice of $\beta$. Fix $(\bar{y},\delta)$ and choose $n\in\mathbb{N}$ and
$k\in\\{1,\ldots,2^{n}-1\\}$ such that

$\delta\in(\pi 2^{-n},\pi 2^{-n+1}],\quad\bar{y}\in[\pi(k-1)2^{-n},\pi
k2^{-n}]$

so that
$[\bar{y},\bar{y}+3\delta]\supseteq[\tilde{y},\tilde{y}+3\tilde{\delta}]$ for
the choice $\tilde{y}=\pi k2^{-n}$, $\tilde{\delta}=\pi 2^{-n-1}$. As a
consequence,

$\begin{array}[]{l}\delta^{-2\beta-3}\inf_{c_{1},c_{2}\in\mathbb{R}}\int_{\bar{y}}^{\bar{y}+3\delta}|\psi(y)-c_{1}-c_{2}y|^{2}\mathrm{d}y\\\
\qquad\gtrsim_{\beta}\tilde{\delta}^{-2\beta-3}\inf_{c_{1},c_{2}\in\mathbb{R}}\int_{\tilde{y}}^{\tilde{y}+3\tilde{\delta}}|\psi(y)-c_{1}-c_{2}y|^{2}\mathrm{d}y\\\
\qquad\gtrsim\,\tilde{\delta}^{-2(\beta-\alpha)}\left(\int_{\tilde{y}}^{\tilde{y}+\tilde{\delta}}|\Delta_{\delta}^{2}\psi(y)|^{-\frac{1}{1+\alpha}}\mathrm{d}y\right)^{-2(1+\alpha)}\\\
\qquad=(\tilde{\delta}^{\varepsilon}G_{\alpha}(\tilde{y},\tilde{\delta}))^{-2(1+\alpha)}\end{array}$

where in the second passage we employed inequality (4.9) and then the
definition of $\beta$. Overall we deduce by the definition of $K$ and the
choice of $(\tilde{y},\tilde{\delta})$ that

$\delta^{-2\beta-3}\inf_{c_{1},c_{2}\in\mathbb{R}}\int_{\bar{y}}^{\bar{y}+3\delta}|\psi(z)-c_{1}-c_{2}z|^{2}\mathrm{d}z\gtrsim_{\beta}K_{\alpha,\varepsilon}(u)^{-2(1+\alpha)};$

taking the infimum over $(\delta,y)$ then yields the conclusion. ∎

###  4.3. Sufficient conditions for stochastic processes

In order to establish prevalence statements, we want to run the same programme
as in Section 3.3, exploiting the properties of LND Gaussian processes and
their fundamental translation invariance from Lemma 2.3. In order for this
strategy to work, we need an equivalent of Proposition 3.10; this is precisely
the aim of this section, cf. Corollary 4.17 below. Its proof requires a few
preparations; we start with the following intermediate, general result.

######  Proposition 4.13.

Let $u:[0,\pi]\rightarrow\mathbb{R}$ be an integrable stochastic process,
$\psi=\int_{0}^{\cdot}u_{s}\mathrm{d}s$ and suppose that there exist
$\lambda,\kappa>0$, $\alpha\in(0,1)$ such that

$\sup_{\delta\in(0,1),\bar{y}\in[0,\pi-\delta]}\mathbb{E}[\exp(\lambda
G_{\alpha}(\bar{y},\delta))]\leqslant\kappa$

for $G$ as defined in (4.10). Then for any $\beta>\alpha$ it holds
$\mathbb{P}(\Gamma_{\beta}(u)>0)=1$.

###### Proof.

By virtue of Lemma 4.12, for $\beta=\alpha+\varepsilon(1+\alpha)$ it holds

$\mathbb{P}(\Gamma_{\beta}(u)>0)\geqslant\mathbb{P}(K_{\alpha,\varepsilon}(u)<\infty),$

so to conclude it suffices to show that
$\mathbb{P}(K_{\alpha,\varepsilon}(u)<\infty)=1$ for all $\varepsilon>0$.
Given $\lambda$ as in the hypothesis, define the random variable

$J:=\sum_{n\in\mathbb{N}}2^{-2n}\sum_{k=1}^{2^{n}-1}\exp(\lambda
G_{\alpha}(\pi k2^{-n},\pi 2^{-n-1})).$

By assumption $\mathbb{E}[J]<\infty$, so that $\mathbb{P}(J<\infty)=1$. For
any $n,k$ it holds

$G_{\alpha}(\pi k2^{-n},\pi
2^{-n-1})\leqslant\frac{1}{\lambda}\log(2^{2n}J)\lesssim\frac{n}{\lambda}(1+\log
J)$

which implies that

$Y:=\sup_{n\in\mathbb{N},1\leqslant k\leqslant
2^{-n}-1}\frac{1}{n}G_{\alpha}(\pi k2^{-n},\pi
2^{-n-1})\lesssim\frac{1}{\lambda}(1+\log
J)<\infty\quad\mathbb{P}\text{-a.s.}$

Finally, for any $\varepsilon>0$ it holds
$K_{\alpha,\varepsilon}(u)\lesssim_{\varepsilon}Y$, which yields the
conclusion. ∎

In order to apply Proposition 4.13 to suitable LND Gaussian processes, we will
need the three Lemmas 4.14-4.16 below.

The next elementary lemma often appears in the probabilistic literature in
connection to so called Krylov or Khasminskii type of estimates, see Lemma 1.1
from [38] for a slightly more general statement. For the sake of completeness,
we give the proof.

######  Lemma 4.14.

Let $X$ be a real valued, nonnegative stochastic process, defined on an
interval $[t_{1},t_{2}]$, adapted to a filtration
$\\{\mathcal{F}_{s}\\}_{s\in[t_{1},t_{2}]}$; suppose there exists a
deterministic $C>0$ such that

$\operatorname{ess}\sup_{\omega\in\Omega}\mathbb{E}\left[\int_{s}^{t}X_{r}|\mathcal{F}_{s}\right]\leqslant
C\quad\forall\,s\in[t_{1},t_{2}].$

Then for any $\lambda\in(0,1)$ it holds

$\mathbb{E}\left[\exp\left(\frac{\lambda}{C}\int_{t_{1}}^{t_{2}}X_{r}\mathrm{d}r\right)\right]\leqslant(1-\lambda)^{-1}.$

