Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field
Carl Johan Peter Johansson Massimo Sorella

Triple trouble with PSR J1618-3921: Mass measurements and orbital dynamics of an eccentric millisecond pulsar
    K. Grunthal 1kgrunthal@mpifr-bonn.mpg.de
          V. Venkatraman Krishnan 1
          P. C. C. Freire 1
          M. Kramer 1
          M. Bailes 8,9
          S. Buchner 7
          M. Burgay 5
          A. D. Cameron 8,9
          C.-H.R. Chen 1
          I. Cognard 2,3
          L. Guillemot 2,3
          M. E. Lower 6
          A. Possenti 5
          G. Theureau 2,3,4
          
    September 9, 2024
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§ ABSTRACT 
We study anomalous dissipation in the context of passive scalars and we   construct a two-dimensional autonomous divergence-free velocity field in C^α (with α∈ (0,1) arbitrary but fixed) which exhibits anomalous dissipation. Our proof employs the fluctuation-dissipation formula, which links spontaneous stochasticity with anomalous dissipation. Therefore, we address the issue of anomalous dissipation by showing that the variance of stochastic trajectories, in the zero noise limit, remains positive.
Based on this result, we answer <cit.> regarding anomalous dissipation for the forced three-dimensional Navier–Stokes equations.






§ INTRODUCTION

We study the evolution of a passive scalar advected by a two-dimensional divergence-free velocity field. More precisely, in the two-dimensional torus  ^2 ≅^2/ ^2, given  a velocity field u : [0,1] ×^2 →^2, we consider the Cauchy problem of the advection-diffusion equation 

    ∂_t θ_κ + u ·∇θ_κ = κΔθ_κ; ADV-DIFF
    θ_κ (0, · ) = θ_ (· ) ∈ L^∞(^d);

where the unknown is the scalar  θ_κ : [0,1] ×^2 →, κ≥ 0 is the diffusivity parameter and θ_ : ^2 → is a given bounded initial datum. If κ =0 this equation reduces to the well-known advection equation (also called transport equation). Supposing that u ∈ L^2((0,1) ×^2; ^2) is divergence-free, there exists a unique solution θ_κ∈ L^∞ ((0,1) ×^2) ∩ L^2((0,1); H^1(^2)), and this solution satisfies the energy equality 

    E1/2∫_^2 |θ_κ (t, x)|^2 dx + κ∫_0^t ∫_^2 | ∇θ_κ (s, x)|^2 dx ds = 1/2∫_^2 |θ_(x)|^2 dx  ,     for  a.e.  t ∈ (0,1)  .

We say that the collection of solutions
{θ_κ}_κ∈ (0,1) exhibits anomalous dissipation if

    lim sup_κ→ 0κ∫_0^1 ∫_^2 | ∇θ_κ (s,x)|^2 dx ds  >0  , AD
 
which directly implies the existence of dissipative
 vanishing diffusivity solutions to the advection equation.
  The main contribution of this paper is the following: 

Let α∈ [0,1) be arbitrary. Then there exists an autonomous and divergence-free velocity field u = ∇^⊥ H ∈ C^α (^2; ^2)  and an initial datum θ_∈ C^∞ (^2) such that  the sequence of unique solutions {θ_κ}_κ >0  to (<ref>) exhibits  anomalous dissipation.
Furthermore, up to not relabelled subsequences, we have 

    κ∫_^2 |∇θ_κ (·, x) |^2 dx *⇀ g ∈ L^∞ ((0,1))  ,

where the convergence is weak* L^∞. Finally, for any converging subsequence θ_κ_n⇀^*θ as κ_n → 0, we have 

    ∫_^2 |θ (· , x )|^2 dx ∈ W^1, ∞ (0,1)    .



 For two-dimensional autonomous velocity fields u = ∇^⊥ H ∈ L^∞(^2), anomalous dissipation cannot occur for all initial data, namely there exists an initial datum θ_∈ W^1, ∞(^2) such that for the corresponding solutions to the advection-diffusion equation θ_κ, we have
 
    lim_κ→ 0κ∫_0^T∫_^2 |∇θ_κ|^2   dx dt = 0   for any fixed T > 0.

Consider θ_ = H and notice that
 
    ∂_t (θ_κ - H) + u ·∇ (θ_κ - H) = κΔθ_κ.

 Multiplying by (θ_κ - H) and integrating in space-time (mollifying and passing to the limit) we find
 
    (θ_κ - H)(T, ·)  _L^2(^2) + 2 κ∫_0^T ∫_^2 |∇θ_κ|^2   dx dt    = 2 κ∫_0^T ∫_^2∇θ_κ·∇ H   dx dt 
       ≤κ∫_0^T ∫_^2 |∇θ_κ|^2   dx dt + κ∫_0^T ∫_^2 |∇ H|^2   dx dt

 from which we conclude (<ref>).

 
For any autonomous, divergence-free velocity field u = ∇^⊥ H ∈ C^0(^2) we say that it satisfies the weak Sard property if and only if 

    ℒ^1 (H (S)) =0,   where   S = {x ∈^2 : ∇ H =0 }.

This is a necessary and sufficient condition for uniqueness of solutions at the level of the advection equation, as proved by Alberti, Bianchini and Crippa in <cit.>. We notice that the autonomous velocity field in Theorem <ref> does not satisfy the weak Sard property. 
Indeed, supposing by contradiction that u satisfies the weak Sard property we find that solutions to the advection equation are unique and hence contradict a result by Rowan <cit.>.


The velocity field u is a steady solution of the two-dimensional Euler equations with a body force, more precisely it follows from (<ref>) and the Leray projector operator that there exists a p ∈ C^1, α (^2) and  a divergence free force F ∈ C^α (^2; ^2)[The α∈ (0,1) is the one in the statement of the Theorem <ref>.] such that 
 
    u ·∇ u + ∇ p = F  .



The problem of anomalous dissipation with autonomous velocity fields was posed by Elgindi and Liss in <cit.> as a problem “of great mathematical, and possibly physical, interest”. Indeed, the advection-diffusion equation in two dimensions with autonomous velocity fields enjoys more rigidity and constraints with respect to the case of two-dimensional time-dependent velocity fields or autonomous velocity fields in higher dimensions. We now give a swift outline of some aspects of this rigidity.

 
  * For two-dimensional time-dependent Lipschitz velocity fields it has been proved by Elgindi, Liss and Mattingly in <cit.> that the optimal enhanced diffusion rate is |lnκ|  whereas in the two-dimensional autonomous case it is κ^-1/3, see the result by Bruè, Coti Zelati and Marconi <cit.>.
 
  * At the level of the advection equation, the mixing scale with time-dependent Lipschitz velocity fields can be exponential-in-time, see the first example by Alberti, Crippa and Mazzucato  <cit.>, whereas in the two-dimensional autonomous case it is at most linear-in-time <cit.> as proved by Bonicatto and Marconi.
 
  * Uniqueness of solutions to the advection equation with two-dimensional autonomous velocity fields is equivalent to having the velocity field satisfying the weak Sard property, see <cit.> by Alberti, Bianchini and Crippa. For time-dependent or autonomous velocity fields in higher dimensions stronger regularity assumptions are required, see for instance <cit.>. 
 
  * Regarding anomalous dissipation, a consequence of the result in <cit.> is that anomalous dissipation does not occur if the velocity field is continuous, divergence-free, autonomous and nowhere vanishing, see <cit.>. This fact is known to be false in the case of time-dependent velocity fields and autonomous velocity fields in higher dimensions <cit.>.


To tackle the two-dimensional autonomous case, the novelty of the present manuscript is twofold.
Firstly, we introduce a new mathematical approach (criterion) relying on ideas from spontaneous stochasticity, a concept that was introduced in <cit.> and, more recently, further developed by Drivas and Eyink in <cit.>. In <cit.>, they introduce the fluctuation-dissipation formula (<ref>) which provides a link between anomalous dissipation and spontaneous stochasticity.
Secondly, we construct a two-dimensional velocity field which exhibits anomalous dissipation. 
Specifically, we construct a velocity field to which the already mentioned criterion can be applied. 
This velocity field cannot satisfy the weak Sard property as noticed in Remark <ref>. 
Examples of velocity fields which do not satisfy the weak Sard property are known <cit.> but it is not clear whether any of those exhibit anomalous dissipation. 
Among the available examples of velocity fields which do not satisfy the weak Sard property in the literature, the one that inspired us the most is the non-divergence free velocity field given in <cit.>.

The anomalous dissipation phenomenon presented in our example is  new and distinct from known techniques, which are detailed at the end of this introduction. We consider an autonomous velocity field with the following property: the forward flow of a smoothed approximation of this field tends to push a certain portion of the torus into a fat Cantor set with positive Lebesgue measure.

The stochastic flow is initially governed by the transport term, up to a certain small scale where diffusion begins to dominate, eventually spreading the flow across the entire fat Cantor set. This “dispersive” phenomenon of forward stochastic trajectories is responsible for the anomalous dissipation. To mathematically prove this, it is more effective to study the backward stochastic flow.
In the context of the backward stochastic flow, this  phenomenon appears as a highly unstable behavior in response to small perturbations (as a small noise), reminiscent of the concept of spontaneous stochasticity, which we discuss in Section <ref>. This perspective is more advantageous for proving anomalous dissipation. Specifically, we will show that backward stochastic trajectories originating from the fat Cantor set can exhibit significantly different behaviors depending on the realization of the Brownian motion. According to the fluctuation-dissipation formula (<ref>), this variability implies anomalous dissipation, as demonstrated by the criterion provided in Proposition <ref>. Finally, we note that the dissipation caused by this phenomenon is continuous over time, due to the autonomous nature of the velocity field, as detailed in Proposition <ref>.




