> <\body> <\hide-preamble> |>|[>>>: >|>|]>|>>>> \; |Galeati & M.Gubinelli>|<\author-affiliation> Institute of Applied Mathematics & Hausdorff Center for Mathematics University of Bonn Germany |>>|<\doc-note> > We study mixing and diffusion properties of passive scalars driven by rough shear flows. Genericity is here understood in the sense of prevalence and (ir)regularity is measured in the Besov\UNikolskii scale >>>, \>. We provide upper and lower bounds, showing that in general inviscid mixing in > holds sharply with rate \t|)>>>, while enhanced dissipation holds with rate |)>\\/+2|)>>>. Our results in the inviscid mixing case rely on the concept of >-irregularity, first introduced by Catellier and Gubinelli (Stoc.Proc.Appl.126, 2016) and provide some new insights compared to the behavior predicted by Colombo, CotiZelati and Widmayer (arXiv:2009.12268, 2020). \; 35Q35, 37C20, 76F25, 76R50. |>-irregularity|Rough Flows.>> <\table-of-contents|toc> Introduction> |.>>>>|> Preliminaries> |.>>>>|> Prevalence |.>>>>|> > Fractional Brownian motion and local nondeterminism |.>>>>|> > Inviscid mixing> |.>>>>|> Lower bounds in terms of regularity |.>>>>|> > Upper bounds in terms of >-irregularity |.>>>>|> > Prevalence statements and proof of Theorem |-0.3em|>|0em||0em|>> |.>>>>|> > Enhanced dissipation> |.>>>>|> Lower bounds in terms of regularity |.>>>>|> > Wei's irregularity condition |.>>>>|> > Sufficient conditions for stochastic processes |.>>>>|> > Prevalence statements and proof of Theorems |-0.3em|>|0em||0em|>>, |-0.3em|>|0em||0em|>> |.>>>>|> > Further comments and future directions> |.>>>>|> Besov spaces> |.>>>>|> A simple extension of a result by Wei> |.>>>>|> |.>>>>|> We are interested in the long time behavior of solutions to <\equation> f+u \f=\ \f>>|=f,>f\x=0>>>>> on the -dimensional flat torus >. The PDE is an advection-diffusion equation associated to a shear flow :\\\>, 0>\\\\> with initial condition \L|)>> and where \> is the diffusion coefficient. Defining >:\\\> as >:=,0|)>>, equation may be written as <\equation> \f+>\\f=\\f which is the equation for a passive scalar advected by the velocity field >>. Note that >> is a divergence-free vector field and a stationary solution to 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 . In particular, shear flows are sufficiently simple to allow explicit calculations, while presenting a highly non trivial behavior, as already observed by Kelvin in in the case of the Couette flow =y>. Observe that for continuous , eq. can be solved explicitly by Feynman\UKac formula, giving <\equation> f=\u> B|)>\s+>B,y+> B|)>|]> where ,B|)>> is a standard 2D Brownian motion(Bm). In the case =0> we obtain <\equation> f=f,y|)>. Both formulas and can then be extended to the case L|)>>,<\footnote> For L|)>>, the formal expression u> B|)>\s> in can be made rigorous using the local time of >; alternatively, equation can be solved by applying the Fourier transform in the -variable and solving the family of equations for =Pf>, see the beginning of Appendix for more details. in which case eq. must be understood in the weak sense, and generate continuous semigroups +\\|)>>> on |)>>. Yet, they do not provide any immediate insight on the long time behavior of the solution , in particular on the decay in time of quantities like |f|\<\|\|\>>>> and |f|\<\|\|\>>>>. \; Following the line of research initiated in,, we consider rough shear flows, in the sense of requiring B>>|)>> for some \>. Here >>|)>> denote the Besov\UNikolskii spaces, see Appendix for their definition. We are interested in understanding the behavior of B>>|)>>, a problem explicitly left open in. For this purpose we adopt the measure-theoretic notion of genericity given by the theory of prevalence, developed by Hunt, Sauer and Yorke to provide an analogous of \PLebesgue almost every\Q on infinite dimensional spaces, see Section for more details. In what follows the expression \Pfor almost every \E>\Q, where is a function space, is understood in the sense of prevalence. The next statement summarizes our main findings. <\theorem> Let \>. The following hold: <\enumerate-roman> For almost every B>>|)>> we have inviscid mixing in the scale |)>>, in the following sense: for any |~>\\>, there exists ,|~>,u|)>> such that, for any \H|)>> satisfying >f|)>\x\0>, it holds <\equation*> |e> f|\<\|\|\>>>\C t|~>>> |f|\<\|\|\>>>\t\0. For almost every B>>|)>> we have enhanced dissipation in the following sense that: for any |~>\\> there exist =C,|~>,u|)>> such that, for any \L|)>> satisfying >f|)>\x\0>, it holds <\equation*> |e+\|)>>f|\<\|\|\>>>\C exp t \|~>||~>+2>>|)>|f|\<\|\|\>>>\t\0,\\. In the above statement, the condition >f|)>\x\0> is necessary, as it naturally ensures that witnesses the effect of the transport operator >; indeed :=>f\x> must solve the standard heat equation g=\\g> and thus cannot exhibit any mixing/enhanced dissipation effect. There is no obvious a priori reason to work with the spaces >>|)>> (e.g. in the authors deal with >|)>=B>,\>|)>>), 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 \>: indeed statements similar to those of Theorem can be given for the (smaller) spaces >|)>> for any choice of |]>>, see Remark below. \; Before moving further, let us heuristically motivate the connection between Points and of Theorem and why it is natural to expect /+2|)>>> to appear, given the decay |f|\<\|\|\>>>\t|)>>>. In fact, the argument can be given in a much more general framework: let >> be a solution to with \0>, >f \z=0> and >:\\\> be a divergence free vector field; then >> satisfies the energy balance <\equation*> |\t>|f>|\<\|\|\>>>=-2\|\f>|\<\|\|\>>>. Now assume the solution to the transport equation f+>\\f=0> to satisfy the decay |f|\<\|\|\>>>>\t s>> for suitable parameters \0,s\> (for 1>, one may reduce to by Riesz\UThorin interpolation theorem). For \1> and sufficiently short times, we expect >> and to stay close and therefore >> to exhibits the same decay as . By the interpolation inequality <\align*> |f>|\<\|\|\>>>\>||f>|\<\|\|\>>>>> |\f>|\<\|\|\>>>>,>>>> we deduce that <\equation> |\t>|f>|\<\|\|\>>>>\\|f>|\<\|\|\>>>|)>>|\f>|\<\|\|\>>>\\ |f>|\<\|\|\>>>>>\\ t>>. Assume for simplicity |f|\<\|\|\>>>=1> and define \0> to be the first time such that |f>|\<\|\|\>>>=1/2>. Integrating over |]>> we obtain <\align*> 2>-1>|\>t>>\t\\ \>>=|\+2>> \|)>|\+2>>.>>>> Namely, in order for the energy |f>|\<\|\|\>>>> to be reduced by half by the dynamics, we need to wait for at most \\/+2|)>>>. Iterating the argument on intervals ,n+1|)>|]>> would then produce an asymptotic decay at least of the form |\+2>>|)>>. While the argument is clearly heuristic, it predicts the correct exponent |\+2>> and works for any choice of the parameter 0> (in particular for as in Theorem) and not only for , 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 and the references therein) and they appear not to be optimal. For instance for >, an application ofCorollary2.3 from would only predict a decay <\equation*> |f|\<\|\|\>>>\exp> t|)>|f|\<\|\|\>>>,q\|\+s+\ s>; in particular =|2\+1>> while =|3\+1>>. \; Understanding the interaction between mixing and diffusion is one of the most fundamental problems in fluid mechanics, dating back to the works of Kelvin and Reynolds. In the pioneering work, such relation has been formalized mathematically by introducing the concept of ; the result has been recently revisited in a more quantitative fashion in the works. The use of weak norms > in order to quantify mixing of passive scalars was first introduced in. 