###### Proof.

Up to rescaling $X$, we may assume $C=1$. It holds

$\mathbb{E}\left[\exp\left(\lambda\int_{t_{1}}^{t_{2}}X_{r}\mathrm{d}r\right)\right]=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{n!}\mathbb{E}\left[\left(\int_{t_{1}}^{t_{2}}X_{r}\mathrm{d}r\right)^{n}\right]=\sum_{n=0}^{\infty}\lambda^{n}I_{n}$

where

$I_{n}=\mathbb{E}\left[\int_{t_{1}<r_{1}<\ldots<r_{n}<t_{2}}X_{r_{1}}\cdot\ldots\cdot
X_{r_{n}}\mathrm{d}r_{1}\ldots\mathrm{d}r_{n}\right].$

By the assumptions and the non-negativity of $X$, it holds

$\displaystyle I_{n}$
$\displaystyle=\int_{t_{1}<r_{1}<\ldots<r_{n-1}<t_{2}}\mathbb{E}\left[X_{r_{1}}\cdot\ldots\cdot
X_{r_{n-1}}\int_{r_{n-1}}^{t}X_{r_{n}}\mathrm{d}r_{n}\right]\mathrm{d}r_{1}\ldots\mathrm{d}r_{n-1}$
$\displaystyle=\int_{t_{1}<r_{1}<\ldots<r_{n-1}<t_{2}}\mathbb{E}\left[X_{r_{1}}\cdot\ldots\cdot
X_{r_{n-1}}\mathbb{E}\left[\int_{r_{n-1}}^{t}X_{r_{n}}\mathrm{d}r_{n}|\mathcal{F}_{r_{n-1}}\right]\right]\mathrm{d}r_{1}\ldots\mathrm{d}r_{n-1}$
$\displaystyle\leqslant\int_{t_{1}<r_{1}<\ldots<r_{n-1}<t_{2}}\mathbb{E}[X_{r_{1}}\cdot\ldots\cdot
X_{r_{n-1}}]\mathrm{d}r_{1}\ldots\mathrm{d}r_{n-1}=I_{n-1}$

which iteratively implies $I_{n}\leqslant 1$. Therefore we obtain

$\mathbb{E}\left[\exp\left(\lambda\int_{s}^{t}X_{u}\mathrm{d}u\right)\right]\leqslant\sum_{n=0}^{\infty}\lambda^{n}=(1-\lambda)^{-1}.\qed$

######  Lemma 4.15.

Let $Z\sim\mathcal{N}(m,\sigma^{2})$ be a real valued Gaussian variable. Then
for any $\theta\in(0,1)$ there exists $c_{\theta}>0$ such that

$\mathbb{E}[|Z|^{-\theta}]\leqslant c_{\theta}\sigma^{-\theta}.$

###### Proof.

Set $Z=\sigma N+m$, then
$\mathbb{E}[|Z|^{-\theta}]=\sigma^{-\theta}\mathbb{E}[|N-x|^{-\theta}]$ for
$x=-m/\sigma$; therefore is sufficed to show that

$\sup_{x\in\mathbb{R}}\mathbb{E}[|N-x|^{-\theta}]=\sup_{x\in\mathbb{R}}\int|x-y|^{-\theta}p(y)\mathrm{d}y=\||\cdot|^{-\theta}\ast
p\|_{L^{\infty}}<\infty$

where $p$ stands for the Gaussian density
$p(x)=(2\pi)^{-1/2}\exp(-|x|^{2}/2)$. By Young’s inequality it holds

$\displaystyle\||\cdot|^{-\theta}\ast p\|_{L^{\infty}}$
$\displaystyle\leqslant\|(|\cdot|^{-\theta}\mathbbm{1}_{|\cdot|<1})\ast
p\|_{L^{\infty}}+\|(|\cdot|^{-\theta}\mathbbm{1}_{|\cdot|\geqslant 1})\ast
p\|_{L^{\infty}}$
$\displaystyle\leqslant\||\cdot|^{-\theta}\mathbbm{1}_{|\cdot|<1}\|_{L^{1}}\|p\|_{L^{\infty}}+\||\cdot|^{-\theta}\mathbbm{1}_{|\cdot|\geqslant
1}\|_{L^{\infty}}\|p\|_{L^{1}}$
$\displaystyle\leqslant(2\pi)^{-1/2}\||\cdot|^{-\theta}\mathbbm{1}_{|\cdot|<1}\|_{L^{1}}+1<\infty$

which gives the conclusion. ∎

######  Lemma 4.16.

Let $Y:[0,\pi]\rightarrow\mathbb{R}$ be a $(1+H)$-SLND Gaussian process with
constant $C_{Y}$, in the sense of Definition 2.2. Then for any $\alpha>H$
there exists $\lambda=\lambda(\alpha,H,C_{Y})>0$ s.t.

$\mathbb{E}\left[\exp\left(\lambda\int_{\bar{y}}^{\bar{y}+\delta}|\Delta^{2}_{\delta}Y_{y}|^{-\frac{1}{1+\alpha}}\mathrm{d}y\right)\right]\leqslant
2\quad\forall\,\delta\in(0,1),\bar{y}\in[0,\pi-\delta].$

###### Proof.

The result follows Lemmas 4.14 and 4.15 applied to the process
$X_{y}=|\Delta^{2}_{\delta}\psi_{y}|^{-\frac{1}{1+\alpha}}$. Indeed, denote by
$\mathcal{F}_{y}$ the natural filtration generated by $\psi$ and let
$\mathcal{G}_{y}:=\mathcal{F}_{y+2\delta}$. It is clear that
$\Delta^{2}_{\delta}\psi_{y}=Y_{y+2\delta}-2Y_{y+\delta}+Y_{y}$ is
$\mathcal{G}_{y}$-adapted; for any $[z,y]\subset[\bar{y},\bar{y}+\delta]$ it
holds

$\operatorname{Var}(\Delta^{2}_{\delta}Y_{y}|\mathcal{G}_{z})=\operatorname{Var}(Y_{y+2\delta}|\mathcal{F}_{z+2\delta})\geqslant
C_{Y}|y-z|^{2(1+H)}.$

As a consequence, we have a decomposition
$\Delta^{2}_{\delta}Y_{y}=Z^{(1)}_{z,y}+Z^{(2)}_{z,y}$ with $Z^{(1)}_{z,y}$
adapted to $\mathcal{G}_{z}$ and $Z^{(2)}_{z,y}$ Gaussian and independent of
$\mathcal{G}_{z}$; therefore