A central motivation for the study of anomalous dissipation comes from fluid dynamics. 
To be precise, we aim at understanding how the cascade of frequencies arises due to a transport term. 
In the incompressible Euler and Navier–Stokes equations, the nonlinear term takes the shape of a transport term and hence understanding anomalous dissipation at the linear level, i.e (<ref>) could allow us to understand how the cascade of frequencies arises due to the nonlinear term in the incompressible Euler and Navier–Stokes equations. 
In recent years, this has been made rigorous is several situations. We refer to <cit.>, in which the authors study the three-dimensional Euler and Navier–Stokes equations via the so-called (2 + 1/2)-dimensional Euler and Navier–Stokes equations. To explain our second main result, we consider the three-dimensional forced Navier–Stokes equations  with a given time independent  force F_ν : ^3 →^3

    ∂_t v_ν + v_ν·∇ v_ν + ∇ P_ν = νΔ v_ν + F_ν ,
    
     (v_ν) =0  ,
    
    
    v_ν (0, · ) = v_, ν (·)  ,

where the unknowns are the velocity v_ν : [0,1] ×^3 →^3 and the pressure P_ν : ^3 →.[If ν =0 these are the 3D forced Euler equations.]
Assuming v_ν = (u_ν^(1), u_ν^(2), θ_ν) depends only on the two first spatial components, the third component of the equations reduces to an advection-diffusion equation with velocity (u_ν^(1), u_ν^(2)). With this assumption, (<ref>) are known as (2 + 1/2)-dimensional Navier–Stokes equations.
Theorem <ref>, together with some key regularity properties of our two-dimensional autonomous velocity field (see Lemma <ref>) yield the result below by considering this setting. This result answers two open questions by Bruè and De Lellis posed in <cit.> for the three-dimensional forced Navier–Stokes equations. Indeed, in Section <ref> we prove the following statement.
 
For any α∈ (0,1), there exist a sequence of viscosity parameters {ν_q }_q ∈,  a sequence of time independent smooth forces  { F_ν_q }_q ∈⊂ C^∞ ( ^3 ) and a sequence of smooth initial data { v_, q}_q ∈⊂ C^∞ ( ^3 ) such that for any q ≥ 1 there exists a unique solution v_ν_q : [0,1] ×^3 →^3 to (<ref>) and it enjoys the following properties: 


  * F_ν_q -  F_0 _C^α +  v_, ν_q - v__C^α→ 0 as ν_q → 0 for some force F_0 ∈ C^α and initial datum v_∈ C^α.

  * Anomalous dissipation holds, i.e. 
    lim sup_ν_q → 0ν_q ∫_0^1 ∫_^3 | ∇ v_ν_q (t, x)|^2 dx dt >0  ,


  * There exists v_0 ∈ L^∞ such that  up to not relabelled subsequences

    v_ν_q*-L^∞⇀ v_0  ∈ L^∞

and v_0 is a solution to the 3D forced Euler equations with  v_ and force F_0 and finally e(t) = 1/2∫_^3  |v_0 (t, x)|^2 dx ∈ W^1, ∞ (0,1).


We now provide a summary of existing works and techniques in the study of anomalous dissipation. 
For the sake of clarity and brevity, we restrict our attention only to advection-diffusion equations.
Subsequently, we go through the techniques while pointing out the challenges that these techniques present in the two-dimensional autonomous case. This motivates and justifies the introduction of the new approach presented in this work. 
The first and pioneering result of anomalous dissipation was given in <cit.> by Drivas, Elgindi, Iyer and Jeong, where the authors use a mixing velocity field in L^1_t C^α_x, with α∈ (0,1) fixed but arbitrary, for which bounded solutions, with initial data close to an eigenfunction of the Laplacian, exhibit anomalous dissipation.
This study attracted the attention of many subsequent mathematical investigations. In <cit.>, Colombo, Crippa and the second author of the present manuscript construct a new mixing velocity field to prove anomalous dissipation in any supercritical Yaglom's regime. More precisely, for any fixed 

    α + 2 β < 1 YAG

the authors construct a velocity field u ∈ L^∞((0,1); C^α (^2)) and an initial datum for which the corresponding solutions {θ_κ}_κ∈ (0,1) exhibit anomalous dissipation and enjoy the regularity 

    sup_κ∈ (0,1)θ_κ_L^2 ((0,1); C^β (^2)) < ∞.

In <cit.>, Armstrong and Vicol construct a velocity field in L^∞ ((0,1); C^α (^2)) with α < 1/3 and prove via a  striking  technique known as quantitative homogenization  that corresponding solutions exhibit anomalous dissipation for all non-constant initial data in H^1(^2). 
Furthermore, dissipation in this scenario occurs continuously in time, as predicted by the theory of scalar turbulence. In constructing this velocity field, at a fixed point in time, all frequencies are activated, and the singular set of this velocity field spans the full space-time dimension in (0,1) ×𝕋^2. Inspired by this construction, in <cit.> Burczak, Székelyhidi and Wu  combined convex integration and quantitative homogenization theory to construct a dense set of solutions to the three-dimensional Euler equations in C^0 for which the corresponding solutions to the advection-diffusion equation exhibit anomalous dissipation for any non-constant initial data in H^1(^3).
In <cit.>, Elgindi and Liss prove anomalous dissipation for any non-constant smooth initial data in any supercritical Yaglom's regime. The construction of their velocity field is a rescaled-in-time and in-space version of the one in <cit.> by Elgindi, Liss and Mattingly. Finally, in the Kraichnan model <cit.>, where the velocity field is a Gaussian random field which is white-in-time and rough-in-space (only Hölder continuous), anomalous dissipation has been proved by Bernard, Gawedzki and Kupiainen <cit.>, see also <cit.>. Recently, Rowan <cit.> provided a PDE-based proof of anomalous dissipation in the Kraichnan model.
Now we go through the techniques mentioned in the summary above and for each technique point out the challenges to overcome in order to apply them in the autonomous two-dimensional setting.

 Balanced growth of norms. The works based on balanced growth of Sobolev norms <cit.> construct smooth velocity fields  which satisfy
 
    θ_0 (t,· ) _H^1^σ∼θ_0 (t,· ) _L^2^σ - 1θ_0 (t, · ) _H^σ    ∀ t ∈ (0,1)   and ∫_0^1 ∇θ_0 (t, ·) _L^2^2 dt = ∞

 for some σ∈ (1,2],
where θ_0 denotes the unique solution to the advection equation.
Such a condition is particularly well-suited for self-similar constructions and alternating shear flows which are smooth for any time less than a fixed singular time T. 
However, solutions to the advection equation may become non-unique with a C^1- divergence-free velocity fields in general and the condition (<ref>) must be adapted accordingly. This means that the conditions must be specified for the solution to the advection equation with a regularized velocity field. If the regularization is sufficiently small (for instance by mollification)  with respect to κ, then the solution to the advection-diffusion equation with the velocity field is close to that with the regularized velocity field. However, the second condition with the regularized velocity field cannot hold.  A possible adapted assumption is to consider a quantitative growth explosion of such an integral depending on the regularization of the velocity field, which is not straightforward to satisfy. Therefore, finding sufficient conditions for anomalous dissipation based on balanced growth, which are satisfied by a two-dimensional autonomous velocity field, seems highly non-trivial. Nevertheless, it is an interesting mathematical problem in the opinion of the authors of this manuscript.
 
  Mixing. The works based on mixing, such as <cit.>, use a condition of the following type:
 
    θ_0 (t_q, ·) - θ_ (λ_q · ) _L^2 ≤1/100θ__L^2    for  some  t_q → 1 , λ_q →∞ .

This condition too is particularly well-suited for velocity fields which generate solutions to the advection equation with a high degree of self-similarity. It should be stressed that although this condition has similarities to the previous one, it is different. Indeed, (<ref>) is not satisfied by the velocity field in <cit.>, but (<ref>) is.
 
  Quantitative homogenization. In the works based on quantitative homogenization <cit.>, one constructs a Cauchy sequence in some Hölder space of smooth divergence-free velocity fields { u_m }_m for which the corresponding solutions {θ_m }_m to the advection-diffusion equation with diffusivity κ_m → 0 exhibit anomalous dissipation.
To do so, it is necessary to control a transport term involving the term

    u_m ·∇χ̃_m , k ,

where χ̃_m , k is a corrector function coming from the homogenization technique.
The term can in fact not be controlled but cancelled by a time derivative. To put this idea into action, one composes u_m with the flow map of u_m-1, which causes the velocity field to become time-dependent. This technical point is also the reason why the velocity field can only be constructed in C^α with α < 1/3.



 §.§ Outline of the paper

We start by introducing some notation and provide preliminaries in Section <ref>. In Section <ref>, we review the concept of spontaneous stochasticity based on which we develop a criterion for anomalous dissipation in Section <ref>. Then, we make a choice of parameters in Section <ref>. Given this choice, we construct the velocity field u of Theorem <ref> in Section <ref>. Subsequently, we prove Theorem <ref> in Section <ref>. Finally, in Section <ref>, we prove Theorem <ref>.



§ ACKNOWLEDGEMENTS
 
MS and CJ are supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00034. The authors are grateful to Maria Colombo for fruitful discussions, to Vlad Vicol for useful observations on continuous in time dissipation and Lucio Galeati for discussions about spontaneous stochasticity and references on the stochastic part.



§ NOTATION AND PRELIMINARIES




 §.§ Notation

In this section, we explain the notation throughout the paper. 
The two-dimensional torus is denoted by ^2. 
The Euclidean norm is denoted by | · |
For any > 0, the -restriction of a set S ⊆^2 is denoted and defined by

    S[] = { x ∈^2 :  (x, S^c) > }

where S^c denotes the complement of S. Similarly the -extension of a set S ⊆^2 is denoted and defined by

    S() = { x ∈^2 :  (x, S) < }.