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, spectral methods and stochastic analysis. Roughly speaking, the main known results for are the following: <\itemize> If C> has a finite number of critical points with maximal order , then enhanced dissipation holds with |)>\\>|)>>, see Theorem1.1 in. There exist C>>, \>, for which enhanced dissipation holds with |)>\\|\+2>>>, see Theorem5.1 from. The above results are sharp, up to logarithmic corrections, in the sense that for C> (resp. C>>) the best possible rate is |)>\\>> (resp. |)>\\|\+2>>>), see Theorem4 in; the proof is based on the Lagrangian Fluctuation Dissipation relation introduced in,. Let us also mention the remarkable stable mixing estimate obtained in for satisfying Assumption(H) therein. Motivated by the above results, the authors of explore the mixing and enhanced dissipation properties of rough shear flows, namely sharply >\UH>lder for \>. In particular, they construct a Weierstrass-type flow such that the following hold (see Theorem1.1 in): <\enumerate-numeric> enhanced dissipation holds with rate |)>\\|\+2>>>, confirming the results from; along suitable sequences \\>, inviscid mixing holds on > with rate \t>>: <\equation*> |eu \>f|\<\|\|\>>>\t>> |f|\<\|\|\>>>. however, to the authors' surprise, there exist other sequences \\> on which inviscid mixing only holds with rate \t>, in the sense that <\equation*> |eu \>f|\<\|\|\>>>\t>> |f|\<\|\|\>>>. In particular, the inviscid mixing rate \t> is the same attained by suitable Lipschitz functions; the authors wonder whether such a discrepancy between Points2. and3. is to be expected for generic flows C>>, see the paragraph \PPerspectives\Q, p.3 in. 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 shows that for generic B>>> (similarly for C>>, see Remark) inviscid mixing holds on > with rate \t>>, uniformly over all 0>. Such a decay is also the best possible, see Theorem below. On the other hand, Theorem confirms the enhanced dissipation rate |)>\\/+2|)>>>, already identified in, as a property of generic shear flows. We believe that the use of less standard spaces >>> and mixing norms > with 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 B>>> satisfy inviscid mixing on > with rate \t>> is still open and goes beyond our current methods. \; As done frequently in the literature, in order to prove Theorem for the PDE, we will pass to study its hypoelliptic counterpart <\equation> \f+u\f=\\f again under the assumption >f\x=0> for all \>. For \\\\>, define the Fourier transform in the -variable as <\equation*> f|)>\>fe \x so that any \\> has a decomposition =f|)>e>. If solves, then for each \> the function \Pf> solves the one dimensional complex valued PDE (harmonic oscillator) <\equation> \f+i k u f=\\f. For \>, \0> and L|)>>, the PDE has an associated semigroup on ;\|)>>, which we denote by \|)>>>; observe that the parameter , up to its sign, may be removed by the rescaling =t>, |~>=\/>. In this way the study of asymptotic behavior of > may be reduced to that of 1>>, which motivates the following definitions.\ Note that whenever we refer to a 0>\\0>>, we always assume it to be a continuous, increasing function. <\definition> A velocity field L|)>> is said to be on the scale ;\|)>>, 0>, with rate , if there exist a constant 0> such that <\equation> |e|\<\|\|\>>\H>\|)>> \k\\,t\1. <\definition> A velocity field L|)>> is said to be on ;\|)>> with rate if there exists a constant 0> such that <\equation> |e\|)>>|\<\|\|\>>\L>\C exp|>|)>t|)> \k\\,\\,t\1. 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 at each Fourier level >. <\theorem> =0>>Let \>. <\enumerate-alpha> Lower bound. Suppose that B>>|)>> is mixing on the scale ;\|)>> with rate , in the sense of Definition; then necessarily \t>>>. Upper bound. Almost every B>>|)>> satisfies the following property: for any |~>\\>, is mixing on the scale ;\|)>> with rate \ t|~>>>>. <\theorem> \0>>Let \>. <\enumerate-alpha> Lower bound. Suppose that B>>|)>> is diffusion enhancing with rate , in the sense of Definition; then necessarily |)>\\|\+2>>>. Upper bound. Almost every B>>|)>> satisfies the following property: for any |~>\\>, is diffusion enhancing with rate |)>\\|~>/|~>+2|)>>>. Theorems and will be proven respectively in Sections and, 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: <\enumerate-numeric> In both cases, the lower bound follows from estimates which explicitly employ the regularity assumption B>>>; in the case \0>, we need to preliminary establish a Lagrangian Fluctuation-Dissipation relation for the PDE \ (see Proposition) similarly in spirit to what was done in. The upper bound is satisfied by any enjoying a suitable analytic property, which encodes its . It turns out that the right properties are given respectively by >-irregularity (see Definition) for =0> and by Wei's irregularity condition (see Definition) for \0>. A shear flow satisfying any of such properties necessarily enjoys only limited regularity in the scales >>> (see Lemma and), confirming that these are the correct spaces to work with. Finally, we show that a.e. B>>> is >-irregular (resp. satisfies Wei's condition), see Section (resp. Section). This is achieved by probabilistic methods, using the law of fractional Brownian motions (see Section for details) to construct a measure witnessing the prevalence of such properties. <\remark> Let us stress that points) of Theorems- holds for B>>>, not only generic elements. Since > is finite, we have the embeddings >\B>>> for any |]>>, thus the lower bound is true for all B>> as well. On the other hand, the proofs of points) of Theorems- can be easily readapted to provide the same statements for almost every B>>, for any choice of |]>>. In particular, one could always work with the spaces >=B>,\>> if desired. There are however several reasons for working with >>> or more generally >> instead of >>. Mathematically, such spaces include genuinely discontinuous functions, as well as (possibly continuous) functions of finite -variation for any |]>>: it holds <\equation*> B\V\V\B>, see Proposition 4.3 from, Proposition 2.3 from for more details. Physically, a simple way to explain singularities in fully developed turbulence is by means of structure functions (see e.g.), which are closely related to the finite difference characterization of Besov spaces >>>. Turbulence is also believed to be closely connected to multifractality (again we refer to the appendix of), a feature which is absent from generic C>> (which are monofractal) but instead manifested by almost every B>>, see. Our results show that the only relevant parameter in understanding mixing and enhanced dissipation rates for a.e. B>> is \>, regardless of the values of ; thus there is no apparent connection between mixing and multifractal features of . In Section we shortly recall some of the main tools we will be working with, specifically the theory of prevalence and the properties of fractional Brownian motion. Sections and contain the proofs of Theorems and and are designed in a similar manner: in both cases we will first prove the lower bound, then introduce the concept of >-irregularity (resp. Wei's condition) and explain its connection to the upper bound, as well as to the irregularity of ; finally, we show by probabilistic means that a.e. B>>> satisfies such property. The end of Section also contains the proof of Theorem. In Appendix we collect some well known results on Besov spaces, while Appendix contains a technical extension of the results from needed to work in our setting. \; The authors were supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Hausdorff Center for Mathematics under Germany's Excellence Strategy \U EXC-2047/1 \U 390685813 and through CRC 1060 - projekt number 211504053. \; \; We will use the notation b> to mean that there exists a constant 0> such that c b>; b> highlights the dependence >. The notation b> stands for b> and a>, similarly for b>. Whenever needed, we will identify the -dimensional torus > with either |]>> or ,\|]>> with periodic boundary condition, and functions :\\\> with >-periodic functions defined on >. We will use >> to denote the canonical distance on the flat torus >, namely >=inf\>k-y|\|>>, where |\|>> denotes the Euclidean distance on >. With a slight abuse, we will keep writing > for \> to denote >>. |)>> denotes classical Lebesgue spaces, >|)>> H>lder spaces and |)>=W|)>> fractional Sobolev spaces. >|)>> denotes Besov spaces on >, see Appendix for their definition and main properties; similarly >|)>> denotes Besov spaces on |]>>. Whenever a stochastic process |)>0>> is considered, if not specified we tacitly assume the existence of an abstract underlying filtered probability space ,,|)>0>,\|)>>, such that the >-algebra > and the filtration |)>0>> satisfy the usual assumptions and |)>0>> is adapted to |)>0>>. Whenever we say that |)>0>> is the natural filtration generated by , then it is tacitly implied that it