$\displaystyle\mathbb{E}\left[\int_{u}^{\bar{y}+\delta}|\Delta^{2}_{\delta}Y_{y}|^{-\frac{1}{1+\alpha}}\mathrm{d}y|\mathcal{G}_{z}\right]$
$\displaystyle=\int_{z}^{\bar{y}+\delta}\mathbb{E}\left[|Z^{(2)}_{z,y}+\cdot|^{-\frac{1}{1+\alpha}}\right](Z^{(1)}_{u,y})\mathrm{d}y.$

By Lemma 4.15, since $\operatorname{Var}(Z^{(2)}_{z,y})\geqslant
C_{Y}{|y-z|^{2(1+H)}}$ and $\theta=(1+\alpha)^{-1}\in(0,1)$, it holds

$\sup_{x\in\mathbb{R}}\mathbb{E}\left[|Z^{(2)}_{z,y}+x|^{-\frac{1}{1+\alpha}}\right]\lesssim_{\alpha}\operatorname{Var}(Z^{(2)}_{z,y})^{-\frac{1}{2(1+\alpha)}}\lesssim_{\alpha,H,C_{Y}}|y-z|^{-\frac{1+H}{1+\alpha}}$

and thus

$\displaystyle\mathbb{E}\left[\int_{z}^{\bar{y}+\delta}|\Delta^{2}_{\delta}X_{y}|^{-\frac{1}{1+\alpha}}\mathrm{d}y|\mathcal{G}_{z}\right]$
$\displaystyle\lesssim\int_{z}^{\bar{y}+\delta}|y-z|^{-\frac{1+H}{1+\alpha}}\mathrm{d}z$
$\displaystyle\lesssim\int_{0}^{1}|r|^{-\frac{1+H}{1+\alpha}}\mathrm{d}r\sim
C(\alpha,H,C_{Y})$

where the estimate is uniform over $z\in[\bar{y},\bar{y}+\delta]$,
$\bar{y}\in\mathbb{T}$ and $\delta\in(0,1)$. Choosing

$\lambda=\frac{1}{2C(\alpha,H,C_{Y})},$

we obtain the conclusion by applying Lemma 4.14. ∎

With Lemmas 4.14-4.16 at hand, we can finally verify that suitable Gaussian
processes verify Wei’s condition with probability $1$; we give the statement
in full generality, but we stress that the most relevant example verifying the
hypothesis below is the canonical process $X$ associated to $\mu^{H}$, as
granted by Lemma 2.4.

######  Corollary 4.17.

Let $X:[0,\pi]\rightarrow\mathbb{R}$ be a Gaussian process such that

$Y_{y}=\int_{0}^{y}X_{z}\mathrm{d}z$

is $(1+H)$-SLND for some $H\in(0,1)$. Then

$\mathbb{P}(\Gamma_{\alpha}(X)>0)=1$

for any $\alpha>H$.

###### Proof.

It follows immediately combining Lemma 4.16 and Proposition 4.13. ∎

###  4.4. Prevalence statements and proof of Theorems 1.5, 1.1

Similarly to Section 3.3, in order to prove prevalence statements in
$B^{\alpha}_{1,\infty}(\mathbb{T})$, we will actually start by establishing
their analogues on $B^{\alpha}_{1,\infty}(0,\pi)$.

######  Theorem 4.18.

Let $\alpha\in(0,1)$; then a.e. $\varphi\in B^{\alpha}_{1,\infty}(0,\pi)$
satisfies $\Gamma_{\beta}(\varphi)>0$ for all $\beta>\alpha$.

###### Proof.

Fix $\alpha\in(0,1)$ and define $\mathcal{A}:=\left\\{\varphi\in
B^{\alpha}_{1,\infty}(0,\pi):\,\Gamma_{\beta}(\varphi)>0\text{ for all
}\beta>\alpha\right\\}$; it holds

$\mathcal{A}=\bigcap_{n=1}^{\infty}\bigcup_{m=1}^{\infty}\mathcal{A}_{n,m}:=\bigcap_{n=1}^{\infty}\bigcup_{m=1}^{\infty}\left\\{\varphi\in
B^{\alpha}_{1,\infty}(0,\pi):\Gamma_{\beta}(\varphi)\geqslant\frac{1}{m}\text{
for }\beta=\alpha+\frac{1}{n}\right\\}.$

The sets $\mathcal{A}_{n,m}$ are closed in the topology of
$B^{\alpha}_{1,\infty}(0,\pi)$ (the map
$\varphi\mapsto\Gamma_{\beta}(\varphi)$ is upper semicontinuous in the
topology of $L^{1}(0,\pi)$), thus $\mathcal{A}$ is Borel measurable. In order
to conclude, it is enough to show that for any fixed $\beta>\alpha$, the set
$\mathcal{A}_{\beta}:=\left\\{\varphi\in
B^{\alpha}_{1,\infty}(0,\pi):\,\Gamma_{\beta}(\varphi)>0\right\\}$ (which is
Borel by the same line of argument) is prevalent.

Now fix $\beta>\alpha$ and choose $H\in(\alpha,\beta)$; denote by $\mu^{H}$
the law of fBm of parameter $H$ on $C([0,\pi])$ and by
$u=\\{u_{y}\\}_{y\in[0,\pi]}$ the associated canonical process. Since
$\mu^{H}$ is supported on $C^{H-\varepsilon}([0,\pi])$ for any $\varepsilon>0$
and $H>\alpha$, it is also a tight probability measure on
$B^{\alpha}_{1,\infty}(0,\pi)$. By Lemma 2.4, the associated process
$\psi=\int_{0}^{\cdot}u(y)\mathrm{d}y$ is $(1+H)$-SLND and so by Lemma 2.3 the
same holds for $f+\psi$, for any measurable $f:[0,\pi]\rightarrow\mathbb{R}$.

In particular, for a given $\varphi\in B^{\alpha}_{1,\infty}(0,\pi)$, taking
$f=\int_{0}^{\cdot}\varphi(y)\mathrm{d}y$, it follows from Corollary 4.17 and
the choice $\beta>H$ that

$\displaystyle\mu^{H}(\varphi+\mathcal{A}_{\beta})$
$\displaystyle=\mu^{H}\left(\left\\{u\in
B^{\alpha}_{1,\infty}(0,\pi):\,\Gamma_{\beta}(u+\varphi)>0\right\\}\right)=1.$

As the reasoning holds for any $\varphi\in B^{\alpha}_{1,\infty}(0,\pi)$, we
deduce that $\mu^{H}$ witnesses the prevalence of $\mathcal{A}_{\beta}$ and we
obtain the conclusion. ∎

As in Section 3.3, we define for
$\tilde{\varphi}:[0,\pi]\rightarrow\mathbb{R}$ the map
$(T\tilde{\varphi})(y)=\tilde{\varphi}(|y|)$; conversely for
$\varphi:\mathbb{T}\rightarrow\mathbb{R}$, $\varphi_{1}(y):=\varphi(y)$,
$\varphi_{2}(y):=\varphi(-y)$. Recall that if $\tilde{\varphi}\in
B^{\alpha}_{1,\infty}\cap L^{\infty}$, then $T\tilde{\varphi}\in
B^{\alpha}_{1,\infty}$.