The cardinality of a set A is denoted by # A. 
For any a,b ∈, we write

    a + b  = { n ∈ : ∃ j ∈ with  n = a + bj }

For vector-valued maps F, we denote the i-th component by F^(i).
The projection map into the j-th component will be denoted by π_j.




 §.§ Preliminaries

We introduce some preliminaries required for the remainder of the paper.
 
Let u ∈ L^∞ ((0,1) ×^2) be a divergence-free velocity field and θ_∈ L^∞ (^2), then for any κ >0 there exists a unique θ_κ∈ L^∞∩ L^2_t H^1_x solution of (<ref>) and it satisfies 

    ∫_^2 |θ_κ  (t,x)|^2 dx + 2 κ∫_0^t ∫_^2 | ∇θ_κ (s,x) |^2 dx ds  = ∫_^2 |θ_  (x)|^2 dx   .


This proposition is a corollary of <cit.>.

 
Let u_1, u_2 ∈ L^∞ ((0,1) ×^2) be two divergence-free velocity fields and θ_∈ L^∞(^2). Let θ_κ,1 be the unique solution of (<ref>) with u_1 and θ_κ,2 be the unique solution of (<ref>) with u_2, then 

    ∫_^2 |θ_κ ,1   (t,x) - θ_κ, 2 (t,x)|^2 dx + κ∫_0^t ∫_^2 | ∇ (θ_κ, 1 - θ_κ, 2) |^2  ≤θ__L^∞^2 /κ∫_0^t ∫_^2 |u_1 (s,x) - u_2 (s,x)|^2 dx ds

for any t ∈ (0,1).


 
The following identity holds in the sense of distribution thanks to the regularity of the solutions

    ∂_t |θ_κ, 1 - θ_κ, 2|^2/2    + u_1 ·∇ (θ_κ,1 - θ_κ, 2) (θ_κ, 1 - θ_κ, 2) + (u_1 - u_2 ) ·∇θ_κ, 2 (θ_κ, 1 - θ_κ, 2)   
        = κΔ ( θ_κ,1  - θ_κ, 2) (θ_κ, 1 - θ_κ, 2 )

and therefore integrating in space-time and using the divergence-free condition on u_1 of the velocity fields and integration by parts we obtain

    ∫_^2 |θ_κ, 1  - θ_κ , 2 |^2 + 2 κ∫_0^t ∫_^2 | ∇ (θ_κ, 1 - θ_κ, 2) |^2 dx ds ≤ 2 ∫_0^t ∫_^2 |u_1 - u_2| |∇ (θ_κ, 1 - θ_κ, 2 )| | θ_κ, 2| dx ds.

Using Young's inequality,  we can can bound the right hand side with 

    κ∫_0^t ∫_^2 | ∇ (θ_κ, 1 - θ_κ, 2) |^2 dx ds + 1/κ∫_0^t ∫_^2 |u_1 - u_2|^2 |θ_κ, 2|^2 dx ds  ,

and using the maximum principle  on θ_κ, 2 we conclude the proof.

We now recall the Strong Markov property and Doob's Maximal inequality which are classical and can be found in <cit.>.

 
Let {W (t) }_t ≥ 0 be a standard Brownian motion, and let τ be a finite stopping time relative to the standard filtration, with associated stopping σ-algebra ℱ_τ . For t ≥ 0, define the post-τ process

    W^⋆ (t) = W(t + τ) - W(τ)
 
and let {ℱ^⋆_t }_t ≥ 0 be the standard filtration for this process. Then, conditional on the event {τ < ∞}, we have


  * W^⋆ (t)_t ≥ 0 is a standard Brownian motion; 

  * for each t > 0, the σ-algebra ℱ^⋆_t is independent of ℱ_τ.



 
Let (Ω, (ℱ_t)_t, ℙ) be a filtered probability space and W_t an adapted ^d-valued Brownian motion. Then, for every c,κ>0, T> T̃≥0 we have

    ℙ ( ω∈Ω : sup_t ∈ [T̃ , T]√(2 κ) | W_t - W_T̃| ≤ c  ) ≥ 1-  2 e^-c^22 κ (T-T̃).



The following ergodic property of the Brownian motion will also be of crucial importance in the proof of Theorem <ref>, see for instance <cit.> for a proof.
 
Let f ∈ L^∞ () ∩ L^1() with f = ∫ f(x) dx ≠ 0 and let (W_t)_t be a standard one dimensional Brownian motion on a probability space (Ω, ℱ, ℙ), then  for any C≥ 0 we have

    lim_t →∞ℙ ( 1/f√(t)∫_0^t f (W_t) dt   ≤ C  ) = √(2/π)∫_0^C exp (- y^2/2) dy   .



We introduce the backward stochastic flow as well as the Feynman-Kac formula which gives a stochastic formulation of solutions to the advection-diffusion equation.

 
Let (W_t)_t be a two-dimensional backward Brownian motion process, u ∈ L^∞ ((0,T) ×^2) be a two-dimensional autonomous divergence free velocity field,  then we define X_T,t^κ the backward stochastic flow on [0,T] ×^2 of u with noise constant √(2 κ) as

    d X_T,t^κ = u (X_T,t^κ) dt + √(2 κ) d W_t  ,
    
    
    X_T,T (x, ω) ≡ x  .



The following representation formula is well-known as Feynman-Kac formula, see for instance <cit.> for a proof.


Let u ∈ C^∞ ([0,1] ×^2; ^2) be a  velocity field. Then the unique solution  θ_κ to the advection-diffusion equation(<ref>)  with diffusivity parameter κ≥ 0 and smooth initial datum θ_∈ C^∞ (^2) can be expressed with the backward stochastic flow map X_t,s^κ as

    θ_κ (x,t) = 𝔼 [θ_ (X_t,0^κ (x, ·)) ].
 



The following lemma is a direct consequence of Grönwall's lemma.
 
Let X_T,t^κ be the backward stochastic flow related to a divergence free velocity field u ∈ L^1_t W_x^1, ∞ and X_T,t be the backward flow map and suppose that 

    | X_T,s^κ (x, ω) - X_T,s (y)| ≤β ,     ∀ s ∈ [0,T]  ,

 then we have the pointwise estimate  

    | X_T,t^κ (x, ω) - X_T,t (y)| ≤ ( |x-y| + √(2 κ)sup_s ∈ [t,T] |W_T  (ω) - W_s (ω) |  )  exp (∫_0^t ∇ u (s, ·  ) _L^∞ (B_β (X_T,s (y) ))ds  ).



Finally, we introduce the so-called fluctuation-dissipation formula. It was introduced in <cit.> and has been used in <cit.> for enhanced dissipation.


Let u ∈ C^∞ ([0,1] ×^2) be a divergence-free velocity field and X_t,0^κ be the backward stochastic flow and θ_κ be the solution to (<ref>) with a bounded initial datum θ_, then

    ∫_^2𝔼[ | θ_ (X_t,0^κ (x, · )) -  𝔼[ θ_ (X_t,0^κ (x, ·))] |^2 ] dx     = 2 κ∫_0^t ∫_^2 |∇θ_κ (x,s)|^2 dx ds. FL-DISS




We have the energy equality

    ∫_^2 |θ_κ (x,t)|^2 dx + 2 κ∫_0^t ∫_^2 | ∇θ_κ (x,s)|^2 dx ds = ∫_^2 |θ_ (x)|^2 dx,

and let h_κ [0,1] ×^2 → be the solution of 

    ∂_t h_κ + u ·∇ h_κ = κΔ h_κ ;
    
    
    h_κ (0, ·) = |θ_ (· )|^2.

Then integrating in space time we obtain

    ∫_^2 h_κ (t, x) dx = ∫_^2 |θ_ (x)|^2 dx

and plugging this identity into (<ref>) we obtain

    ∫_^2 h_κ (t, x) dx -  ∫_^2 |θ_κ (x,t)|^2 dx =  2 κ∫_0^t ∫_^2 | ∇θ_κ (x,s)|^2 dx ds.

Finally, using the Feynman-Kac formula, we have 

    h_κ (x,t)= 𝔼[ | θ_  (X_t,0^κ (x, · )) |^2 ]   and   | θ_κ (x,t) |^2 = | 𝔼 [ θ_ (X_t,0^κ (x, · )) ] |^2

which concludes the proof.

 
  
 

§ SPONTANEOUS STOCHASTICITY

The proof of Theorem <ref> strongly relies on the idea of strong separation of different realizations of the stochastic flow in finite time due to a chaotic behaviour of the velocity field. 
This phenomenon reminds  
 spontaneous stochasticity. This section is devoted to making clear the connection between our proof of Theorem <ref> and spontaneous stochasticity.
 
Even though spontaneous stochasticity has been studied over the last few decades, see for instance <cit.>, we could not find a precise definition of spontaneous stochasticity. Therefore, consistently to all the previous papers, we give Definition <ref>. 
Then, we state a sufficient condition proving spontaneous stochasticity based on separation in finite time of different realizations of the stochastic flow, see Proposition <ref>. This is essentially the condition we will prove to conclude Theorem <ref>, see Proposition <ref>.




Spontaneous stochasticity is a macroscopic effect  caused by  the explosive dispersion of particle pairs in a turbulent flow predicted by Richardson <cit.> with arbitrarily small stochastic perturbation. This phenomenon leads, in the zero noise limit, to non-uniqueness of Lagrangian trajectories. In <cit.>, spontaneous stochasticity has been proved for the Kraichnan model <cit.>, where the
velocity is a Gaussian random field irregular in space and white-noise correlation in time,  see <cit.> for a review.
Spontaneous stochasticity has been  further developed in <cit.>, where the authors introduced the fluctuation dissipation formula and studied the relation between anomalous dissipation and spontaneous stochasticity both for passive and active scalars, as for instance Burgers equation.