is actually its right continuous, normal augmentation wrt. >. We denote by > integration (equiv. expectation) wrt. the probability >. The theory of prevalence has been developed by Hunt, Sauer and Yorke in in order to provide a measure theoretic notion of genericity in infinite dimensional spaces. It is a natural generalization of the concept of \Pfull Lebesgue measure sets\Q from the finite dimensional setting. We follow here the exposition given in. For our purposes however it will be enough to work with Banach spaces . <\definition> Let be a complete metric vector space. A Borel set E> is said to be if there exists a measure > such that: <\enumerate-roman> There exists a compact set E> such that \\\>. For every E>, =0>. In this case, the measure > is said to be to . More generally, a subset of is shy if it is contained in a shy Borel set. The complement of a shy set is called a set. Sometimes it is said informally that the measure > \Pwitnesses\Q the prevalence of >. It follows immediately from Point of Definition that, if needed, one can assume > to be a compactly supported probability measure on . If is separable, then any probability measure on is tight and therefore Point is automatically satisfied. \; The following properties hold for prevalence (all proofs can be found in): <\enumerate-numeric> If is finite dimensional, then a set is shy if and only if it has zero Lebesgue measure. If is shy, then so is for any E>. Prevalent sets are dense. If =+\>, then compact subsets of are shy. 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. E>, we mean that the set of elements of for which the statement holds is a prevalent set. Property1. states that this convention is consistent with the finite dimensional case. In the context of a function space , it is natural to consider as probability measure the law induced by an valued random variable. Namely, given stochastic process defined on a probability space ,,\|)>> taking values in a separable Banach space , in order to show that a property > holds for a.e. E>, it suffices to show that <\equation*> \ satisfies property >>|)>=1,\f\E. Clearly, we are assuming that the set E : satisfies property >>|}>> is Borel measurable; if is not separable, we need to additionally require that the law of is tight, so as to satisfy Point of Definition. As a consequence of properties4. and5., the set of all possible realizations of a probability measure > 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 \PProperty > holds for a.e. (in the sense of prevalence)\Q and \PProperty > holds for >-a.e. \Q; 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 \Pmany more\Q than just the realizations of a given measure >. In order to prove prevalence statements, we will use the law of fractional Brownian motion (fBm) as a suitable transverse measure. The material presented here on fBm is mostly classical and can be found in the monograph. Given an interval > and a probability measure > on |)>>, denote by > the associated canonical process, |)>=\> for \C|)>> and by =\,s\t|}>|)>> the associated natural filtration. The law of fBm of Hurst parameter > is defined as the unique Gaussian measure > on =C|)>> such that <\equation*> >X|)> \\|)>=0,>X|)>X|)> \\|)>=+-|)>. For , the law of fBm corresponds to the classical Wiener measure; instead for 1/2>, the associated canonical process is not a semimartingale nor a Markov process. The support of > in terms of Besov spaces is well understood, with sharp results going back to (see also for a modern proof which extends to the vector valued case): it holds <\equation*> \>|)>=1\\\0,\>|)>=1\p\|)>, while <\equation*> \|)>=0,\|)>=0 \p,q\|)>. In particular fBm trajectories are sharply not -H>lder continuous, but by Ascoli\UArzelà > is a tight probability measure on >>> for any \0> and any |]>>. \; A fundamental feature of fBm which we will exploit heavily is its (LND for short). There are several non-equivalent concepts of LND in the literature, see for a review; here we will adopt the definition from. <\definition> Given \0>, a real valued Gaussian process is with parameter >, >-SLND for short, if there exists a constant > such that <\equation> Var\||)>\C*> uniformly over > with t>. In above, \||)>> denotes the conditional variance; equivalently, Definition amount to the condition that, for any t>, there is a decomposition =X>+X>> where >> is Gaussian and adapted to > while >> is Gaussian, independent of >, with variance >|)>\C >>. \; There are three fundamental facts involving LND we will need, which can be found in and in the references therein: <\itemize> If is the canonical process associated to >, then is -SLND. Given as above, define the process =X \s>; then is >-SLND. If is the canonical process associated to a Gaussian measure > on |)>> and is >-SLND, then so is the process =X+f> for any deterministic, measurable function \\>. This section contains the proof of Theorem, which we split in several steps. Recall the setting: in order to study the transport equation f+u\f=0>, we pass to Fourier modes =f|)>>, solving f+i k u f=0>; namely =e>f>. It is then natural to take a slightly more general perspective and study maps of the form eu>g> with \\>, H|)>>. We show here that the regularity of , measured in the Besov\UNikolskii scale >>>, necessarily implies a lower bound on the decay of solutions in the >-norm. The proof is partly inspired by that of Proposition3.2 from. <\lemma> Let B>>|)>> for some \>. Then for any H|)>> there exists a constant ,g|)>> such that <\equation> |eu> g|\<\|\|\>>>\C|u|\<\|\|\>>>>>|)>>> |\|>>>\|\|>\1. <\proof> Fix > with |\|>\1> and set >\eu>g>; we claim that >\B/2>>>. By Sobolev and Besov embeddings, L>\B/2>>>; u>\L>>, so it's enough to show that u>\B/2>>>. By the basic estimate -e|\|>\ >, it holds <\align*> |eu+y|)>>-eu+|)>>|\<\|\|\>>>>||\|>|u+y|)>-u+|)>|\<\|\|\>>>>>|||\|>|u|\<\|\|\>>>>> d>|)>/2>.>>>> By the equivalent characterization of Besov\UNikolskii spaces, this implies <\equation*> |eu>|\<\|\|\>>/2>>>\1+|\|>|u|\<\|\|\>>>>>\|u|\<\|\|\>>>>>|)>|\|> and so by Proposition in Appendix we conclude that >\B/2>>> with <\equation> |>|\<\|\|\>>/2>>>\|g|\<\|\|\>>>|u|\<\|\|\>>>>>|)>|\|>. Clearly |>|\<\|\|\>>>=|g|\<\|\|\>>>>. Using the interpolation inequality from Corollary in Appendix (for the choice =1/2>, =\/2>) we obtain <\equation> |g|\<\|\|\>>>=|>|\<\|\|\>>>\|>|\<\|\|\>>>|1+\>> |>|\<\|\|\>>/2>>>>>. Rearranging now the terms in and applying the estimate we find <\equation> |>|\<\|\|\>>>>|>||>|\<\|\|\>>/2>>>>> |g|\<\|\|\>>>>>>>||>||g|\<\|\|\>>>>>|g|\<\|\|\>>>>>|u|\<\|\|\>>>>>|)>>>|\|>>>>>>>> where the hidden constant in only depends on >. Using the definition of >> and relabelling the constant to include the -dependent terms yields the conclusion. <\corollary> Let B>>|)>> be mixing with rate , in the sense of Definition. Then there exists a constant ,u|)>> such that <\equation*> r\C \ t>>. <\proof> Choose any H|)>> with |g|\<\|\|\>>>=1>; then by Definition applied for the choice and Lemma for =-t>, it holds <\align*> >\>||e|\<\|\|\>>\H>\|eg|\<\|\|\>>>\,g>|u|\<\|\|\>>>>>|)>>> \ t>>;>>>> up to relabelling constants, this yields the conclusion. <\remark> In fact, the statement of Lemma can be generalized as follows. For \>, B>>|)>>, H|)>> and any 0> there exists a constant ,g,s|)>> such that <\equation*> |eu> g|\<\|\|\>>>\C|u|\<\|\|\>>>>>|)>>> |\|>>>\|\|>\1. Then arguing as in Corollary one can conclude that the best possible rate for inviscid mixing on the scale |)>> is \t>>. Taking provides the rate >>, which is in line with Proposition3.2 from. >-irregularity> The concept of >-irregularity was first introduced in in the study of regularization by noise phenomena. Its applications to PDEs have been subsequently explored in. <\definition> Let \>, \0>; a measurable map |]>\\> is said to be ,\|)>>-irregular> if there exists a constant 0> such that <\equation> e u>\z|\|>\C> |\|>>\\\\,I\|]> where stands for a subinterval of |]>> and > denotes its length. A similar definition holds for \\>; a map \\> is said to be ,\|)>>-irregular if it is so once it is identified with a >-periodic map \\>. In all cases, the optimal constant in is denoted by |\|\<\|\|\>>,\>>. We say that is >-irregular> for short if there exists \1/2> such that it is ,\|)>>-irregular. The property of >-irregularity may be rephrased in the following form, more suited for our purposes. <\lemma> Let \\> be ,\|)>>-irregular, then <\equation*> |eu>|\<\|\|\>>-1>,\>>\|\|\<\|\|\>>,\> |\|>>\\\\. <\proof> For >\,\|]>> and \\>, define the function <\align*> >>|)>>|>>>eu>\y->+\|2\>|)>>>eu>\y;>>>> by periodicity can be identified with a function on >. Then by definition of ,\|)>>-irregularity it holds |v>|\<\|\|\>>>>\|\|\<\|\|\>>,\> |\|>>> and so by Proposition we deduce that <\equation*> |eu>|\<\|\|\>>-1>\>>>|||>|)>+>>>eu>\y|\<\|\|\>>-1>,\>>>>||>||v>|\<\|\|\>>>>+>>>eu>\y|\|>>>||>||\|\<\|\|\>>,\> |\|>>.