######  Corollary 4.19.

Almost every $\varphi\in B^{\alpha}_{1,\infty}(\mathbb{T})$ satisfies
$\Gamma_{\beta}(\varphi)>0$ for all $\beta>\alpha$.

###### Proof.

The proof is very similar to that of Corollary 3.13, again employing measures
of the form $\nu^{H}=T_{\sharp}\mu^{H}$ for suitable $H\in(0,1)$;
specifically, once we fix $\beta>\alpha$ and we define a subset
$\mathcal{A}_{\beta}$ of $B^{\alpha}_{1,\infty}(\mathbb{T})$ as in the proof
of Theorem 4.18, it suffices to choose $H\in(\alpha,\beta)$. In this way
$\mu^{H}$ is tight on $B^{\alpha}_{1,\infty}(0,\pi)\cap L^{\infty}(0,\pi)$, so
$\nu^{H}$ is tight on $B^{\alpha}_{1,\infty}(\mathbb{T})$; the verification
that

$\nu^{H}(\varphi+\mathcal{A}_{\beta})=1\quad\forall\,\varphi\in
B^{\alpha}_{1,\infty}(\mathbb{T})$

is almost identical to that of Corollary 3.13, only this time invoking Lemma
4.10 and Theorem 4.18. ∎

At this point we have all the ingredient to close the dissipative case.

###### Proof of Theorem 1.5.

The lower bound comes from Proposition 4.1, while the upper bound from a
combination of Theorem 4.6 and Corollary 4.19. ∎

The main result of the paper, Theorem 1.1, is now a direct consequence of
Theorems 1.4 and 1.5. In fact, let us record here a slightly sharper estimate.
Given $f\in L^{2}(\mathbb{T}^{2})$, for any $s\in\mathbb{R}$ define

$\|f\|_{L^{2}_{x}H^{s}_{y}}^{2}:=\sum_{k\in\mathbb{Z}}\|P_{k}f\|_{H^{s}(\mathbb{T};\mathbb{C})}^{2}=\sum_{(k,\eta)\in\mathbb{Z}^{2}}(1+|\eta|^{2})^{s}|\hat{f}(k,\eta)|^{2};$

it’s clear that, for $s\geqslant 0$,
$\|f\|_{L^{2}_{x}H^{s}_{y}}\leqslant\|f\|_{H^{s}(\mathbb{T}^{2})}$ and
$\|f\|_{H^{-s}(\mathbb{T}^{2})}\leqslant\|f\|_{L^{2}_{x}H^{-s}_{y}}$.

######  Theorem 4.20.

Almost every $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ satisfies the following
property: for any $\tilde{\alpha}>\alpha$, there exists
$C=C(\alpha,\tilde{\alpha},u)$ such that, for any $f_{0}\in
H^{1/2}(\mathbb{T}^{2})$ with $P_{0}f_{0}\equiv 0$, it holds

$\|e^{tu\partial_{x}}f_{0}\|_{{L^{2}_{x}}H^{-1/2}_{y}}\leqslant
Ct^{-\frac{1}{2\tilde{\alpha}}}\|f_{0}\|_{L^{2}_{x}H^{1/2}_{y}}.$ (4.11)

Almost every $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$ satisfies the following
property: for any $\tilde{\alpha}>\alpha$ there exist
$C_{i}=C(\alpha,\tilde{\alpha},u)$ such that, for any $f_{0}\in
L^{2}(\mathbb{T}^{2})$ with $P_{0}f_{0}\equiv 0$, it holds

$\|e^{-t(u\partial_{x}-\nu\Delta)}f\|_{L^{2}(\mathbb{T}^{2})}\leqslant
C_{1}\exp\left(-C_{2}t\nu^{\frac{\tilde{\alpha}}{\tilde{\alpha}+2}}\right)\|f_{0}\|_{L^{2}(\mathbb{T}^{2})}.$
(4.12)

###### Proof.

By Theorem 1.4 b), for almost every $u\in B^{\alpha}_{1,\infty}(\mathbb{T})$
and any $\tilde{\alpha}>\alpha$ it holds

$\displaystyle\|e^{-tu\partial_{x}}f_{0}\|_{{L^{2}_{x}}H^{-1/2}_{y}}^{2}$
$\displaystyle=\sum_{k\in\mathbb{Z}_{0}}\|P_{k}(e^{-tu\partial_{x}}f_{0})\|_{H^{-1/2}}^{2}\lesssim\sum_{k\in\mathbb{Z}_{0}}(t|k|)^{-\frac{1}{\tilde{\alpha}}}\|P_{k}f_{0}\|_{H^{-1/2}}^{2}\lesssim
t^{-\frac{1}{\tilde{\alpha}}}\|f_{0}\|_{{L^{2}_{x}}H^{-1/2}_{y}}^{2}$

proving (4.11). Denote $\mathcal{L}_{\nu}=-u\partial_{x}+\nu\partial_{y}^{2}$,
so that $-u\partial_{x}+-\nu\Delta=\mathcal{L}_{\nu}+\nu\partial_{x}^{2}$,
where the operators $\mathcal{L}_{\nu}$ and $\nu\partial_{x}^{2}$ commute;
also observe that $P_{k}(e^{t\partial_{x}^{2}}f)=e^{-tk^{2}}P_{k}f$.