Using the description given in <cit.>  we consistently define spontaneous stochasticity as follows.
Notice that we consider a continuous modification of the Brownian motion which is C^β with β < 1/2 and  X_1,s^κ is the pathwise unique solution (see for instance <cit.>) which is also continuous in time.

 
Let u ∈ L^∞ ((0,T) ×^d ; ^d) be a divergence-free velocity field and let (Ω, ℱ, ℙ) be a probability space and X_t,s^κ be the backward stochastic flow. For any x ∈^d we have  X_T, ·^κ (x, ·) : (Ω, ℙ) → ( C([0,T]; ^d) , η_x^κ ), where η_x^κ = X_T, ·^κ (x, ·)_# (ℙ) ∈ℳ ( C([0,T]; ^d) ) is the pushforward of the backward stochastic flow  and finally we set η^κ (d γ, d x) = η_x^κ (d γ) ⊗ dx ∈ℳ (C([0,T]; ^d) ×^d ), namely 

    ∫_C([0,T]; ^d) ×^d f(γ, x)  η^κ (d γ, d x) = ∫_^d∫_C([0,T]; ^d ) f(γ, x)  η^κ_x (d γ) d x  ,

for any continuous  f :  C([0,T]; ^d) ×^d →. We say that u  exhibits spontaneous stochasticity if there exist A ⊂^d with ℒ^d (A) >0 and a narrowly converging subsequence η^κ_q⇀η  such that

    η   =  η_x ⊗ℒ^d

and for all x ∈ A, η_x is not a Dirac delta (meaning that there exists no continuous curve γ_x such that η_x = δ_γ_x).


Related to the definition above, we point out the recent paper <cit.> where the author considers the inviscid case via regularisation by mollification of the velocity field.

We cannot employ the Riesz representation theorem over the space C([0,T]; ^d) ×^d to characterize the weak* convergence in the space of measures, since C([0,T]; ^d) is not locally compact. Therefore, we use Prokhorov's theorem to deduce the compactness result of Lemma <ref>.



Let (X, d) be a Polish space and let {μ_n }_n⊂ℳ (X) be a sequence of probability measures, then the two facts are equivalent:


  * (Tightness). There exists F: X → such that F(x) →∞ as d(x, 0) →∞ so that  the sublevel sets { x ∈ X: F(x) ≤ L } are precompact and 

    ∫_X F d μ_n ≤ 1     ∀ n ∈ .
 

  * There exists a subsequence {μ_n_j}_j narrowly converging μ_n_j⇀μ, i.e. for any continuous bounded function φ X → we have 

    ∫_X φ d μ_n_j→∫_X φ d μ




 
Let η^κ∈ℳ ( C([0,T]; ^d) ×^d ) defined as above  η^κ (d γ, d x) = η_x^κ (dγ) ⊗ℒ^d (dx) where η_x^κ = X_T, ·^κ (x, ·)_# (ℙ) ∈ℳ ( C([0,T]; ^d) ). Then {η^κ}_κ >0 admits a narrowly converging subsequence.



We need to check the tightness hypothesis. Let β < 1/2 and observe that

    𝔼 | X_T,t^κ (x, ω) - X_T,s^κ (x, ω) | ≤ ( 2   u _L^∞ + √(2 κ) c ) |t-s|^β≤ C |t-s|^β

for any |t-s| ≤ 1 and κ∈ (0,1) for some C independent of κ. 
We define 

    F(γ , x ) =  1/Cγ_C^β

which is set = + ∞ if γ∈ C([0,T] ) ∖ C^β ([0,T]). Thanks to Ascoli-Arzela's theorem we can show that the sublevel sets of the functional F are compact. Finally,  by a direct computation we have 

    ∫_C([0,T]; ^d) ×^d F(x, γ) η^κ (d γ, dx) = 1C∫_^d∫_Ω X_T, ·^κ (x, ω) _C^β d ℙ(ω) dx ≤ 1

for any κ∈ (0,1), concluding the proof.


 

Let  d≥ 2 and u ∈ L^∞ ((0,T) ×^d ; ^d) be a divergence free velocity field and  X_t,s^κ be its backward stochastic flow as in Definition  <ref>. Suppose that there exist a constant c >0 and {κ_q }_q ∈ with κ_q → 0 as q →∞ such that 

    inf_κ∈{κ_q}_q ∫_^d  𝔼 [  | X_T, 0^κ (x, · )  - 𝔼 [  X_T, 0^κ (x, · )  ] |^2  ] dx ≥ c  ,

then u exhibits spontaneous stochasticity.



In all the proof we characterize the measures η∈ℳ (C([0,T]; ^d) ×^d) considering the pushforward of through the evaluation map (e_t, Id) : C([0,T]; ^d)  ×^d →^d ×^d, where e_t (γ) = γ (t). It is straightforward to check that ν∈ℳ (C[0,T]) is such that ν = δ_γ∈ℳ (C([0,T]; ^d)) if and only if for any t∈ [0,T] there exists x ∈^d such that  (e_t)_# (ν) = δ_x∈ℳ (^d). 
Let us fix the sequence {κ_q }_q  for which the assumption holds. We suppose by contradiction that there is no spontaneous stochasticity, i.e. for any converging  subsequence η^κ_j⇀^* η we have η (d γ, dx)= δ_γ_x (d γ) ⊗ dx.

  Thanks to Lemma <ref> there exists a further subsequence, not relabelled, converging narrowly, namely η^κ_q⇀η.
For such a subsequence {κ_q }_q we have that   for any f, g ∈ C^0 (^d) and  ∀
 t ∈ [0,T] we have
 
    ∫_^d∫_C([0,T]; ^d) g(x) f (γ (t)) η_x ( d γ) dx     = lim_κ_q → 0∫_^d∫_C([0,T]; ^d)  g(x) f (γ (t))  η_x^κ_q (d γ) dx 
     
        = lim_κ_q → 0∫_^d∫_Ω g(x) f(X_T, t^κ_q (x, ω)) d ℙ(ω)dx   .

 Since we are supposing by contradiction that u does not have spontaneous stochasticity  we have η_x = δ_γ_x for a.e. x ∈^d. Testing with f (x) = x_i for i =1, …, d (by approximating these functions with continuous function on the torus) we deduce component-wise that
 
    lim_κ_q → 0∫_^d  g(x) ( γ_x (t) -  𝔼 [ X_T, t^κ_q (x, · ))]  )dx =0.

 From this we deduce that for any t ∈ [0,T] there exists N_t ⊂^d with ℒ^d (N_t)=0 such that 
 
    lim_κ_q → 0δ_𝔼 [X_T, t^κ_q (x, · ) ] = (e_t)_#η_x
 
 for any x ∈ N_t^c (where the limit is intended in the sense of narrow convergence). 
 Considering N = ⋃_t ∈∩ [0,T] N_t and using the continuity of curves we deduce that 

    η (d γ, dx)= lim_κ_q → 0δ_𝔼 [X_T, ·^κ_q (x, · ) ] (d γ) ⊗ dx  .
 
Therefore, choosing g ≡ 1 and f(x) = |x|^2 (by approximating these functions with continuous functions on the torus) we deduce
 
    lim_κ_q → 0∫_^d𝔼 [|X_T, t^κ_q  (x, ·)|^2] dx= lim_κ_q → 0∫_^d | 𝔼 [X_T, t^κ_q  (x, ·)]  |^2 dx  ,

 for any t ∈ [0,T], 
 which implies that 
 
    lim_κ_q → 0 ∫_^d  𝔼 [  | X_T, 0^κ_q (x, · )  - 𝔼 [  X_T, 0^κ_q (x, · )  ] |^2  ] dx =0

 contradicting the hypothesis.


We use  the fluctuation-dissipation formula (<ref>) given in Lemma <ref>  and we give a simple sufficient criterion to prove anomalous dissipation.



§ A CRITERION FOR ANOMALOUS DISSIPATION AND CONTINUOUS IN TIME DISSIPATION

In this section, we state and prove a criterion for anomalous dissipation which considers the stochastic trajectories of suitable approximations of the velocity field. The criterion essentially states that anomalous dissipation occurs if trajectories evaluated at time one (or any other fixed time for that matter) remain stochastic in the vanishing noise limit.
In addition, we prove that anomalous dissipation with autonomous velocity fields necessarily occurs continuously by which we mean that up to nonrelabeled subsequences

    κ∫_^d |∇θ_κ|^2   dx *⇀ f ∈ L^∞ (0,T)

 as κ→ 0, where the convergence is L^∞ weak*.
We start with a criterion for anomalous dissipation.
 
Let u ∈ L^∞ ((0,T) ×^d; ^d) be a divergence free velocity field. We say that u exhibits anomalous dissipation if there exists θ_∈ C^∞ (^d) for which  the following holds

    lim sup_κ→ 0κ∫_0^T ∫_^d | ∇θ_κ (x, s) |^2 dx ds >0  .

Furthermore, for future purposes, we will also say that the subsequence {θ_κ_q}_q for which the lim sup is attained exhibits anomalous dissipation.

The following is a new criterion to prove anomalous dissipation, which is a new approach with respect to the previous approaches.
 
Let α∈ (0,1) and  u ∈ C^α ((0,T) ×^d; ^d) be a divergence free velocity field such that there exist { u_q }_q ∈⊂ C^∞ ( [0,T] ×^d; ^d) divergence free and {κ_q }_q ∈  with the following properties


  * 1/√(κ_q) u - u_q_L^2_t,x→ 0 as q →∞,

  * we denote the stochastic backward flow  of u_q with noise parameter √(2 κ_q) as X_t, 0^q, κ_q.  We suppose that there exist a constant c>0 and { D_q }_q ∈⊂^d with ℒ^d (D_q)≥ c for any q ∈ with the following property 

    inf_q ∈inf_x ∈ D_q𝔼 [  | X_T, 0^q, κ_q (x, ω)  - 𝔼 [  X_T, 0^q, κ_q (x, · )  ] |^2  ] ≥ c  .