>>>>> \; The relation between >-irregularity and inviscid mixing comes from the next result. <\lemma> Let \\> be ,\|)>>-irregular for some \1/2>. Then there exists a constant |)>> such that <\equation> |e u> g|\<\|\|\>>>\C |\|\<\|\|\>>,\> |\|>>|g|\<\|\|\>>> \\\0,g\H. As a consequence is mixing on the scale > with rate =t>> in the sense of Definition. <\proof> The proof of the estimate relies on several properties of Besov spaces, for which we refer the reader to Appendix. By assumption +1/2\1>, thus by Proposition and Lemma it holds <\align*> |eu>g|\<\|\|\>>-1>>>>||eu>|\<\|\|\>>-1>,\>>|g|\<\|\|\>>>>>>|||\|\<\|\|\>>,\> |\|>>|g|\<\|\|\>>>>>|||\|\<\|\|\>>,\> |\|>>|g|\<\|\|\>>>.>>>> Again by the hypothesis -1\-1/2> and so by Besov embeddings -1>>\H>, yielding the first claim. Applying estimate for \>, =-t k> gives <\equation*> |e|\<\|\|\>>\H>\|\|\<\|\|\>>,\> ||)>>> and thus the conclusion. The property of >-irregularity implies roughness of , as the name suggests. To quantify this precisely, we recall the concept of H>lder roughness, as presented in. <\definition> A measurable map \\> is said to be >-H>lder if there exists >> such that: for any \0> and any >\\>, there exists \> satisfying <\equation*> d>>,z|)>\\>|)>-u|\|>\L> \>. The optimal constant >> is called the modulus of >-H>lder roughness of . Definition is equivalent to requiring <\equation> L>=inf>\\,\\0> supB>>|)>>-u>|)>|\|>|\>>\0. A detailed study of analytic properties of >-irregular paths was carried out in Section5 of; let us recall the following result. <\proposition> Let \\> be ,\|)>>-irregular and define >\|)>/\>. Then: <\enumerate-alpha> is >-H>lder rough for any \\>> with >=+\>. has infinite -variation on any subinterval \> and for any 1/\>>. <\proof> For functions \\>, both statements are proved in, see Corollary65 and Corollary68 therein. Up to identifying \\> with a function defined on |]>>, the result carries over to this setting as well. The parameter >> is also linked to the regularity of in the Besov\UNikolskii scales >>>. <\lemma> Let \\> be ,\|)>>-irregular and set >=|)>/\>; then does not belong to >>> for any \\>>. <\proof> Fix \\>> and choose |~>\>,\|)>>; going through the same computations as in Theorem63 and Corollary65 from, it can be shown that <\equation*> lim\0> inf\> \|)>:-u|\|>\\|~>>|)>=1, where > denotes the Lebesgue measure on >. It follows that, for all \0> sufficiently small, it must hold <\align*> >|>\|)>:-u|\|>\\|~>>|)> \y>>||>\|~>>>-u|\|>\h \y>>|||~>>>|u+h|)>-u|)>|\<\|\|\>>> \h>>||\-|~>> |u|\>>>>>>>> where in the second passage we used Markov's inequality. Since \|~>>, letting \0> we conclude that |u|\>>>>=+\>. <\remark> If is >-irregular, then Lemma implies that does not belong to >>> for any \|)>>. Conversely, if B>>>, then it can only be >-irregular for parameters > satisfying \|)>>. > Given the results of Sections\U, it is natural to wonder whether generic elements of >>> are \Palmost as irregular as possible\Q, in the sense of being >-irregular for any \|)>>; we provide here a positive answer. The proof requires a few simple preparations. Differently from before, in this section we will denote functions on > by >, while the letter shall be used for the canonical process associated to the measure > involved in the proofs of prevalence statements. We identify the torus > with the interval ,\|]>>, up to \\>; thus any measurable function :\\\> can be identified with :,\|]>\\> such that |)>=\|)>>. Any such > is in a 1-1 correspondence with a pair ,\|)>> a measurable functions defined on |]>>, given by :=\>, :=\>; they satisfy the constraint |)>=\|)>>. <\lemma> A measurable function :\\\> is ,\|)>>-irregular if and only if the functions ,\:|]>\\> are so. <\proof> The proof is elementary. Given ,\|]>>, setting =I\|]>>, =I\,0|]>> it holds |\|>,|\|>|}>\\2 max|\|>,|\|>|}>>, so that <\equation*> max|\>|\<\|\|\>>,\>,|\>|\<\|\|\>>,\>|}>\|\>|\<\|\|\>>,\>\2 max|\>|\<\|\|\>>,\>,|\>|\<\|\|\>>,\>|}>. Conversely, given a measurable |~>:|]>\\>, we can associate it another function =T |~>:\\\> by setting |~>=|~>|)>>, which corresponds to |~>|)>=|~>|)>=|~>>. It immediately follows from Lemma that |~>> is ,\|)>>-irregular if and only if |~>> is so; it is also easy to check that, if |~>\B>>|)>\L>|)>>, then |~>\B>>|)>>. Next, we give the following result, which is of independent interest. <\theorem> Let \>; then a.e. \B>>|)>> is >-irregular for every \|)>>. <\proof> Given \0>, define the set\ <\equation*> >=\B>>|)> :\>-irregular>|}>; it holds\ <\equation*> >=>,n,m>, with <\equation*> ,n,m>\\B>>|)> :\,\|)>>-irr. for >\=+,|\>|\<\|\|\>>,\>\m|}>. The sets ,n,m>> are closed in the topology of >>|)>> (the map \|\>|\<\|\|\>>,\>> is lower semicontinuous in the topology of |)>>), thus >> is Borel measurable. If we show that >> is prevalent in >>|)>> for any \|)>>, then the same holds for <\equation*> =\B>>|)> :\>-irregular for every >\\>|}>=>>-> providing the conclusion. Now fix \|)>> and choose > such that \>, \>; denote by > the law of fractional Brownian motion on |]>|)>> and by ,y\|]>|}>> the associated canonical process. Since > is supported on >|]>|)>> for any \0> and \>, it is also a tight probability measure on >>|)>>. Since is a -SLND process and \>, it follows from Pointa) of Proposition51 from that, for any measurable :|]>\\>, >-a.s. +u> is >-irregular. Taking \B>>|)>> we deduce that <\equation*> \+>|)>=1\\\B>>|)>, namely that > witnesses the prevalence of >> in >>|)>>. <\corollary> Let \>, then a.e. \B>>|)>> is >-irregular for any \|)>>. <\proof> The proof that the set <\equation*> \\B>>|)> :\\\\> |}> is Borel in the topology of >>|)>> is identical to that of Theorem. Now let > be a measure on >>|)>> witnessing the prevalence statement of Theorem; since in that proof we can take =\> for suitable \>, we may assume > to take values in >>|)>\L>|)>>. Therefore we can define a measure on >>|)>> by =T>\>, where |~>|)>=|~>|)>> for |]>>; also recall the notation ,\> from Lemma. For any \B>>|)>> it holds <\align*> +|)>>|B>>|)> :T u+\\\\> |}>|)>>>||B>>|)> : u+\\\\> |}>|)>>>||>>> where we used the fact that the > witnesses the prevalence of the statement of Theorem and that the intersection of sets of full measure is still of full measure. Thus > witnesses the prevalence of the set >. We are now ready to complete the <\render-proof|Proof of Theorem> The lower bound comes from Corollary, while the upper bound from a combination of Lemma and Corollary. This section contains the proof of Theorem split in several steps. Recall the setting: we want to study the asymptotic behavior of the family of complex-valued PDEs, equivalently obtain upper and lower bounds on <\equation*> |e>>|\<\|\|\>>;\|)>\L;\|)>>t\\, where >\-i k u+\\.