Combining these properties with Theorem 1.5, for almost every $u\in
B^{\alpha}_{1,\infty}(\mathbb{T})$ and any $\tilde{\alpha}>\alpha$ it holds

$\displaystyle\|e^{t(-u\partial_{x}+\nu\Delta)}f\|_{L^{2}}^{2}$
$\displaystyle=\sum_{k\in\mathbb{Z}_{0}}\|P_{k}(e^{t\partial_{x}^{2}}e^{t\mathcal{L}_{\nu}}f)\|_{L^{2}}^{2}=\sum_{k\in\mathbb{Z}_{0}}e^{-2tk^{2}}\|P_{k}(e^{t\mathcal{L}_{\nu}}f)\|_{L^{2}}^{2}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}_{0}}\exp\left(-2t|k|^{2}-Ct\nu^{\frac{\alpha}{\alpha+2}}|k|^{\frac{2}{\alpha+2}}\right)\|P_{k}f\|_{L^{2}}^{2}$
$\displaystyle\lesssim\exp\left(-Ct\nu^{\frac{\alpha}{\alpha+2}}\right)\sum_{k\in\mathbb{Z}_{0}}\|P_{k}f\|_{L^{2}}^{2}$

which yields (4.12). ∎

##  5\. Further comments and future directions

We have shown in this paper that generic rough shear flows satisfy both
inviscid mixing and enhanced dissipation properties, with rates sharply
determined by the regularity parameter $\alpha\in(0,1)$ in the Besov scale
$B^{\alpha}_{1,\infty}$. In the enhanced dissipation case, this confirms the
intuition from [10]; instead in the inviscid mixing one, it shows that the
behavior presented by Weierstrass-type functions constructed therein is not
generic in the sense of prevalence. Our results provide a connection to the
property of $\rho$-irregularity, which was never observed in this context, and
highlight the importance of working with mixing scales $H^{-s}$ with $s\neq
1$.

We conclude by presenting a few additional remarks and open problems arising
from this work.

  1. 1.

We are currently unable to determine whether there is a clear connection
between the properties of $\rho$-irregularity and Wei’s condition. Lemma 3.6,
together with the trivial estimate
$\|f_{t}\|_{H^{-1}}\leqslant\|f_{t}\|_{H^{-1/2}}$, imply that for
$\alpha\in(0,1/2)$ the shear flows $u\in C^{\alpha}$ constructed in [10]
satisfy $\Gamma_{\alpha}(u)>0$ but are not $\rho$-irregular with
$\rho\sim(2\alpha)^{-1}$. Heuristically, this fact is similar to the existence
of flows with small dissipation time which are not mixing, like the cellular
flows presented in [30].

  2. 2.

The above argument also implies the existence of Weierstrass type functions
which are not $\rho$-irregular, for suitable values $\rho$. We believe this
problem was open in the probabilistic community, although never been
explicitly addressed in the literature.

  3. 3.

Even without establishing a direct connection to Wei’s condition, it would be
interesting to show that functions $u$ which are $\rho$-irregular for
$\rho\sim(2\alpha)^{-1}$ are diffusion enhancing with rate
$r_{\text{dif}}(\nu)\sim\nu^{\alpha/(\alpha+2)}$, in line with the heuristic
argument presenting in the introduction. Since such $u$ are mixing, they are
indeed diffusion enhancing with a suitable rate by [11]; however the
quantitative results from [13] only imply the worsened rate
$\nu^{3\alpha/(1+3\alpha)}$ and it is rather unclear how to “bridge the gap”.

  4. 4.

Going through the same proof as in Lemma 3.6, one can show that if $u$ is
$(\gamma,\rho)$-irregular with $\gamma>1-s$, then $u$ is mixing on the scale
$H^{s}$ with rate $r_{s\text{-mix}}(t)=t^{\rho}$. In the case $\gamma=0$ an
even simpler proof based on duality and integration by parts provides mixing
on the scale $H^{1}$ with the same rate. In fact, since
$H^{1}(\mathbb{T};\mathbb{C})$ is an algebra, by integration by parts it holds

$\begin{array}[]{ll}|\langle e^{i\xi
u}f,g\rangle|&=\left|\int_{-\pi}^{\pi}e^{i\xi
u(y)}f(y)g(y)\mathrm{d}y\right|\\\
&\leqslant|(fg)(-\pi)|\left|\int_{-\pi}^{\pi}e^{i\xi
u(y)}\mathrm{d}y\right|\\\
&\qquad+\int_{-\pi}^{\pi}|(fg)^{\prime}(y)|\left|\int_{-\pi}^{y}e^{i\xi
u(z)}\mathrm{d}z\right|\mathrm{d}y\\\
&\lesssim(\|fg\|_{L^{\infty}}+\|(fg)^{\prime}\|_{L^{1}})\|\Phi^{u}\|_{0,\rho}|\xi|^{-\rho}\\\
&\lesssim\|f\|_{H^{1}}\|g\|_{H^{1}}\|\Phi^{u}\|_{0,\rho}|\xi|^{-\rho}\end{array}$

which by duality implies $\|e^{i\xi
u}f\|_{H^{-1}}\lesssim\|f\|_{H^{1}}\|\Phi^{u}\|_{0,\rho}|\xi|^{-\rho}$ and so
the claim.

  5. 5.

It is however an open problem to provide examples of $(0,\rho)$-irregular
functions $u$, for any $\rho<1$. See Remark 69 in [24] for a deeper
discussion. There are several examples of $u:[0,\pi]\rightarrow\mathbb{R}$
which are $(0,1)$-irregular, including the choice $u(y)=y$; by Proposition 1.4
from [7], it is enough to require the existence of $\delta>0$ such that

$\frac{1}{\delta}\leqslant|u^{\prime}(y)|,\quad\frac{|u^{\prime\prime}(y)|}{|u^{\prime}(y)|}\leqslant\delta\quad\forall\,y\in[0,\pi];$
(5.1)

observe the similarity of condition (5.1) with Assumption (H) from [12].

  6. 6.

The property of $(\gamma,\rho)$-irregularity can be reformulated in terms of
the (Fourier transform of) occupation measure of $u$, namely the family
$\\{\mu^{u}_{I},I\subset\mathbb{T}\\}$ given by
$\mu^{u}_{I}=u_{\sharp}\mathcal{L}_{I}$ where $I$ are subintervals of
$\mathbb{T}$ and $\mathcal{L}_{I}$ stands for the Lebesgue measure on $I$; see
Section 2.3 from [24] for more details. Closely related to the occupation
measure of $u$ is its local time, namely the Radon–Nikodym derivative
$\mathrm{d}\mu^{u}_{\mathbb{T}}/\mathrm{d}\mathcal{L}_{\mathbb{T}}$, which has
been intensively studied for stochastic processes, see the review [25]. The
following question arises naturally: is it possible to link the mixing
properties of $u$ to the regularity of its local time?

  7. 7.