Then u exhibits anomalous dissipation. 

 

The proof of this proposition requires to construct an initial datum for which anomalous dissipation occurs. We suppose, without loss of generality, that the first component of X_T,0^q, κ_q = ( x_T,0^q, κ_q , y_T,0^q, κ_q) satisfies (<ref>) with constant c/2, namely

    inf_q ∈inf_x ∈ D_q𝔼 [  | x_T, 0^q, κ_q (x, ω)  - 𝔼 [  x_T, 0^q, κ_q (x, · )  ] |^2  ] ≥c/2 ,

We now define  θ_ (x,  y) = x for any x, y ∈ and let θ_κ_q^(q) be the solution of the advection diffusion equation with velocity field u_q and diffusivity parameter κ_q. Applying Lemma <ref> we have
 
    2 κ_q∫_0^T ∫_^d |∇θ_κ_q^(q) (x,s)|^2 dx ds    ≥∫_^d𝔼[ | θ_ (X_T,0^q, κ_q (x, · )) -  𝔼[ θ_ (X_T,0^q, κ_q (x, ·))] |^2 ] dx 
     
        = ∫_^d𝔼[ |  x_T,0^q, κ_q (x, · ) -  𝔼[ x_T,0^q, κ_q (x, ·) ] |^2 ] dx 
     
       ≥∫_D_q𝔼[ |  x_T,0^q, κ_q (x, · ) -  𝔼[ x_T,0^q, κ_q (x, ·) ] |^2 ] dx 
     
       ≥c/2 ,     ∀ q ∈ .

 Let θ_κ_q be the solution of the advection diffusion with velocity field u and diffusivity parameter κ_q. Applying Proposition <ref> and Lemma <ref> we finally have 
 
    lim sup_q →∞κ_q∫_0^T ∫_^d |∇θ_κ_q (x,s)|^2   dx ds = lim sup_q →∞κ_q∫_0^T ∫_^d |∇θ_κ_q^(q) (x,s)|^2 dx ds ≥c/2 .

 Mollifying the initial datum θ_, γ = θ_⋆φ_γ with a parameter γ >0 so that 
 
    θ_ - θ_, γ_L^2 < c/2

 we conclude that  anomalous dissipation holds for the advection diffusion equation with the smooth initial datum θ_, γ∈ C^∞.


We now show that anomalous dissipation for autonomous velocity fields is necessarily continuous.

 
 Let d ≥ 2 and u ∈ L^∞ (^d) an autonomous divergence free velocity field. Let θ_κ : [0,T] ×^d → be the solution of the advection diffusion with a θ_∈ C^2 (^d). Then 
 
    sup_κ∈ (0,1)∂_t θ_κ_L^∞ ((0,T) ×^d)) ≤ u _L^2θ__C^1 + θ__C^2 .

 In particular, up to not relabelled subsequence, 
 
    κ∫_^d |∇θ_κ (x,·)|^2 dx *⇀ f ∈ L^∞ (0,T)

 as κ→ 0, where the convergence is L^∞ weak*. Finally, if θ_κ⇀^* θ as κ→ 0 (up to not relabelled subsequence), then 
 
    e(t) = ∫_^d |θ (x,t)|^2 dx ∈ W^1, ∞ (0,T)  .

 
 
 
 We notice that 
 
    ADV-DIFF-t∂_t  g_κ +  u  ·∇  g_κ = κΔ  g_κ,
    
    
     g_κ (0, · ) = - u ·∇θ_ (· ) + κΔθ_ (· )   .

satisfies

    g_κ_L^∞ ((0,T) ×^d ) ≤ u _L^2θ__C^1 + θ__C^2 ,

from the maximum principle. We now claim that the last bound holds also for ∂_t θ_κ. 
First, we introduce θ_κ^ε to be the solution to

    ∂_t  θ^ε_κ +  u_ε·∇θ^ε_κ = κΔθ^ε_κ,
    
    θ^ε_κ (0, · ) = θ_ (·)

with u_ε = u ⋆φ_ε a space mollification. It is possible to prove that θ_κ^ε is smooth  and ∂_t θ_κ^ε (t, ·) → - u ·∇θ_ (·) + κΔθ_ (·) in L^2(^d) as t → 0^+. Furthermore, from standard energy estimate we have 

    sup_t ∈ [0,1]θ_κ (t, ·) - θ_κ^ε (t, ·) _L^2 (^d)≤ u - u_ε_L^2 ( (0,T) ×^d )^2/κθ__L^∞^2  ,

and therefore θ_κ^ε→θ_κ in C([0,T); L^2( ^d)) as ε→ 0.  Since ∂_t θ_κ^ε solves (<ref>) and it enjoys ∂_t θ_κ^ε_L^∞ ((0,1) ×^d ) ≤ u _L^2θ__C^1 + θ__C^2 which implies

    θ_κ^ε→θ_κ     C([0,T); L^2( ^d)) ,
    
       ∂_t θ_κ^ε⋆⇀∂_t θ_κ   weak  *-L^∞ .

In particular, we have obtained the bound 

    sup_κ∈ (0,1)∂_t θ_κ_L^∞ ((0,T) ×^d)≤ u _L^2θ__C^1 + θ__C^2 .

 From this and  a standard energy balance  we observe that 

    2 κ∫_^d |∇θ_κ (x,·)|^2 dx = - d/dt∫_^d |θ_κ (x,t)|^2 dx = ∫_^d∂_t θ_κ (x,t) θ_κ (x,t)   dx ≤∂_t θ_κ_L^∞θ_κ_L^∞

and the last is bounded independently on κ thanks to the previous computations.
Similarly, if up to not relabelled subsequences we have

    θ_κ*⇀θ∈ L^∞
    ∂_t θ_κ*⇀∂_t θ∈ L^∞

we  can bound

    | d/dt∫_^d |θ (x,t)|^2 dx  | = 2  | ∫_^d∂_t θ (x,t ) θ(x,t) dx  | ≤∂_t θ_L^∞θ_L^∞

where the last term is bounded since ∂_t θ_L^∞≤ u _L^2θ__C^1 + θ__C^2 thanks to the weak* convergence.
 



§ CHOICE OF THE PARAMETERS

In this section we define the parameters we will use in the construction of the velocity field. Let us fix α <1 as in the statement of Theorem <ref>, then we can find δ ,  >0 such that 


    12+δ >  + α (1 + δ)^4 ( 3  + 3 δ + (1+δ)^2/2 + δ) 
    
    1 > (2 + δ)  + (1 + α) (1 + δ)^5 ( 3  + 3 δ + (1+δ)^2/2 + δ) 
    1 + 2(1 + δ)^52 + δ >  + (2 + α) (1 + δ)^4 ( 3  + 3 δ + (1+δ)^2/2 + δ)
 

noticing that for = δ = 0 the inequalities reduce to α <1 and therefore (since the previous inequalities depend continuously on , δ) we can find a ball of radius r>0, i.e. B_r (0) = { (, δ) ∈^2: || + |δ| < r}, such that  the inequalities are true. 
 We also require  that ≪δ, more precisely we require  

    2 δ^2 + δ^3 ≥ (1+ δ)^2 + 2  .



We now inductively define the following sequence  of parameters, needed for the construction of the velocity field in Section <ref>.
 

    a_q+1   = a_q^1+ δ , 
    
    n_q+1   = a_q/2 a_q+1 ,
    
    A_q+1   = A_q/2 n_q+1 , 
    
    A_q+1   =  A_q+1 a_q^δ - δ/2 + δ , 
    B_q+1   = L_q - n_q+1A_q+1 -2 A_q/ n_q+1 , 
    
    L_q+1   = B_q/2 - 4 A_q+1 -  B_q/2 a_q^δ , 
    
    v_q+1   = v_q A_q+1/A_q+1
    κ_q    = B_q^2  ,

 
with initial conditions  A_0 = a_0, v_0 =1 and L_0 =1/4 and  B_0 = 1/4.  We now choose 
a_0 sufficiently small in terms of δ and  so that 

    ∑_k =q^∞ a_k^^2≤ 2 a_q^^2    ∀ q ∈ ,
 
which is verified if we choose a_0 such that  a_0^^2 δ≤ 1/2
We finally define the sequence of diffusivity parameters  as

    κ_q = B_q^2.


The following lemma gives us the asymptotic behaviour of the sequences defined above as q →∞.


Given the definition of the sequences in (<ref>), 
the following bounds hold true for every q ≥ 0
 

    n_q = a_q-1^- δ/2  , 
    
    N_q = a_0 a_q^-1 , 
    
    A_q = A_0 a_0^-  - 1+ δ/2+ δ a_q^ + 1+ δ/2+δ
    A_q = A_0 a_0^-  - 1+ δ/2 + δ a_q-1^ + δ + 1+ δ/2 + δ
    1/2 B_0 a_0^- 1+ δ/2+δ  a_q^1+ δ/2+δ≤B_q≤ B_0 a_0^- 1+ δ/2+δ  a_q^1+ δ/2+δ , 
    ∀ q ≥ 1   1/4 B_0 a_0^- 1+ δ/2+δ  a_q^1/2+δ≤ L_q≤1/2 B_0 a_0^- 1+ δ/2+δ  a_q^1/2+δ , 
    1/4 B_0^2 a_0^-2+ 2 δ/2+δ  a_q^2+ 2 δ/2+δ≤κ_q≤ B_0^2 a_0^-2+ 2 δ/2+δ  a_q^2+ 2 δ/2+δ
    
    v_q = v_0 a_0^ - 1/2+ δ a_q^-  + 1/2+ δ

 


The equality (<ref>) follows directly from the choice of parameters. Note that

    N_q = ∏_j =1^q 2 n_j = ∏_j=1^q a_j-1^- δ =a_0 a_q^-1,
 
proving (<ref>), where in the last identity we used the relation a_j= a_0^(1+ δ)^j and an explicit computation using the power series identity. 
We now prove (<ref>). The equation clearly holds for q = 0. 
The case q ≥ 1 follows by induction combined with (<ref>) and (<ref>).
Equation (<ref>) then follows from (<ref>), (<ref>) and (<ref>).
We now prove estimate (<ref>) by induction. 
We start by noticing that (<ref>) holds for q=0,1. 
Now, assume that it is true for j ≤ q and using (<ref>), we have

    B_q+1   = 2L_q a_q^δ - 5 A_0 a_0^- - 1 + δ/2 + δ a_q^ + 1 + δ/2 + δ a_q^δ + δ(1 + δ)/2 + δ.