> We show here that if has regularity of degree \>, as measured in a suitable Besov\UNikolskii scale, then the its best possible diffusion enhancing rate is |)>\\/|)>>>. The precise statement goes as follows. <\proposition> Let B>>|)>> be diffusion enhancing with rate , in the sense of Definition; then there exists a constant 0> such that <\equation*> r|)>\C \|\+2>> for all \>. In order to provide estimates for >>> it is convenient to study more generally the properties of solutions \\> to <\equation> \g+i\u g=\ \ g in function of the parameters \\>, \> and the shear flow . The proof of Proposition follows a similar strategy to and is based on deriving a Lagrangian Fluctuation-Dissipation relation (FDR) for the PDE, which is a result of independent interest. <\proposition> Let L|)>>, be a solution to with initial data \L;\|)>>; for any \\0>\\>, define the complex random variable <\equation*> Z=expu>B|)>\s|)> g>B|)> where is a standard real-valued BM. Then we have the following Lagrangian FDR: <\equation> |g|\<\|\|\>>>-|g|\<\|\|\>>>=>Var|)> \y. <\proof> Without loss of generality we can assume and > to be smooth, as identity in the general case follows from an approximation argument (the definition of > is meaningful for any L|)>> thanks to the properties of the local time of a Brownian motion). By the Feynman\UKac formula, the solution to is given by =\|]>>. Moreover since is real valued, we have the energy balance <\equation*> |g|\<\|\|\>>>-|g|\<\|\|\>>>=2\|\g|\<\|\|\>>> \s; and more generally, the map \> satisfies <\equation*> \=\ \-2\g|\|>. Now let to be a solution of h=\ \h> with initial data =|\|>>. It holds <\equation*> |\t>>-h|]> \x=-2\|\g|\<\|\|\>>>, which implies that <\equation*> |g|\<\|\|\>>>-|g|\<\|\|\>>>=2\|\g|\<\|\|\>>>=>-|\|>|]>\y. Finally, since by Feynman\UKac, =\>B|)>|\|>|]>>, we obtain <\equation*> |g|\<\|\|\>>>-|g|\<\|\|\>>>>||>>B|)>|\|>|]>-|]>|\|>|)> \x>>|||>|\|>|]>-|]>|\|>|)>\x>>>>> which gives the conclusion. <\lemma> Let \H;\|)>>, B>>|)>> for some \> and \\>. Then there exists |)>\0> such that the solution to satisfies <\equation*> |g|\<\|\|\>>>-|g|\<\|\|\>>>\C|g|\<\|\|\>>> t+|u|\>>>>|\|> \|2>> t|2>>|)>\t,\\0. <\proof> Recall the elementary identity =\|\|>|]>> for > being an i.i.d. copy of . In our setting, we take <\equation*> =expu>|)>\s|)> g>|)> where > is another BM independent of . Therefore <\align*> |g|\<\|\|\>>>-|g|\<\|\|\>>>>| >\-|\|>|]> \y>>||\>>B|)>-g>|)>|\|>\y|]>>>|||g|\<\|\|\>>>> \>u>B|)>\s>-eu>|)>\s>|\|>\y|]>.>>>> Using the inequality a>-eb>|\|>\ |\|>> and the characterization of Besov spaces in terms of finite differences (see Appendix), we deduce <\align*> |g|\<\|\|\>>>-|g|\<\|\|\>>>>|\|g+> B|)>-g+> B|)>|\<\|\|\>>>|]> >>||+ |g|\<\|\|\>>>> |\|>\>>B|)>-u>|)>|\|>\s \y|]>>>|||g|\<\|\|\>>> \-|\|>|]>+|\|>\|u+>B|)>-u+>|)>|\<\|\|\>>>|]> \s|]>>>|||g|\<\|\|\>>> t + |u|\>>>>|\|> \|2>>\-|\|>>|]> \s|]>;>>>> computing the last expectation yields the conclusion. We are now ready to complete the <\render-proof|Proof of Proposition> The proof goes along the same lines as Lemma2 from. We argue by contradiction. Assume there exists no such constant , then it must hold <\equation> liminf\0> \|\+2>> r|)>=+\. Now fix \H> with |g|\<\|\|\>>>=1>; by Definition and Lemma applied to =1> we deduce that there exist constants ,C\0> such that, for any \1> and 1>, it holds <\align*> e|)>t>>|1-|e> >|\<\|\|\>>>\1-|g|\<\|\|\>>>>>||C|g|\<\|\|\>>> t+|u|\>>>> \|2>> t|2>>|)>>>||C|g|\<\|\|\>>>|u|\>>>>|)>\|2>> t|2>>.>>>> Let \ 0> be a sequence realizing the liminf in and choose <\equation*> t=|)> \/+2|)>>|)>; then we obtain <\equation*> 1-C exp|\+2>> r|)>|)>|)>\,u>|\+2>> r|)>|)>+2|4>>. Taking the limit as \> on both sides we find 0> which is absurd. A major role in the analysis of dissipation enhancement by rough shear flows is played by the following condition, first introduced in. <\definition> We say that L> satisfies Wei's condition with parameter \0> if, setting =u\z>, it holds <\equation> \>\\,>\|]>> \-3>inf,c\\>>>>+\>-c-cy|\|> \y|]>\0. A similar definition holds for \ L|)>>; L|)>> is said to satisfy Wei's condition once it is identified with a >-periodic map on >. <\remark> Note that the condition is independent of the choice of the primitive >. Denoting by > the set of all polynomials of degree at most one, for L|)>> the definition is equivalent to <\equation*> \>=\,\1>-3>inf>-P|\|> \y|)>\0; this highlights its \Pcomplementarity\Q to the seminorm |\|\>+3>>> associated to the higher order Campanato space +3>>, as defined in. Observe that >> is homogeneous, i.e. >u|)>=\\>> for all \0>. The importance of condition comes from the following result. <\theorem> Let L|)>> be such that >\0> for some \0>. Then there exist positive constants ,C>, depending on > and >>, such that <\equation> |e>>|\<\|\|\>>\L>\C exp \|\+2>> +2>> t|)>\\\,k\\,t\0. Namely, is diffusion enhancing with rate \x/+2|)>>>, in the sense of Definition. The statement comes from Theorem5.1 from; therein is required to be continuous, but this restriction is not necessary, see Appendix for the proof. Following the same approach as in Section, we proceed to show that the condition >> implies irregularity of ; we start by relating it to the property of >-H>lder roughness, in the sense of Definition. <\lemma> Let L|)>> be such that >\0> for some \0>. Then is >H>lder rough and it holds >\ \>>. <\proof> Fix \0>, >\,\|]>>; it holds <\align*> ,c\\>>>>+\>-c-cy|\|> \y>|>>>+\>-\>|)>-\>|)>>|)>|\|>\y>>||>>>+\>>>-u>|)>|\|>\z|)>\y>>||\+3> B>>|)>>-u>|)>|\|>|\>>|)>.>>>> As the inequality holds for all > and >>, we obtain >\L>> and the conclusion. We can also relate Wei's condition to regularity in the Besov\UNikolskii scales >>>. <\lemma> Let L|)>> be such that >\0> for some \>. Then does not belong to |~>>>> for any |~>\\> and does not belong to >> for any \>. <\proof> For any >\,\|]>> and \0> it holds <\align*> +3> \>>|>>>+\>>>-u>|)>|]>\z|\|>\y>>||>>>+\>>>>+\>-u>|)>|\|>\z|)>\y>>>> thus implying that <\equation> inf>\\>>>+h|)>-u>|)>|\|>\h\\> \>\\\. Now fix |~>\\>; starting from and arguing as in the proof of Lemma (with > replaced by >), one obtains <\equation*> 2\ \>\\|~>-\>|u|\>|~>>>>, which implies the first claim by letting \0>. Integrating over >\\> yields <\equation> >|u+h|)>-u|)>|\<\|\|\>>> \h\\> \>\\\; now assume by contradiction that B>> for some \>, then by its equivalent characterization (see Appendix) and the uniform integrability of hq>|u+h|)>-u|)>|\<\|\|\>>>> it must hold <\equation> lim\0>>|u+h|)>-u|)>|\<\|\|\>>>|q>> \h=0. On the other hand, by estimate and Jensen's inequality, it holds <\align*> >|u+h|)>-u|)>|\<\|\|\>>>|q>> \h>|\q>>|u+h|)>-u|)>|\<\|\|\>>> \h>>||\|)>>>|u+h|)>-u|)>|\<\|\|\>>> \h|)>>>||\>\0>>>> uniformly in \>, contradicting. <\remark> It follows from Lemma and the construction presented Section 2 from that, for any \\> as in Lemma2.1 therein, there exists a Weierstrass-type function which belongs to >|)>>, satisfies Wei's condition with parameter > and does not belong to >> for any |]>,q\|)>>, nor to any |~>>> with |~>\\>. In light of Theorem, in order to show that almost every shear flow enhances dissipation, it will suffice to show that almost every satisfies Wei's condition. We therefore need to find sufficient conditions in order for >\0> to hold. We start with the following simple fact, whose proof is almost identical to that of Lemma. <\lemma> A map \\> satisfies >\0> if and only if the maps :|]>\\> defined by =u>, =u> do so. In this way, we can reduce the task to identifying sufficient conditions for functions defined on a standard interval |]>>. For any \0>, we denote by >> the discrete Laplacian operator >f=f|)>-2f|)>+f>. <\lemma> For any \0> and any >,\|)>> it holds <\equation> -3>inf,c>>>>+3\>-c-cy|\|>\y>>|\ >>>+\>>\|\|>>> \y|)>|)>>>>>>> \; <\proof> First observe that >+cy|)>\0> for any >, > and that for any it holds <\equation*> >>>+3\>|\|>\y\>>>+\>>f|\|> \y. Next, applying Jensen inequality for =x|)>>>>, which is convex on |)>>, it holds <\equation*> >>>>+\>>f|\|> \y|)>|)>>>\>>>>+\>>f|\|>>>\y. Algebraic manipulations of this inequality and the choice =\-c-cy> yield. In view of Lemma, given \0> and an integrable |]>\\>, we define <\equation> G>|)>:=>>>+\>>\|\|>>> \y, where > is as usual a primitive of (defined up to constant). <\lemma> For any \> and \0>, define :=\+\|)>> and <\equation*> K,\>:=sup\,1\k\2-1>2> G>k2,\2|)>. Then there exists a constant ,\|)>> such that <\equation*> \>\C ,\>|)>>. <\proof> First observe that, for any \,> <\equation*> >|\|>\>inf\,>\|]>> \-3>inf,c\\>>>>+3\>-c-cy|\|> \y so to conclude it suffices to provide a lower bound on the latter for our choice of >. Fix >,\|)>> and choose \> and ,2-1|}>> such that <\equation*> \\2,\2|]>,>\2,\k2|]> so that >,>+3\|]>\,+3|~>|]>> for the choice =\k2>, |~>=\2>. As a consequence, <\equation*> -3>inf,c\\>>>>+3\>-c-cy|\|> \y>>|\>|~>-3>inf,c\\>>+3|~>>-c-cy|\|> \y>>|\|~>-\|)>> >+|~>>>\|\|>>> \y|)>|)>>>>|=|~>> G>,|~>|)>|)>|)>>>>>>> where in the second passage we employed inequality and then the definition of >. Overall we deduce by the definition of and the choice of ,|~>|)>> that <\equation*> \-3>inf,c\\>>>>+3\>-c-cz|\|> \z\>K,\>|)>>; taking the infimum over ,y|)>> then yields