In the paper we have always focused on the scales
$B^{\alpha}_{1,\infty}(\mathbb{T})$ with $\alpha\in(0,1)$. If one is instead
interested in the mixing properties of generic $u\in C(\mathbb{T})$, much
faster rates are available. Indeed for any $\beta>1$ it’s possible to
construct $\tilde{u}^{\beta}\in C([0,\pi])$ satisfying

$\left|\int_{y_{1}}^{y_{2}}e^{i\xi
u^{\beta}(z)}\mathrm{d}z\right|\lesssim_{\gamma,\beta}|y_{2}-y_{1}|^{\gamma}\exp\left(-C_{\gamma,\beta}|\xi|^{\frac{2}{1+\beta}}\right)\quad\forall\,\xi\in\mathbb{R},\quad
0\leqslant y_{1}\leqslant y_{2}\leqslant\pi$ (5.2)

and so by symmetrization the same holds for $u^{\beta}:=T\tilde{u}^{\beta}$.
Such $\tilde{u}^{\beta}$ are given by typical realization of the so called
$(2\beta)$-log Brownian motion, see [27] for its definition and Propositions
48 and 49 from [24] for the proof of (5.2). In fact, one could use the law of
such process to prove that a.e. $u\in C(\mathbb{T})$ satisfies (5.2) for any
$\beta>1$ (the value $\beta=1$ can only be attained by Caratheodory functions,
which are naturally discontinuous). Arguing as in the proof of Lemma 3.6 one
can deduce that such $u$ are exponentially mixing, in the sense that they
satisfy the estimate

$\|e^{i\xi
u}g\|_{H^{-1}}\lesssim\exp\left(-C_{\gamma,\beta}|\xi|^{\frac{2}{1+\beta}}\right)\|g\|_{H^{1}}\quad\forall\,g\in
H^{1}(\mathbb{T};\mathbb{C})$ (5.3)

and so that

$\|e^{-tu\partial_{x}}f\|_{L^{2}_{x}H^{-1}_{y}}\lesssim\exp\left(-C_{\gamma,\beta}t^{\frac{2}{1+\beta}}\right)\|f\|_{L^{2}_{x}H^{1}_{y}}$
(5.4)

for all $f\in H^{1}(\mathbb{T}^{2})$ satisfying $P_{0}f\equiv 0$.

  8. 8.

Finally, let us point out that the property of $\rho$-irregularity also holds
for generic vector valued functions $u:[0,1]\rightarrow\mathbb{R}^{d}$ (resp.
$u:\mathbb{T}\rightarrow\mathbb{R}^{d}$), for any $d\in\mathbb{N}$, see [24].
In particular, similar statements to part i. of Theorem 1.1 can be established
for “higher dimensional” shear flows of the form

$\partial_{t}f+\bar{u}\cdot\nabla f=\nu\Delta f$

for $f:\mathbb{T}^{d+1}\rightarrow\mathbb{R}$,
$\bar{u}(x_{1},\ldots,x_{d+1}):=(u(x_{d+1}),0)^{T}$; observe that for $d=2$,
$\bar{u}$ is a stationary solution to $3$D Euler equations. In light of [11],
the vector field $\bar{u}$ constructed by a $\rho$-irregular $u$ is diffusion
enhancing; thus can be applied in the study of suppression of blow-up by
mixing phenomena similarly to what was done in [33, 3, 30].

##  Appendix A Besov spaces

In this appendix we record fundamentals on Besov spaces $B^{s}_{p,q}$ on the
torus $\mathbb{T}^{d}$, although in the paper we only need the case $d=1$. For
a gentle introduction on spaces on $\mathbb{R}^{d}$ we refer to the monograph
[1]; see also the classical paper [41] for spaces on an interval
$I\subset\mathbb{R}$. All their properties transfer to the analogous spaces on
$\mathbb{T}^{d}$ by a clever use of Poisson summation formula, see [26], [36].
Alternatively, periodic Besov spaces have been treated in Chapter 3 of [40].

Given a dyadic partition of the unity $(\chi,\varphi)$ and the associated
Littlewood–Paley blocks $\\{\Delta_{j}\\}_{j\geqslant-1}$, the (inhomogeneous)
Besov spaces $B^{s}_{p,q}(\mathbb{T}^{d})$ with $s\in\mathbb{R}$, $p,q$ are
defined as the set of tempered distributions
$f\in\mathcal{S}^{\prime}(\mathbb{T}^{d})$ such that

$\|f\|_{B^{s}_{p,q}}:=\|2^{sj}\|\Delta_{j}f\|_{L^{p}}\|_{\ell^{q}}=\left(\sum_{j=-1}^{\infty}2^{jsq}\|\Delta_{j}f\|_{L^{p}}^{q}\right)^{1/q}<\infty$

with the usual conventions when $q=\infty$.
$(B^{s}_{p,q}(\mathbb{T}^{d}),\|\cdot\|_{B^{s}_{p,q}})$ are Banach spaces and
the definition is independent of the choice of the partition of unity
$(\chi,\varphi)$. Besov spaces are handy to use due to their many properties,
including functional embeddings and behavior under derivation and
multiplication; we recall them briefly.

######  Proposition A.1 ([1], Prop. 2.71).

Let $1\leqslant p_{1}\leqslant p_{2}\leqslant\infty$ and $1\leqslant
q_{1}\leqslant q_{2}\leqslant\infty$. Then for any $s\in\mathbb{R}$, the space
$B^{s}_{p_{1},q_{1}}$ continuously embeds in
$B^{s-d\left(\frac{1}{p_{1}}-\frac{1}{p_{2}}\right)}_{p_{2},q_{2}}$.

Also recall the following basic facts: for any $\varepsilon>0$ and any
$p,q\in[1,\infty]$, the space $B^{s}_{p,q}$ continuously embeds in
$B^{s-\varepsilon}_{p,1}$, as can be checked using the definition; for any
$p\in[1,\infty]$, we have the embeddings

$B^{0}_{p,1}\hookrightarrow L^{p}\hookrightarrow B^{0}_{p,\infty}$

see for instance Remark A.3 from [36] for the second statement.

######  Proposition A.2 ([36], Prop. A.5).

Let $s\in\mathbb{R}$, $p,q\in[1,\infty]$, $i\in\\{1,\ldots,n\\}$. Then the map
$f\mapsto\partial_{i}f$ is a continuous linear operator from $B^{s}_{p,q}$ to
$B^{s-1}_{p,q}$.

######  Proposition A.3 ([36], Prop. A.7).

Let $s_{1},s_{2}\in\mathbb{R}$ and $p,p_{1},p_{2},q\in[1,\infty]$ be such that

$s_{1}<0<s_{2},\quad
s_{1}+s_{2}>0,\quad\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}};$

then $(f,g)\mapsto fg$ is a well-defined continuous bilinear map from
$B^{s_{1}}_{p_{1},q}\times B^{s_{2}}_{p_{2},q}$ to $B^{s_{1}}_{p,q}$.