Using (<ref>) and (<ref>) we continue

    B_q+1   = B_q-1 a_q^δ (1- a_q-1^δ) - 8 a_q^δ A_0 a_0^-  - 1 + δ/2 + δ a_q-1^ + δ + 1 + δ/2 + δ - 5 a_q^δ A_0 a_0^- - 1 + δ/2 + δ a_q^ + 1 + δ/2 + δ a_q^δ + δ(1 + δ)/2 + δ - δ
       ≥ a_q^δ[ B_q-1  (1- a_q-1^δ) - 16 A_0 B_0^-1 B_q-1 a_0^-  a_q-1^ + δ - 10 A_0 B_0^-1 B_q-1 a_0^- a_q-1^ (1 + δ)^2] 
       ≥ a_q^δ B_q-1( 1 - 2 a_q-1^δ)

where we used 40 a_0^δ≤ A_0^-1 B_0 in the last inequality.
Iterating this inequality yields

    B_q ≥ B_0 ∏_j = 1^q2 a_2j - 1^δ( 1 - 2 a_2j - 2^δ)  if q is even and     B_q ≥ B_1 ∏_j = 1^q-1/2 a_2j^δ( 1 - 2 a_2j - 1^δ)  if q is odd.

Now, using that 1- x ≤^-2 x for any x ∈ (0, 1/2) we have

    ∏_j = 1^q (1 - 2 a_j^δ) ≥∏_j = 1^q^- 4 a_j^δ =  ^- 4 ∑_j = 1^q a_j^δ≥12  ∀q≥ 1.

Hence for q even

    B_q    ≥12 B_0 ∏_j = 1^q2 a_2j - 1^δ = 12 B_0 ∏_j = 1^q2 a_0^δ(1 + δ)^2j - 1 = 12 B_0 a_0^δ (1 + δ)^-1∑_j = 1^q2 (1 + δ)^2j = 12 B_0 a_0^δ (1 + δ)^-1∑_j = 1^q2 (1 + δ)^2j
       ≥12 B_0 a_0^δ (1 + δ)^-1 (1 + δ)^2 (1 + δ)^q - 1/(1 + δ)^2 - 1≥12 B_0 a_0^δ (1 + δ)^-1 (1 + δ)^2 (1 + δ)^q - 1/δ^2 + 2 δ≥12 B_0 a_0^- 1 + δ/2 + δ a_0^(1 + δ)^q+1/2 + δ = 12 B_0 a_0^- 1 + δ/2 + δ a_q^1 + δ/2 + δ

Similarly, when q is odd, with similar computations and using the fact that B_1 = B_0 a_0^- 1 + δ/2 + δ a_1^1+δ/2 + δ we can prove that

    B_q ≥12 a_0^- 1 + δ/2 + δ a_q^1 + δ/2 + δ.

Hence, we have proved the lower bound of (<ref>).
The upper bound for B_q+1 follows from the observation that

    B_q+1≤L_q/n_q+1≤B_q-1/2 n_q+1  = a_q^δ B_q-1

which in turn leads to

    B_q ≤ B_0 ∏_j = 1^q2 a_2j - 1^δ if q is even and    B_q ≤ B_1 ∏_j = 1^q-1/2 a_2j^δ if q is odd.

By means of explicit computations together with the facts that a_q = a_0^(1 + δ)^q and B_1 = B_0 a_0^- 1 + δ/2 + δ a_1^1+δ/2 + δ we obtain that

    B_q ≤ B_0 a_0^- 1 + δ/2 + δ a_q^1 + δ/2 + δ  ∀ q ≥ 1.

Thus, we have proved the upper bound of (<ref>).
Inequalities (<ref>) follow from (<ref>), (<ref>), (<ref>) and (<ref>).
Inequalities (<ref>) follows immediately from (<ref>) due to the definition of κ_q in (<ref>).
Finally, from (<ref>), (<ref>) and (<ref>) we obtain

    v_q+1 = v_q a_q^- δ + δ/2+δ.
 
An explicit computation using that a_q= a_0^(1+δ)^q yields (<ref>).



























§ CONSTRUCTION OF THE VELOCITY FIELD

In this section, we construct the velocity field u of Theorem <ref> as the limit of a sequence of smooth velocity field { u_q }_q. The construction is carried out in several steps. First we construct a special rotating pipe in Subsection <ref>. We develop this construction further in Subsection <ref>. Then we build what will become the building block of our velocity fields in Subsection <ref>. In Subsection <ref>, we use this building block to build a sequence of L^∞ divergence free velocity fields { b_q }_q. Finally, we construct the sequence of velocity fields { u_q }_q as well as the velocity field u ∈ C^α in Subsection <ref> by a  mollification argument.



 §.§ Construction of an enlarging rotating pipe

The purpose of this subsection is to construct a velocity field whose integral curves make a 90 degree rotation and the entering velocity is different from the exiting velocity (see Figure <ref>). 


 Let 0 < r < R < ∞ and v > 0, λ≥ 1. Then there exists an autonomous velocity field u = u_r, R, λ, v∈ BV([0, λ R] × [0, R]) such that
 
  
  * u = v ^1 |_{ 0 }× [r,R] - v/λ^1 |_ [λ r, λ R] ×{ 0 }; 
  
  * the trace of u on { 0 }× [0,R] is given by
  
    u(0,y) = 
       
        (v,0)     if y ∈ [r, R]
    
        (0,0)     otherwise.
 
  
  * the trace of u on [0,λ R] ×{ 0 } is given by
  
    u(0,y) = 
       
        (0,-v/λ)     if x ∈ [λ r, λ R]
    
        (0,0)     otherwise.
 
  
  * there exists a constant C > 0 such that for any 0 <  < r/2 and any integral curve γ [0,T] → [0, λ R] × [0, R] starting from a point in { 0 }× [r + , R - ] with T such that γ(T) ∈ [0,λ R] ×{ 0 } we have
  
    ∫_0^T  ∇ u _L^∞(B_(γ(t)))  dt ≤ C.

  
 
 Moreover, for extensions and subsequent mollifications of this velocity field[By extensions, we mean that we extend u to the whole ^2 and set u = (v, 0) if (x,y) ∈ (- ∞, 0) × [0, R], u = (0, - v/λ) if (x,y) ∈ [0, λ R] × (- ∞, 0) and u = (0,0) if (x,y) ∉[0, λ R] × [0,R] ∪ (- ∞, 0) × [0, R] ∪ [0, λ R] × (- ∞, 0).] we have
  
  
  * there exists a constant C > 0 such that for any , δ> 0 satisfying the inequality 
  
    0 < 5 λδ <  < r/2.

  it holds that
  any integral curve γ [0,T] →^2 of u_δ starting from a point in { - δ}× [r + 3 , R - 3] and γ(T) ∈ [0,λ R] ×{ - δ} we have
  
    ∫_0^T  ∇ u_δ_L^∞(B_(γ(t)))  dt ≤ C.
 
 



 Let 0 < r < R < ∞ and v, λ∈ as in the statement of the lemma.
 Let 0 < r^' < R^' and α∈ be real numbers (to be selected depending on r, R, v and λ) and define a velocity field u_α [0, R^']^2 →^2 as
 
    u_α(x,y) = 
      α√(x^2 + y^2) (y, -x)    if (r^')^2 ≤ x^2 + y^2 ≤ (R^')^2; 
    
       (0,0)    otherwise.

 We note that for any (x,y) ∈ [0, R^']^2 such that (r^')^2 < x^2 + y^2 < (R^')^2 we have
 
    |u_α(x, y)| = α  and   |∇ u_α(x,y)| = α|(x,y)|.

 We will then rescale this velocity field in a suitable way (and then select α, r^' and R^' depending on r, R, v and λ) to obtain the desired velocity field. Before doing this, let us make a few observations.
 Firstly, we note that the distributional divergence of u_α is given by
 
    u_α = α |_{ 0 }× [r^',R^'] - α |_ [r^', R^'] ×{ 0 }.

 Secondly, let γ [0,T] → [0, R^']^2 be any integral curve starting from a point in { 0 }× [r^',R^'] with T such that γ(T) ∈ [r^', R^'] ×{ 0 }. Then, a computation yields:
 
    T = π |γ(0)|2 α  and  ∫_0^T |∇ u_α(γ(t))|   dt = α∫_0^T 1|γ(t)|  dt = π/2.

 More precisely, we can prove that
 
    ∇ u_α_L^∞(B_(γ(t)))≤2 α|γ(t)|

 under the assumption that < r^'2.
 Therefore, under this assumption, we obtain
 
    ∫_0^T ∇ u_α_L^∞(B_(γ(t)))  dt ≤ 2 α∫_0^T 1|γ(t)|  dt ≤π.

 We now rescale the velocity field.
 Define the matrix
 
    A =
      [   λ^12      0;      0 λ^- 12;        ]∈^2 × 2

 and define w_α [0, λ^12 R^'] × [0, λ^- 12 R^'] →^2 as
 
    w_α(x,y) = A u_α (A^-1 (x,y)).