the conclusion. In order to establish prevalence statements, we will sample as a suitable stochastic process. Lemma readily gives the following intermediate, general result. <\proposition> Let |]>\\> be an integrable stochastic process, =>u \s> and suppose that there exist ,\\0>, \> such that <\equation*> sup\,>\-\|]>>\ G>>,\|)>|)>|]>\\ for as defined in. Then for any \\> it holds >\0|)>=1>. <\proof> By virtue of Lemma, for =\+\|)>> it holds <\equation*> \>\0|)>\\,\>\\|)>, so to conclude it suffices to show that ,\>\\|)>=1> for all \0>. Given > as in the hypothesis, define the random variable <\equation*> J:=\>2-1>exp G>k2,\2|)>|)>. By assumption \\>, so that \|)>=1>. For any it holds <\equation*> G>k2,\2|)>\> log J|)>\ > which implies that <\equation*> Y\sup\,1\k\2-1>G>k2,\2|)>\>\\\ Finally, for any \0> it holds ,\>\>Y>, which yields the conclusion. We want to apply Proposition to a specific family of Gaussian processes; this requires a few preparations in terms of the three Lemmas- below. The next elementary lemma often appears in the probabilistic literature in connection to so called Krylov or Khasminskii type of estimates, see Lemma1.1 from for a slightly more general statement. For the sake of completeness, we give the proof. <\lemma> Let be a real valued, nonnegative stochastic process, defined on an interval ,t|]>>, adapted to a filtration |}>,t|]>>>; suppose there exists a deterministic 0> such that <\equation*> ess sup\\> \X|]>\C\s\,t|]>. Then for any \> it holds <\equation*> \|C>>>X \r|)>|]>\|)>. <\proof> Up to rescaling , we may assume . It holds <\equation*> \>>X \r|)>|]>=>|n!> \>>X \r|)>|]>=>\I where <\equation*> I=\\r\\\r\t>X>\\\X> \r\\r|]>. By the assumptions and the non-negativity of , it holds <\align*> >|\r\\\r\t>\>\\\X>>X>\r|]>\r\\r>>||\r\\\r\t>\>\\\X>\>X>\r>|]>|]>\r\\r>>|| \r\\\r\t>\>\\\X>|]>\r\\r= I>>>> which iteratively implies \1>. Therefore we obtain <\equation*> \X \u|)>|]>\>\=|)>. <\lemma> Let |)>> be a real valued Gaussian variable. Then for any \> there exists >\0> such that <\equation*> \>|]>\c> \>. <\proof> Set N+m>, then >|]>=\> \>|]>> for >; therefore is sufficed to show that <\equation*> sup\>\> |]>=sup\>> p \y=||\|>>\p|\<\|\|\>>>>\\ where stands for the Gaussian density =|)>exp/2|)>>. By Young's inequality it holds <\align*> ||\|>>\p|\<\|\|\>>>>>|||\|>>|\|>\1>|)>\p|\<\|\|\>>>> + ||\|>>|\|>\1>|)>\p|\<\|\|\>>>>>>||||\|>>|\|>\1>|\<\|\|\>>>|p|\<\|\|\>>>>+||\|>>|\|>\1>|\<\|\|\>>>> |p|\<\|\|\>>>>>|||)>||\|>>|\|>\1>|\<\|\|\>>>+ 1\\>>>> which gives the conclusion. <\lemma> Let |]>\\> be a >-SLND Gaussian process with constant >, in the sense of Definition. Then for any \H> there exists =\,H,C|)>\0> s.t. <\equation*> \>>>+\>> Y|\|>>> \y|)>|]>\2\\\,>\-\|]>. <\proof> The result follows Lemmas and applied to the process => \|\|>>>>. Indeed, denote by > the natural filtration generated by > and let \>>. It is clear that >\=Y>-2Y>+Y> is >-adapted; for any \>,>+\|]>> it holds <\equation*> Var>Y\||)>=Var>\|>|)>\C>. As a consequence, we have a decomposition >Y=Z>+Z>> with >> adapted to > and >> Gaussian and independent of >; therefore <\align*> >+\>>Y|\|>>>\y|]>>|>+\>\>+\|\|>>>|]>>|)> \y.>>>> By Lemma, since >|)>\C>> and =|)>\>, it holds <\equation*> sup\> \>+x|\|>>>|]>\> Var>|)>|)>>>\,H,C>>> and thus <\align*> >+\>>X|\|>>>\y|]>>|>+\>>> \z>>||>> \r\ C,H,C|)>>>>> where the estimate is uniform over >,>+\|]>>, >\\> and \>. Choosing <\equation*> \=,H,C|)>>, we obtain the conclusion by applying Lemma. <\corollary> Let |]>\\> be a Gaussian process such that <\equation*> Y=X \z is >SLND for some >. Then <\equation*> \>\0|)>=1 for any \H>. <\proof> It follows immediately combining Lemma and Proposition. ,> Similarly to Section, we define for |~>:|]>\\> the map |~>|)>=|~>|)>>; conversely for :\\\>, :=\>, :=\>. Recall that if |~>\B>>\L>>, then |~>\B>>>. We are now ready to provide a prevalence statement which is of interest on its own. <\theorem> Let \>; then a.e. \B>>|)>> satisfies >|)>\0> for all \\>. <\proof> Fix \> and define :=\B>>|)> :\>|)>\0\\\|}>>; it holds\ <\equation*> =>>\>>\B>>|)> :\>|)>\\=\+|}>. The sets > are closed in the topology of >>|)>> (the map \\>|)>> is upper semicontinuous in the topology of |)>>), thus > is Borel measurable. In order to conclude, it is enough to show that for any \\>, the set >:=\B>>|)> :\>|)>\0|}>> (which is Borel by the same line of argument) is prevalent. Now fix \\> and choose ,\|)>>; denote by > the law of fractional Brownian motion of parameter on |]>|)>> and by |}>|]>>> the associated canonical process. Since > is supported on >|]>|)>> for any \0> and \>, it is also a tight probability measure on >>|)>>. As recalled in Section, the associated process =>u\y> is >-SLND and so is > for any measurable |]>\\>. In particular, for a given \B>>|)>>, taking >\\y>, it follows from Corollary and the choice \H> that <\align*> +>|)>>|B>>|)> :\>|)>\0|}>|)>=1.>>>> As the reasoning holds for any \B>>|)>>, we deduce that > witnesses the prevalence of >> and we obtain the conclusion. <\corollary> Almost every \B>>|)>> satisfies >|)>\0> for all \\>. <\proof> The proof is almost identical to that of Corollary, only this time employing the measure =T>\> for |~>|)>=|~>|)>>, > being a measure witnessing the prevalence statement from Theorem; as in Corollary, we may assume > to be a tight probability measure on >>|)>\L>|)>>, so that > is tight on >>|)>>. At this point we have all the ingredient to close the dissipative case. <\render-proof|Proof of Theorem> The lower bound comes from Proposition, while the upper bound from a combination of Theorem and Corollary. The main result of the paper, Theorem, is now a direct consequence of Theorems> and.\ \; In fact, let us record here a slightly sharper estimate. Given L|)>>, for any \> define <\equation*> |f|\<\|\|\>>H>\\>|P f|\<\|\|\>>;\|)>>=|)>\\>|\|>|)> |)>|\|>; it's clear that, for 0>, |f|\<\|\|\>>H>\|f|\<\|\|\>>|)>>> and |f|\<\|\|\>>|)>>\|f|\<\|\|\>>H>>. <\theorem> Almost every B>>|)>> satisfies the following property: for any |~>\\>, there exists ,|~>,u|)>> such that, for any \H|)>> with f\0>, it holds <\equation> |e>f|\<\|\|\>>H>\C t|~>>>|f|\<\|\|\>>H> . Almost every B>>|)>> satisfies the following property: for any |~>\\> there exist =C,|~>,u|)>> such that, for any \L|)>> with f\0>, it holds <\equation> |e-\\|)>>f|\<\|\|\>>|)>>\C exp t \|~>||~>+2>>|)>|f|\<\|\|\>>|)>>. <\proof> By Theorem b), for almost every B>>|)>> and any |~>\\> it holds <\align*> |e>f|\<\|\|\>>H>>|\>|P>f|)>|\<\|\|\>>>\\>|)>|~>>>|Pf|\<\|\|\>>>\t|~>>>|f|\<\|\|\>>H>>>>> proving. Denote >=-u\+\\>, so that +-\\=>+\\>, where the operators >> and \> commute; also observe that >f|)>=e> Pf>. Combining these properties with Theorem, for almost every B>>|)>> and any |~>\\> it holds <\align*> |e+\\|)>>f|\<\|\|\>>>>|\>|P> e>>f|)>|\<\|\|\>>>=\>e>|P>>f|)>|\<\|\|\>>>>>||\>exp-C t \|\+2>>+2>>|)>|P f|\<\|\|\>>>>>||exp|\+2>>|)>\>|P f|\<\|\|\>>>>>>> which yields. 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 \> in the Besov scale >>>. In the enhanced dissipation case, this confirms the intuition from; 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 >-irregularity, which was never observed in this context, and highlight the importance of working with mixing scales > with 1>. \; We conclude by presenting a few additional remarks and open problems arising from this work. <\enumerate-numeric> We are currently unable to determine whether there is a clear connection between the properties of >-irregularity and Wei's condition. Lemma, together with the trivial estimate |f|\<\|\|\>>>\|f|\<\|\|\>>>>, imply that for \> the shear flows C>> constructed in satisfy >\0> but are not >-irregular with \|)>>. Heuristically, this fact is similar to the existence of flows with small dissipation time which are not mixing, like the cellular flows presented in. The above argument also implies the existence of Weierstrass type functions which are not >-irregular, for suitable values >. We believe this problem was open in the probabilistic community, although never been explicitly addressed in the literature. Even without establishing a direct connection to Wei's condition, it would be interesting to show that functions which are >-irregular for \|)>> are diffusion enhancing with rate |)>\\/+2|)>>>, in line with the heuristic argument presenting in the introduction. Since such are mixing, they are indeed diffusion enhancing with a suitable rate by; however the quantitative results from only imply the worsened rate /|)>>> and it is rather unclear how to \Pbridge the gap\Q. Going through the same proof as in Lemma, one can show that if is ,\|)>>-irregular with \1-s>, then is mixing on the scale > with rate =t>>. In the case =0> an even simpler proof based on duality and integration by parts provides mixing on the scale > with the same rate. In fact, since ;\|)>> is an algebra, by integration by parts it holds <\equation*> eu> f,g\|\|>>|>>eu> fg\y|\|>>>|||)>|\|>>>eu>\y|\|>>>||+>>|\|>>e u>\z |\|>\y>>|||f g|\<\|\|\>>>>+||\<\|\|\>>>|)> |\|\<\|\|\>>> |\|>>>>|||f|\<\|\|\>>>|g|\<\|\|\>>>|\|\<\|\|\>>> |\|>>>>>>> which by duality implies |eu> f|\<\|\|\>>>\|f|\<\|\|\>>>|\|\<\|\|\>>> |\|>>> and so the claim. It is however an open problem to provide examples of |)>>-irregular functions , for any \1>. See Remark69 in for a deeper discussion. There are several examples of |]>\\> which are >-irregular, including the choice =y>; by Proposition1.4 from, it is enough to require the existence of \0> such that <\equation> >\|\|>,|\|>||\|>>\\\y\|]>; observe the similarity of condition with Assumption(H) from. The property of ,\|)>>-irregularity can be reformulated in terms of the (Fourier transform of) of , namely the family ,I\\|}>> given by =u>> where are subintervals of > and > stands for the Lebesgue measure on ; see Section2.3 from for more details. Closely related to the occupation measure of is its , namely the Radon\UNikodym derivative \>/\>>, which has been intensively studied for stochastic processes, see the review. The following question arises naturally: is it possible to link the mixing properties of to the regularity of its local time? In the paper we have always focused on the scales >>|)>> with \>. If one is instead interested in the mixing properties of generic C|)>>, much faster rates are available. Indeed for any \1> it's possible to construct >\C|]>|)>> satisfying <\equation> >>eu>>\z|\|>\,\>-y|\|>> exp,\> |\|>>>|)>\\\\,0\y\y\\ and so by symmetrization the same holds for >\T >>. Such >> are given by typical realization of the so called |)>>-log Brownian motion, see for its definition and Propositions48 and49 from for the proof of. In fact, one could use the law of such process to prove that a.e. C|)>> satisfies for any \1> (the value =1> can only be attained by Caratheodory functions, which are naturally discontinuous). Arguing as in the proof of Lemma one can deduce that such are exponentially mixing, in the sense that they satisfy the estimate <\equation> |e u> g|\<\|\|\>>>\exp,\> |\|>>>|)> |g|\<\|\|\>>>\g\H;\|)> and so that <\equation> |e>f|\<\|\|\>>H>\exp,\> t>>|)>|f|\<\|\|\>>H> for all H|)>> satisfying f\0>. Finally, let us point out that the property of >-irregularity also holds for generic vector valued functions \\> (resp. \\>), for any \>, see. In particular, similar statements to part of Theorem can be established for \Phigher dimensional\Q shear flows of the form <\equation*> \f+>\\f=\\f for \\>, >,\,x|)>\|)>,0|)>>; observe that for , >> is a stationary solution to D Euler equations. In light of, the vector field >> constructed by a >-irregular is diffusion enhancing; thus can be applied in the study of suppression of blow-up by mixing phenomena similarly to what was done in. In this appendix we record fundamentals on Besov spaces > on the torus >, although in the paper we only need the case . For a gentle introduction on spaces on > we refer to the monograph; see also the classical paper for spaces on an interval \>. All their properties transfer to the analogous spaces on > by a clever use of Poisson summation formula, see,. Alternatively, periodic Besov spaces have been treated in Chapter3 of. \; Given a dyadic partition of the unity ,\|)>> and the associated Littlewood\UPaley blocks |}>-1>>, the (inhomogeneous) Besov spaces |)>> with \>, are defined as the set of tempered distributions |)>> such that <\equation*> |f|\<\|\|\>>>\|2 |\f|\<\|\|\>>>|\<\|\|\>>>=>2 |\f|\<\|\|\>>>|)>\\ with the usual conventions when >. |)>,|\|\<\|\|\>>>|)>> are Banach spaces and the definition is independent of the choice of the partition of unity ,\|)>>. 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> , Prop. 2.71>Let p\p\\> and q\q\\>. Then for any \>, the space ,q>> continuously embeds in >->|)>>,q>>. Also recall the following basic facts: for any \0> and any |]>>, the space > continuously embeds in >>, as can be checked using the definition; for any |]>>, we have the embeddings <\equation*> B\L\B> see for instance RemarkA.3 from for the second statement. <\proposition> , Prop. A.5>Let \>, |]>>, ,n|}>>. Then the map \f> is a continuous linear operator from > to >. <\proposition> , Prop. A.7>Let ,s\\> and ,p,q\|]>> be such that <\equation*> s\0\s,s+s\0,=>+>; then \f g> is a well-defined continuous bilinear map from >,q>\B>,q>> to >>. <\proposition> , Cor. 2.86>For any 0> and |]>>, the space \L>> is an algebra and there exists a constant > such that <\equation*> |f g|\<\|\|\>>>\C|f|\<\|\|\>>>>|g|\<\|\|\>>>+|f|\<\|\|\>>>|g|\<\|\|\>>>>|)>\f,g\B\L>. Another key property of Besov spaces is that they include several other classical function spaces: <\itemize> for \>, |)>> coincide with the Sobolev spaces |)>>, with equivalent norms; for >, ,\>|)>> coincide with |)>>, the space of periodic -H>lder continuous functions (w.r.t. the canonical distance >>), with equivalent norms; for > and |)>>, the spaces >|)>>, often referred to as Besov\UNikolskii spaces, can be characterized by the equivalent norm <\equation*> \<\|\|\>f|~>>>\|f|\<\|\|\>>>+supy\\>|f+x|)>-f+y|)>|\<\|\|\>>>|d>>. for >, |)>> the space |)>> has equivalent norm <\equation*> \<\|\|\>f|~>>\|f|\<\|\|\>>>+>|f+ x|)>-f|)>|\<\|\|\>>>|d>>|)> >> \x. \; We conclude this appendix by proving some interpolation inequalities, which played a fundamental role in the proofs in Section. <\lemma> Let |]>>, ,s\\> with \s> and \>. Then there exists a constant -s,\|)>> such that <\equation> |f|\<\|\|\>>s+|)>s>>\C |f|\<\|\|\>>>>>> |f|\<\|\|\>>>>>> \f\B>>. <\proof> The result is well known, see Theorem 2.80 from for the statement on >; let us provide a self-contained proof. We may assume |f|\<\|\|\>>>>>=1>; for any 0> it holds <\align*> |f|\<\|\|\>>>>>>|N>2s+|)>s|)>>|\f|\<\|\|\>>>+N>2s+|)>s|)>>|\f|\<\|\|\>>>>>|||f|\<\|\|\>>>>>N>2|)>-s|)>>+|f|\<\|\|\>>>>>N>2-s|)>>>>|||f|\<\|\|\>>>>>2|)>-s|)>>+2-s|)>>.>>>> Choosing such that |f|\<\|\|\>>>>>\2-s|)>>> the conclusion then follows. <\corollary> For any ,s\0> there exists a constant ,s|)>> such that <\equation> |f|\<\|\|\>>>\C |f|\<\|\|\>>>>/+s|)>> |f|\<\|\|\>>>>>/+s|)>>\f\B>>. <\proof> Applying Lemma for the choice , =s/-s|)>> and using Besov embeddings we find <\equation*> |f|\<\|\|\>>>\|f|\<\|\|\>>>\|f|\<\|\|\>>>>>> |f|\<\|\|\>>>>>>\|f|\<\|\|\>>>>>|f|\<\|\|\>>>>>>. Theorem5.1 from requires the restriction to work with C|)>>, but we show here that such a restriction is not necessary and in fact the result holds for any L|)>>, as stated in Theorem. Let us recall the setting: we are interested in the decay of solutions to complex valued PDEs of the form <\equation> \f+i u f=\ \f. Equation is well-posed (in the weak sense) for any L|)>> and \L|)>>. Indeed, for smooth , any solution to satisfies <\equation*> \|f|\<\|\|\>>>+2\|\f|\<\|\|\>>>=0, thus implying that it belongs to |)>|)>\L|)>|)>>; therefore we have uniform estimates for L|)>|)>> only depending on |u|\<\|\|\>>>>. Arguing by weak compactness one can then easily construct weak solutions to for any L|)>>, establish their uniqueness, and show that they are the strong limit in ;L|)>|)>> of those to smooth . Overall, this defines the semigroup e \-i u|)>>> on |)>> for any L|)>> and \0>. \; Identifying L|)>> with a >-periodic function, its primitive > is a (non periodic) element of |)>>, well defined up to additive constant; for given \>, define <\equation*> \,u|)>:=inf,c\\>>>-c-c \|\|>\y. Denote by 0>\/2|]>> the inverse of 36 x tan x>, which is a one-to-one increasing function. The next statement summarizes some of the main findings from. <\lemma> Let C|)>> and \0> be fixed; then for all \> and 0> it holds <\equation> |e-i u|)>>|\<\|\|\>>\L>\exp|2>-t \ \F \,u|)>|)>|)>|)>. <\proof> By time rescaling, the solution to is given by =f>,y|)>> where >> solves f>+i u>f>=\f>> where >=u/\>; applying the Gearhart\UPr>ss theorem (Theorem1.3 from) to >>, it holds <\equation*> |f|\<\|\|\>>>\exp|2>-t \ \>|)>|)> where > is defined as in Section4 from. By Lemma4.3 therein and the 2-homogeneity of \,u|)>>, for any \> it holds <\equation*> \>|)>\\ F,u>|)>|)>|)>=\ F \,u|)>|)>|)> which gives the conclusion. We can now give the <\render-proof|Proof of Theorem> By time rescaling, we can restrict to the case . Now let L|)>> be a function satisfying >\0> for some \> and consider a family >,\\0|}>> of continuous functions satisfying |u>-u|\<\|\|\>>>\\>. Denote by >> the primitive of >>; by the basic inequality \b/2->, for any > it holds <\equation*> ,u>|)>>| \,u|)>-sup\>>>>-\|\|> \y>>|| \,u|)>-2\|u-u>|\<\|\|\>>|)>>.