######  Proposition A.4 ([1], Cor. 2.86).

For any $s>0$ and $p,q\in[1,\infty]$, the space $B^{s}_{p,q}\cap L^{\infty}$
is an algebra and there exists a constant $C=C(s)$ such that

$\|fg\|_{B^{s}_{p,q}}\leqslant
C(\|f\|_{L^{\infty}}\|g\|_{B^{s}_{p,q}}+\|f\|_{B^{s}_{p,q}}\|g\|_{L^{\infty}})\quad\forall\,f,g\in
B^{s}_{p,q}\cap L^{\infty}.$

Another key property of Besov spaces is that they include several other
classical function spaces:

  * •

for $s\in\mathbb{R}$, $B^{s}_{2,2}(\mathbb{T}^{d})$ coincide with the Sobolev
spaces $H^{s}(\mathbb{T}^{d})$, with equivalent norms;

  * •

for $s\in(0,1)$, $B^{s}_{\infty,\infty}(\mathbb{T}^{d})$ coincide with
$C^{s}(\mathbb{T}^{d})$, the space of periodic $s$-Hölder continuous functions
(w.r.t. the canonical distance $d_{\mathbb{T}^{d}}$), with equivalent norms;

  * •

for $s\in(0,1)$ and $p\in[1,\infty)$, the spaces
$B^{s}_{p,\infty}(\mathbb{T}^{d})$, often referred to as Besov–Nikolskii
spaces, can be characterized by the equivalent norm

$\|f\tilde{\|}_{B^{s}_{p,\infty}}:=\|f\|_{L^{p}}+\sup_{x\neq
y\in\mathbb{T}^{d}}\frac{\left\|f\left(\cdot\,+x\right)-f\left(\cdot\,+y\right)\right\|_{L^{p}}}{d_{\mathbb{T}^{d}}(x,y)^{s}}$
(A.1)

  * •

for $s\in(0,1)$, $p,q\in[1,\infty)$ the space $B^{s}_{p,q}(\mathbb{T}^{d})$
has equivalent norm

$\|f\tilde{\|}_{B^{s}_{p,q}}:=\|f\|_{L^{p}}+\int_{\mathbb{T}^{d}}\left(\frac{\left\|f\left(\cdot\,+x\right)-f(\cdot)\right\|_{L^{p}}}{d_{\mathbb{T}^{d}}(x,0)^{s}}\right)^{q}\frac{1}{d_{\mathbb{T}^{d}}(x,0)^{d}}\mathrm{d}x.$
(A.2)

We conclude this appendix by proving some interpolation inequalities, which
played a fundamental role in the proofs in Section 3.1.

######  Lemma A.5.

Let $p\in[1,\infty]$, $s_{1},s_{2}\in\mathbb{R}$ with $s_{1}<s_{2}$ and
$\theta\in(0,1)$. Then there exists a constant $C=C(p,s_{2}-s_{1},\theta)$
such that

$\|f\|_{B^{\theta s_{1}+(1-\theta)s_{2}}_{p,1}}\leqslant
C\|f\|_{B^{s_{1}}_{p,\infty}}^{\theta}\|f\|^{1-\theta}_{B^{s_{2}}_{p,\infty}}\quad\forall\,f\in
B^{s_{2}}_{p,\infty}.$ (A.3)

###### Proof.

The result is well known, see Theorem 2.80 from [1] for the statement on
$\mathbb{R}^{d}$; let us provide a self-contained proof.

We may assume $\|f\|_{B^{s_{2}}_{p,\infty}}=1$; for any $N\geqslant 0$ it
holds

$\displaystyle\|f\|_{B^{s_{\theta}}_{p,1}}$
$\displaystyle=\sum_{j<N}2^{j(\theta
s_{1}+(1-\theta)s_{2})}\|\Delta_{j}f\|_{L^{p}}+\sum_{j\geqslant N}2^{j(\theta
s_{1}+(1-\theta)s_{2})}\|\Delta_{j}f\|_{L^{p}}$
$\displaystyle\leqslant\|f\|_{B^{s_{1}}_{p,\infty}}\sum_{j<N}2^{j(1-\theta)(s_{2}-s_{1})}+\|f\|_{B^{s_{2}}_{p,\infty}}\sum_{j\geqslant
N}2^{-j\theta(s_{2}-s_{1})}$
$\displaystyle\lesssim\|f\|_{B^{s_{1}}_{p,\infty}}2^{N(1-\theta)(s_{2}-s_{1})}+2^{-N\theta(s_{2}-s_{1})}.$

Choosing $N$ such that $\|f\|_{B^{s_{1}}_{p,\infty}}\sim 2^{-N(s_{2}-s_{1})}$
the conclusion then follows. ∎

######  Corollary A.6.

For any $s_{1},s_{2}>0$ there exists a constant $C(s_{1},s_{2})$ such that

$\|f\|_{L^{2}}\leqslant
C\|f\|_{H^{-s_{1}}}^{s_{2}/(s_{1}+s_{2})}\|f\|_{B^{s_{2}}_{2,\infty}}^{s_{1}/(s_{1}+s_{2})}\quad\forall\,f\in
B^{s_{2}}_{2,\infty}.$ (A.4)

###### Proof.

Applying Lemma A.5 for the choice $p=2$, $\theta=s_{2}/(s_{1}-s_{2})$ and
using Besov embeddings we find

$\|f\|_{L^{2}}\leqslant\|f\|_{B^{0}_{2,1}}\lesssim\|f\|_{B^{-s_{1}}_{2,\infty}}^{\theta}\|f\|^{1-\theta}_{B^{s_{2}}_{2,\infty}}\lesssim\|f\|_{H^{-s_{1}}}^{\theta}\|f\|^{1-\theta}_{B^{s_{2}}_{2,\infty}}.\qed$

##  Appendix B A simple extension of a result by Wei

Theorem 5.1 from [43] requires the restriction to work with $u\in
C(\mathbb{T})$, but we show here that such a restriction is not necessary and
in fact the result holds for any $u\in L^{1}(\mathbb{T})$, as stated in
Theorem 4.6. Let us recall the setting: we are interested in the decay of
solutions to complex valued PDEs of the form

$\partial_{t}f+iuf=\nu\partial_{y}^{2}f.$ (B.1)

Equation B.1 is well-posed (in the weak sense) for any $u\in
L^{1}(\mathbb{T})$ and $f_{0}\in L^{2}(\mathbb{T})$. Indeed, for smooth $u$,
any solution $f$ to (B.1) satisfies

$\partial_{t}\|f\|_{L^{2}}^{2}+2\nu\|\partial_{y}f\|_{L^{2}}^{2}=0,$

thus implying that it belongs to $L^{2}(0,T;H^{1}(\mathbb{T}))\hookrightarrow
L^{2}(0,T;C(\mathbb{T}))$; therefore we have uniform estimates for $iuf\in
L^{2}(0,T;L^{1}(\mathbb{T}))$ only depending on $\|u\|_{L^{1}}$. Arguing by
weak compactness one can then easily construct weak solutions to (B.1) for any
$u\in L^{1}(\mathbb{T})$, establish their uniqueness, and show that they are
the strong limit in $C([0,T];L^{2}(\mathbb{T}))$ of those to smooth $u$.
Overall, this defines the semigroup $t\mapsto e^{t(\nu\partial_{y}^{2}-iu)}$
on $L^{2}(\mathbb{T})$ for any $u\in L^{1}(\mathbb{T})$ and $\nu>0$.