 We get that
 
    ∇ w_α(x,y) = A A^-1∇ u_α (A^-1(x,y)) = ∇ u_α (A^-1(x,y))

 and hence
 
    w_α(x,y) =  (∇ w_α)(x,y) =  (∇ u_α)(A^-1(x,y)) =  u_α(A^-1(x,y)) = 0

 for all (x,y) ∈ (0, λ^12 R^') × (0, λ^- 12 R^').
 Now select R^' = λ^12 R and r^' = λ^12 r and fix α = v/√(λ). Then w_α is the desired velocity field, which we from now on denote by u. We already know that <ref>, <ref> and <ref> are fulfilled. We now verify <ref>.
 Let η [0,T] → [0, λ R] × [0, R] be an integral curve starting from a point on { 0 }× [r,R] of u with T such that η(T) ∈ [λ r, λ R] ×{ 0 }.
 Our goal is to estimate
 
    ∫_0^T ∇ u _L^∞(B_(η(t)))  dt.

 It is not difficult to see that γ(t)  A^-1η(t) is an integral curve of u_α and hence we can apply the observation we already made for the velocity field u_α. Indeed, we find
 
    ∫_0^T ∇ u _L^∞(B_(η(t)))  dt    = ∫_0^T ∇ w_α_L^∞(B_(A γ (t)))  dt 
       ≤∫_0^T ∇ u_α_L^∞(B_λ^12(γ (t)))  dt ≤π

 under the assumption that λ^12 < r^'2 which is equivalent to < r2. Hence we have proved <ref> and the proof for the velocity field u is finished. 
 We now treat the case of the mollified velocity field. Let , δ > 0 be real numbers such that (<ref>) holds and extend u to ^2 so that 
 
    u(x,y) = ( v 1_[r,R](y), 0 )  if x < 0  and   u(x,y) = ( 0, - v/λ1_[λ r, λ R](x) )  if y < 0.

 Then, consider the mollified velocity field u_δ = u ⋆ρ_δ. Let (x_0, y_0) ∈{ - δ}× [r, R] and let γ respectively γ_δ be two integral curves starting from (x_0, y_0) with respect to u respectively u_δ.
 Geometric arguments yield
 
    |γ(t) - γ_δ(t)| ≤ 4 λδ  ∀ t ≥ 0.

 Then
 
    ∫_0^T∇ u_δ_L^∞(B_(γ_δ(t)))  dt    ≤∫_0^T∇ u _L^∞(B_ + δ(γ_δ(t)))  dt 
       ≤∫_0^T∇ u _L^∞(B_ + 5 λδ(γ(t)))  dt 
       ≤∫_0^T∇ u _L^∞(B_2 (γ(t)))  dt
     ≤ C

 where the last inequality follows from <ref>. This proves <ref> and finishes the proof of Lemma <ref>.



 In the remainder of the paper, when using the velocity fields from Lemma <ref> we will always use R = 2r.




 §.§ Construction of an twice rotating and enlarging pipe


It is possible to construct, thanks to the previous construction of a rotating pipe, a pipe with a velocity field which starts out horizontal with magnitude v and the width of the pipe A. 
Then, it turns and becomes a vertical  pipe with width A^' and the magnitude of the velocity field is v^'. To ensure the divergence-free condition we have the relation v^' = v A/A^'.
 Afterwards, the pipe again turns and becomes horizontal with width A and the magnitude of the velocity field is again v (see Figure <ref>).  




 §.§ Building blocks: Branching-merging pipe

In this subsection, we will use the velocity field of the previous subsection in order to build a velocity field whose fundamental property is that it starts out as a unique pipe of width A and velocity v and then branches into 2n (with n large) smaller pipes of width 

    A = A/2n
 
 and each of this pipe width enlarges to A^' but with reduced intensity of the velocity which is given by 
 
    v^' = v A/A^'≪ v  .

  Afterwards, all these smaller pipes shrink back into small pipes of width A with corresponding velocity v and  merge back into one pipe of width A as in Figure <ref>.

The quantity L corresponds to the length of the whole construction, (A+ B) corresponds to the width of the whole construction
and B^' corresponds to the distance in between smaller pipes. Note that these parameters are linked by the following relation:

    L = n A^' + n B^' + 2 A.

Moreover, note that the length of the thinner pipes is given in terms of B and A as

    B - 3 A2.

This velocity field is our building block, needed in Subsection <ref>, and we denote it as

    W = W_L, A, B, A^', B^', n, v.

We will regularly use rotated (by some multiple of π2) versions of this velocity field. This will always be clear from the context and the properties we would like to achieve, hence we will slightly abuse the notation using always the same notation independently of whether the velocity field is rotated or not. With the twice rotating enlarging pipe given in Section <ref> we can indeed construct our building block W. 
The key properties of W are stated in the following lemma.

Let n ≥ 10 be an integer and let A, B, L, A^', B^', v ∈ (0,1) be real numbers such that

    A ≪ B,    A^'≪ B^',   and   L = n A^' + n B^' + 2 A.

Let L^'∈ (0,1) be an any real number such that

    L^' < B - 3 A - A/2.

Then there exists W = W_L, A, B, A^', B^', n, v : R = [0,L] × [- A+ B/2, A+ B/2] →^2 with the following properties:


  * W _L^∞ = v.

  * W =  (W 1_R) = v ^1 |_{ 0 }×[ - A/2, A/2] - v ^1 |_{ L }×[ - A/2, A/2].

  * There are  2n rectangles { R_i }_i =1^2n such that, up to a rotation of π/2 or -π / 2 and a translation,  

    R_i = [0, L^'] ×[ - A^' + B^'/2, A^' + B^'/2]
 
and in the coordinates of R_i

    W (x, y)  = [ v^'1_[- A^'/2, A^'/2](y);                        0 ]    ∀ (x, y) ∈ R_i  ,

   where v^' = v  A/A^'.
   
  * It holds that W _L^∞ (R_i)≤ v^' and in the coordinates of R_i 
   
    (W 1_R_i) =  v^'^1 |_{ 0 }×[ - A^'/2, A^'/2] - v^'^1 |_{ L^'}×[ - A^'/2, A^'/2]
 for any i =1 , ... , 2n.
   
  * The collection of rectangles { R_i }_i = 1^2n can be reordered so that
   
    R_i = R_1 + (i-1)(A^' + B^', 0)    R_i+n = R_n+1 + (i-1)(A^' + B^', 0)   ∀ i = 1, …, n.






 §.§ Construction of the L^∞ velocity fields

In this subsection, we use the velocity field from the previous subsection as a building block in the construction of a sequence of velocity fields { b_q }_q ≥ 1. The sequence is constructed iteratively by adding more and more suitably rescaled copies of branching-merging pipes at each step. To do this we simultaneously define a sequence of sets of rectangles {ℛ_q }_q, where 
ℛ_q = { R_1, … , R_N_q} and R_i is a rectangle for each i = 1, …, N_q. Given  the velocity field  b_q and the collection of mutually disjoint rectangles ℛ_q, we define b_q+1 as b_q, redefining it on R_i as a branching merging pipe keeping the divergence free property.
For the construction we identify ^2 with [0,1]^2.
 Recall the parameters selected in Section <ref>, that satisfy (<ref>) and  will be the parameters involved in our construction. Firstly, we define b_0 = ∇^⊥ H_0 ^2 →^2 such that for all (x,y) ∈ (A_0, 1 - A_0)^2

    b_0(x,y) = 
     
      (v_0, 0)    if |y - 1/2| < A_0/2;
    
      (0,0)    otherwise.

In other words, b_0 is a straight pipe of velocity v_0 and width A_0 inside the square (A_0, 1 - A_0)^2.  We now define the set of rectangles at step 0 as ℛ_0 = { (A_0, 1 - A_0)^2 }.
 We now define b_1 as b_0 and redefine it on each rectangle of the collection ℛ_0   to be a branching-merging constructed in Subsection <ref> with the parameters L_0, A_0, B_0, A_1, B_1, n_1, v_0 from Section <ref>.  In particular, on each rectangles of the collection ℛ_0 (which in this case has a single element) we replace the straight pipe given by b_0 with a consistent branching merging pipe.
 
 
 
Then we note that for b_1 there exists a collection of 2 n_1 = N_1 rectangles { R_i }_i = 1^N_1 so that up to rotations and translations

    R_i = [0, L_1] ×[- A_1/2 - B_1/2, A_1/2 + B_1/2]

and in these rectangles the velocity field b_1 is composed by a single straight pipe of width A_1 (see Figure <ref>).
 Then we define this new collection of rectangles ℛ_1 = { R_1, … , R_N_1 }. 
To construct b_q+1 from b_q for any q, the procedure is as follows. Assume that there exists a collection of rectangles ℛ_q such that #ℛ_q = N_q and for any R ∈ℛ_q it holds that up to a rotation and a translation

    R = [0, L_q] ×[ - A_q + B_q/2, A_q + B_q/2]

and in the coordinates of R

    b_q(x,y) = 
     [ v_q 1_[- A_q /2, A_q/2](y);                          0 ]  ∀ (x,y) ∈ R.