>>>>> Combined with the fact that by definition ,u|)>\2+3> \+3> \>>, we deduce <\equation> \,u>|)>\2+2> \+3> \> -2 \ \\\\,\\0. Now fix \0> and define =e/2>>, =2+2> \>>; applying Lemma to >>, exploiting the fact that is increasing, and choosing =\+2|)>>>, we obtain <\equation> |e-i u>|)>>|\<\|\|\>>\L>\C exp|\+2>> F-2 \+1|\+2>> \|)>|)>\t\0 where the estimate holds for all \0> small enough such that -2 \+1|)>/+2|)>> \\0>. Since the semigroup \-i u>|)>>> pointwise converges to \-i u>|)>>> as \0>, passing to the limit on both sides of gives the conclusion. <\bibliography|bib|tm-plain|myBiblio> <\bib-list|44> Hajer Bahouri, Jean-Yves CheminRaphaël Danchin. , 343. Springer Science & Business Media, 2011. Jacob BedrossianMichele Coti Zelati. Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. , 224(3):1161\U1204, 2017. Jacob BedrossianSiming He. Suppression of Blow-Up in Patlak\UKeller\USegel Via Shear Flows. , 49(6):4722\U4766, 2017. Sergio Campanato. Proprietà di una famiglia di spazi funzionali. , 18(1):137\U160, 1964. Rémi Catellier. Rough linear transport equation with an irregular drift. , 4(3):477\U534, 2016. Rémi CatellierMassimiliano Gubinelli. Averaging along irregular curves and regularisation of ODEs. , 126(8):2323\U2366, 2016. K.ChoukM.Gubinelli. Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations. , 40(11):2047\U2081, 2015. 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Properties of local-nondeterminism of Gaussian and stable random fields and their applications. , 15, 157\U193. 2006. MicheleCoti ZelatiTheodoreD.Drivas. A stochastic approach to enhanced diffusion. , 2019. \; <\initial> <\collection> <\attachments> <\collection> <\associate|bib-bibliography> <\db-entry|+1CLJhZmi1u3N0Sa9|article|kelvin> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0Sao|article|wei> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0Sah|article|colombozelati> <|db-entry> Michele Coti Klaus > <\db-entry|+1CLJhZmi1u3N0Sal|article|huntsauer> <|db-entry> Tim James A. > <\db-entry|+1CLJhZmi1u3N0SaO|article|zelati2020relation> <|db-entry> Matias G. Tarek M. > <\db-entry|+1CLJhZmi1u3N0Sa7|article|reynolds> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaH|article|constantin2008> <|db-entry> Alexander Lenya Andrej > <\db-entry|+1CLJhZmi1u3N0Sa8|article|thiffeault> <|db-entry> Jean-Luc Charles R. > <\db-entry|+1CLJhZmi1u3N0SaL|article|bedrossian2017enhanced> <|db-entry> Michele > <\db-entry|+1CLJhZmi1u3N0SaN|article|zelati2020stable> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaM|article|zelati2020separation> <|db-entry> Michele > >> <\db-entry|+1CLJhZmi1u3N0Sap|article|zelatidrivas> <|db-entry> Theodore D. > <\db-entry|+1CLJhZmi1u3N0Saq|article|drivas1> <|db-entry> Gregory L. > <\db-entry|+1CLJhZmi1u3N0Sar|article|drivas2> <|db-entry> Gregory L. > <\db-entry|+1CLJhZmi1u3N0Sac|article|liu2020> <|db-entry> David Josef > <\db-entry|+1CLJhZmi1u3N0Sab|article|friz2021> <|db-entry> Benjamin > <\db-entry|+1CLJhZmi1u3N0SaD|article|frisch1980> <|db-entry> Giorgio > <\db-entry|+1CLJhZmi1u3N0SaE|article|jaffard2000> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaG|article|fraysse2006> <|db-entry> Stéphane > <\db-entry|+1CLJhZmi1u3N0SaF|article|fraysse2010> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaC|book|nualart> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaS|article|ciesielski1993> <|db-entry> Gérard Bernard > <\db-entry|+1CLJhZmi1u3N0SaR|article|veraar2009> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaB|inproceedings|xiao> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0Saj|misc|galeati2020prevalence> <|db-entry> Massimiliano > <\db-entry|+1CLJhZmi1u3N0SaZ|article|catellier2016averaging> <|db-entry> Massimiliano > <\db-entry|+1CLJhZmi1u3N0Saf|article|choukgubinelli1> <|db-entry> M. > <\db-entry|+1CLJhZmi1u3N0Sag|article|choukgubinelli2> <|db-entry> Massimiliano > <\db-entry|+1CLJhZmi1u3N0SaX|article|catellier2016rough> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0Saa|book|frizhairer> <|db-entry> Martin > <\db-entry|+1CLJhZmi1u3N0SaQ|article|campanato1964> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaP|book|portenko1990> <|db-entry> > <\db-entry|+1CLJhZmi1u3N0SaI|article|iyer2021> <|db-entry> Xiaoqian Andrej > <\db-entry|+1CLJhZmi1u3N0Sak|article|geman> <|db-entry> Joseph > <\db-entry|+1CLJhZmi1u3N0SaA|article|perkowski> <|db-entry> Nicolas > >> regularization of odes perturbed by noise> <\db-entry|+1CLJhZmi1u3N0SaJ|article|kiselev2016> <|db-entry> Xiaoqian > <\db-entry|+1CLJhZmi1u3N0SaK|article|bedrossian2017suppression> <|db-entry> Siming > <\db-entry|+1CLJhZmi1u3N0SaW|book|bahouri> <|db-entry> Jean-Yves Raphaël > <\db-entry|+1CLJhZmi1u3N0SaV|article|gubinelli2015lectures> <|db-entry> Nicolas > <\db-entry|+1CLJhZmi1u3N0SaU|article|mourrat> <|db-entry> Hendrik > > Model Comes Down from Infinity> <\db-entry|+1CLJhZmi1u3N0SaT|book|schmeisser1987> <|db-entry> Hans > <\db-entry|+1CLJhZmi1u3N0Saw|misc|TeXmacs:website> <|db-entry> > > <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> TeXmacs:website kelvin wei colombozelati colombozelati huntsauer colombozelati zelati2020relation zelati2020relation kelvin reynolds constantin2008 zelati2020relation fengiyer thiffeault bedrossian2017enhanced zelati2020stable zelati2020separation wei zelatidrivas bedrossian2017enhanced wei zelatidrivas drivas1 drivas2 zelati2020stable colombozelati colombozelati wei colombozelati wei colombozelati zelatidrivas liu2020 friz2021 frisch1980 frisch1980 jaffard2000 fraysse2006 fraysse2010 wei huntsauer huntsauer huntsauer nualart ciesielski1993 veraar2009 xiao galeati2020prevalence galeati2020prevalence colombozelati colombozelati catellier2016averaging choukgubinelli1 choukgubinelli2 galeati2020prevalence catellier2016rough frizhairer galeati2020prevalence galeati2020prevalence galeati2020prevalence galeati2020prevalence zelatidrivas zelatidrivas wei campanato1964 wei colombozelati portenko1990 colombozelati colombozelati iyer2021 constantin2008 zelati2020relation galeati2020prevalence choukgubinelli1 zelati2020stable galeati2020prevalence geman perkowski galeati2020prevalence galeati2020prevalence constantin2008 kiselev2016 bedrossian2017suppression iyer2021 bahouri simon gubinelli2015lectures mourrat schmeisser1987 bahouri mourrat mourrat mourrat bahouri bahouri wei wei wei wei <\associate|toc> |math-font-series||1.Introduction> |.>>>>|> |math-font-series||2.Preliminaries> |.>>>>|> |2.1.Prevalence |.>>>>|> > |2.2.Fractional Brownian motion and local nondeterminism |.>>>>|> > |math-font-series||3.Inviscid mixing> |.>>>>|> |3.1.Lower bounds in terms of regularity |.>>>>|> > |3.2.Upper bounds in terms of |\>-irregularity |.>>>>|> > |3.3.Prevalence statements and proof of Theorem |->|-0.3em|>|0em||0em|>> |.>>>>|> > |math-font-series||4.Enhanced dissipation> |.>>>>|> |4.1.Lower bounds in terms of regularity |.>>>>|> > |4.2.Wei's irregularity condition |.>>>>|> > |4.3.Sufficient conditions for stochastic processes |.>>>>|> > |4.4.Prevalence statements and proof of Theorems |->|-0.3em|>|0em||0em|>>, |->|-0.3em|>|0em||0em|>> |.>>>>|> > |math-font-series||5.Further comments and future directions> |.>>>>|> |math-font-series||Appendix A.Besov spaces> |.>>>>|> |math-font-series||Appendix B.A simple extension of a result by Wei> |.>>>>|> |math-font-series||Bibliography> |.>>>>|>