Identifying $u\in L^{1}(\mathbb{T})$ with a $2\pi$-periodic function, its
primitive $\psi$ is a (non periodic) element of $C(\mathbb{R})$, well defined
up to additive constant; for given $\delta\in(0,1)$, define

$\omega_{1}(\delta,u):=\inf_{x,c_{1},c_{2}\in\mathbb{R}}\int_{x-\delta}^{x+\delta}|\psi(y)-c_{1}-c_{2}\delta|^{2}\mathrm{d}y.$

Denote by $F:\mathbb{R}_{\geqslant 0}\rightarrow[0,\pi/2]$ the inverse of
$x\mapsto 36x\tan x$, which is a one-to-one increasing function. The next
statement summarizes some of the main findings from [43].

######  Lemma B.1.

Let $u\in C(\mathbb{T})$ and $\nu>0$ be fixed; then for all $\delta\in(0,1)$
and $t\geqslant 0$ it holds

$\|e^{t(\partial_{y}^{2}-iu)}\|_{L^{2}\rightarrow
L^{2}}\leqslant\exp\left(\frac{\pi}{2}-t\nu\delta^{-2}F(\delta\nu^{-2}(\omega_{1}(\delta,u)))^{2}\right).$
(B.2)

###### Proof.

By time rescaling, the solution $f$ to (B.1) is given by
$f(t,y)=f^{\nu}(t\nu,y)$ where $f^{\nu}$ solves
$\partial_{t}f^{\nu}+iu^{\nu}f^{\nu}=\partial_{y}^{2}f^{\nu}$ where
$u^{\nu}=u/\nu$; applying the Gearhart–Prüss theorem (Theorem 1.3 from [43])
to $f^{\nu}$, it holds

$\|f_{t}\|_{L^{2}}\leqslant\exp\left(\frac{\pi}{2}-t\nu\psi_{1}(u^{\nu})\right)$

where $\psi_{1}(u)$ is defined as in Section 4 from [43]. By Lemma 4.3 therein
and the 2-homogeneity of $u\mapsto\omega(\delta,u)$, for any $\delta\in(0,1)$
it holds

$\psi_{1}(u^{\nu})\geqslant\delta^{-2}F(\delta(\omega_{1}(\delta,u^{\nu})))^{2}=\delta^{-2}F(\delta\nu^{-2}(\omega_{1}(\delta,u)))$

which gives the conclusion. ∎

We can now give the

###### Proof of Theorem 4.6.

By time rescaling, we can restrict to the case $k=1$. Now let $u\in
L^{1}(\mathbb{T})$ be a function satisfying $\Gamma_{\alpha}(u)>0$ for some
$\alpha\in(0,1)$ and consider a family $\\{u^{\varepsilon},\varepsilon>0\\}$
of continuous functions satisfying
$\|u^{\varepsilon}-u\|_{L^{1}}\leqslant\varepsilon$. Denote by
$\psi^{\varepsilon}$ the primitive of $u^{\varepsilon}$; by the basic
inequality $a^{2}\geqslant b^{2}/2-(a-b)^{2}$, for any $\delta(0,1)$ it holds

$\begin{array}[]{ll}\omega_{1}(\delta,u^{\varepsilon})&\geqslant\frac{1}{2}\omega_{1}(\delta,u)-\sup_{x\in\mathbb{R}}\int_{x-\delta}^{x+\delta}|\psi^{\varepsilon}(y)-\psi(y)|^{2}\mathrm{d}y\\\
&\geqslant\frac{1}{2}\omega_{1}(\delta,u)-2\delta\|u-u^{\varepsilon}\|_{L^{1}(\mathbb{T})}^{2}.\end{array}$

Combined with the fact that by definition $\omega_{1}(\delta,u)\geqslant
2^{2\alpha+3}\delta^{2\alpha+3}\Gamma_{\alpha}(u)^{2}$, we deduce

$\omega_{1}(\delta,u^{\varepsilon})\geqslant
2^{2\alpha+2}\delta^{2\alpha+3}\Gamma_{\alpha}(u)^{2}-2\delta\varepsilon^{2}\quad\forall\,\delta\in(0,1),\,\varepsilon>0.$
(B.3)

Now fix $\nu>0$ and define $C_{1}=e^{\pi/2}$,
$C_{2}=2^{2\alpha+2}\Gamma_{\alpha}(u)^{2}$; applying Lemma B.1 to
$u^{\varepsilon}$, exploiting the fact that $F$ is increasing, and choosing
$\delta=\nu^{1/(\alpha+2)}$, we obtain

$\|e^{t(\partial_{y}^{2}-iu^{\varepsilon})}\|_{L^{2}\rightarrow
L^{2}}\leqslant
C_{1}\exp\left(-t\nu^{\frac{\alpha}{\alpha+2}}F\left(C_{2}-2\nu^{-2\frac{\alpha+1}{\alpha+2}}\varepsilon^{2}\right)^{2}\right)\quad\forall\,t\geqslant
0$ (B.4)

where the estimate holds for all $\varepsilon>0$ small enough such that
$C_{2}-2\nu^{-2(\alpha+1)/(\alpha+2)}\varepsilon^{2}>0$.

Since the semigroup $e^{t(\nu\partial_{y}^{2}-iu^{\varepsilon})}$ pointwise
converges to $e^{t(\nu\partial_{y}^{2}-iu^{\varepsilon})}$ as
$\varepsilon\rightarrow 0^{+}$, passing to the limit on both sides of (B.4)
gives the conclusion. ∎

Declarations. The authors have no relevant financial or non-financial
interests to disclose.

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