Now we define b_q+1 by redefining the velocity field b_q on each R ∈ℛ_q. Precisely we define b_q+1  to be a branching-merging pipe with parameters L_q, A_q, B_q, A_q+1, B_q+1, n_q+1, v_q i.e.

    b_q+1 (x,y) = W_L_q, A_q, B_q, A_q+1, B_q+1, n_q+1, v_q(x,y)

in the coordinates of R ∈ℛ_q.  Thanks to Lemma <ref> and (<ref>) we have that b_q+1 is still divergence free.
In a short-formula, we have
 
    b_q+1 (x,y) =
     
     O_q^T W_L_q, A_q, B_q, A_q+1, B_q+1, n_q+1, v_q (τ_q ∘ O_q (x,y))
        ∀ (x,y) ∈⋃ℛ_q
     
    
     b_q (x,y)     otherwise

where  O_q is the identity or a rotation by a multiple of π/2 and τ_q is a translation so that b_q+1 remains divergence free.  We now have to define the new collection ℛ_q+1 of rectangles.
Thanks to Lemma <ref>, for each R ∈ℛ_q, there exists a collection ℛ_q+1(R) of 2 n_q+1 mutually disjoint rectangles such that for all R∈ℛ_q+1(R) we have that R⊆ R and

    R = [0, L_q+1] ×[ - A_q+1 + B_q+1/2, A_q+1 + B_q+1/2]

up to a translation and a rotation of π/2 and in these coordinates

    b_q+1(x,y) = 
     [ v_q+11_[- A_q+1/2, A_q+1/2](y);                              0 ]  ∀ (x,y) ∈R.

Note that thanks to Lemma <ref>  #ℛ_q+1(R) = 2 n_q+1 for all R ∈ℛ_q.
Finally, we define

    ℛ_q+1 = ⋃_R ∈ℛ_qℛ_q+1(R)  ,

hence #ℛ_q+1 = 2n_q+1· N_q = N_q+1.

Before proving further properties of the velocity fields, we define some additional objects useful for the proof of Theorem <ref>. First note that for any q ≥ 2 we can define the ancestor map 𝔞_q ℛ_q→ℛ_q-1 defined such that 𝔞_q(R) is the unique rectangle belonging to ℛ_q-1 such that R ⊆𝔞_q(R) ∈ℛ_q-1.
Let q ≥ 2 be arbitrary and let R ∈ℛ_q-1 be arbitrary.
Then we define ℰ_q(R) ⊆ℛ_q(R) by removing the two first and the two last rectangles in the branching structure (see Figure <ref>). 


Hence

    #ℰ_q(R) = 2n_q - 4   ∀ R ∈ℛ_q-1,   ∀ q ≥ 2.

We define ℰ_1 ⊆ℛ_1 to be the sub-collection of rectangles obtained by removing the two first and the two last rectangles in the branching structure of b_1. Hence #ℰ_1 = #ℛ_1 - 4 = 2n_1 - 4 = N_1 - 4. 
Then, we define ℰ_q inductively by the following relation

    ℰ_q = ⋃_R ∈ℰ_q-1ℰ_q(R)   ∀ q ≥ 2.

Hence

    #ℰ_q = ∏_j = 1^q (2 n_j - 4) = N_q ≥N_q/2.

Again, let q ≥ 2 be arbitrary and let R ∈ℛ_q-1. Define ℳ_q(R) ⊆ℛ_q(R) by removing on each side of the branching structure the ⌈n_q4⌉ first and ⌈n_q4⌉ last rectangles from ℛ_q(R).
Hence #ℳ_q(R) = 2 n_q - 4 ⌈n_q4⌉≥n_q/2.
 
Then set

    ℳ_q = ⋃_R ∈ℛ_q-1ℳ_q(R)

and note that #ℳ_q ≥N_q/2.
Finally, we will define a sub-collection of rectangles denoted by 𝒢_q ⊆ℛ_q. The meaning and usefulness of this sub-collection is not clear at this stage all we can say is that it is a sub-collection of “good” rectangles. Precisely, we define 𝒢_q as

    𝒢_q = { R ∈ℰ_q ∩ℳ_q : 
     𝔞_q(R) ∈ℳ_q-1,     (𝔞_q-1∘𝔞_q)(R) ∈ℳ_q-2, 
     (𝔞_q-2∘𝔞_q-1∘𝔞_q)(R) ∈ℳ_q-3,     (𝔞_q-3∘𝔞_q-2∘𝔞_q-1∘𝔞_q)(R) ∈ℳ_q-4
    }.

We observe that 

    #𝒢_q ≥ 4^-5#ℰ_q ≥ 4^-5Ñ_q ≥ 2^-11 N_q.

 
For the sequence of velocity fields { b_q }_q ≥ 1 it holds that:
 
    b_q - b_q-1_L^∞(^2)   ≤ 2 v_q-1≤ C a_q-1^-  + 1/2 + δ
     b_q - b_q-1_BV(^2)   ≤ C a_q-1^-  - 2 δ

 where C is a constant that depends only on a_0.
 
 
 By the construction above, there exists a collection ℛ_q-1 of N_q-1 rectangles such that up to a translation and a rotation
 
    R = [0, L_q-1] ×[ - A_q-1 + B_q-12, A_q-1 + B_q - 12]   ∀ R ∈ℛ_q-1.

 and the velocity field b_q-1 is a straight pipe whenever restricted to any of the rectangles in ℛ_q-1.  
 Equivalent to the above definition we have 
 
    b_q = b_q-1 - ∑_R ∈ℛ_q-1 b_q-11_R_straight pipes     + ∑_R ∈ℛ_q-1 W_L_q-1, A_q-1, B_q-1, A_q, B_q, n_q, v_q-1_new branching-merging pipes .

  By construction b_q-1_L^∞ (R)≤  v_q-1 for any R ∈ℛ_q-1  and the new  velocities of the branching-merging pipes are bounded by v_q thanks to Lemma <ref>. As a consequence,
  
    b_q - b_q-1_L^∞(^2)≤ 2 v_q-1.

In order to prove (<ref>), we observe that b_q - b_q-1 is identically zero except in the N_q-1 rectangles belonging to ℛ_q-1. Then, in order to prove (<ref>), it suffices to bound b_q _BV(R_i) and b_q-1_BV(R_i). From the construction, we directly see that

    b_q-1_BV(R)   ≤ 2 L_q-1 v_q-1; 
     b_q_BV(R)   ≤ 2 n_q (2 L_q-1 + 4 L_q) v_q-1.

for all R ∈ℛ_q-1.
Hence

    b_q - b_q-1_BV(^2)≤ N_q-1[ 2 L_q-1 v_q-1 + 2 n_q (2 L_q-1 + 4 L_q) v_q-1] ≤ C a_q-1^-  - 2 δ

 where C is a constant that depends only on a_0. This ends the proof of Lemma <ref>.

In the next lemma, we recall some of the most important properties of the velocity fields { b_q }_q ≥ 1

 The sequence of velocity fields { b_q }_q ≥ 1 constructed above satisfies the following properties for all q ≥ 2
 
  
  *  
  For all R ∈ℛ_q-1 it holds that up to a rotation and a translation
  
    R = [0, L_q-1] ×[ - A_q-1 + B_q-12, A_q-1 + B_q-12]

  and in the coordinates of R we have
  
    b_q-1(x,y) 
       =
       [ v_q-11_[- A_q-1/2, A_q-1/2](y);                              0 ]  ∀ (x,y) ∈ R.

  
  *  Each R ∈ℛ_q-1 contains a collection of 2 n_q rectangles given by ℛ_q(R) such that for all R∈ℛ_q(R) 
  
    R = [0, L_q] ×[ - A_q + B_q2, A_q + B_q2]

  up to a rotation and a translation and in these coordinates
  
    b_q(x,y) 
       =
       [ v_q1_[- A_q/2, A_q/2](y);                        0 ]  ∀ (x,y) ∈R.

  
  *  Each R ∈ℛ_q contains a collection of rectangles denoted by ℱ_q (R) of cardinality #ℱ_q (R) ≤ 6 n_q+1 such that for all R∈ℱ_q (R) up to a rotation and a translation
  
    R = [0, L_q+1] ×[ - A_q+1, A_q+1]

  and
  
    b_q+2^(2)_L^∞(R)≤ v_q+1  ∀R∈ℱ_q (R)   and   b_q+2^(2)≡ 0  in R ∖⋃_R∈ℱ_q (R)R.

  Moreover,
  
    inf_R_1, R_2 ∈ℱ_q (R)( R_1, R_2 ) ≥B_q+1/4.

  


The points <ref> and <ref> follow immediately from the construction so we only need to prove <ref>. 
In order to prove <ref>, we note that each R ∈ℛ_q contains a collection of 2 n_q+1 rectangles given by ℛ_q+1(R) such that for all R∈ℛ_q+1(R)

    R = [0, L_q+1] ×[ - A_q+1 + B_q+1/2, A_q+1 + B_q+1/2]

and in these coordinates

    b_q+1(x,y)
     =
     [ v_q+11_[ - A_q+1/2, A_q+1/2](y);                               0 ]  ∀ (x,y) ∈R.

In order to construct b_q+2 inside R from b_q+1 we replace the straight pipes in the rectangles of ℛ_q+1(R) by branching-merging pipes. For any R ∈ℛ_q and any R∈ℛ_q+1(R) we define (in the coordinates of R)

    F_R, R^1    = [0, L_q+1] ×[ - A_q+1 + B_q+1/2 , - B_q+1/2]; 
    
     F_R, R^2    = [0, L_q+1] ×[ - A_q+1, A_q+1]; 
    
     F_R, R^3    = [0, L_q+1] ×[ B_q+1/2 , A_q+1 + B_q+1/2].

From the construction, it is clear that

    ( b_q+2^(2)) ∩ (R ∩R) ⊆⋃_ξ = 1^3 F_R, R^ξ.

Now note that F_R, R^2 is a rectangle of length L_q+1 and width 2 A_q+1 whereas F_R, R^1 and F_R, R^3 are rectangles of length L_q+1 and width A_q+1. Moreover, F_R, R^1 and F_R, R^3 are situated at the boundary of R. Hence, by merging sets F_R, R^1 and F_R, R^3 where R∈ℛ_q+1(R) with each other when adjacent we obtain 2 n_q+1 - 1 rectangles of length L_q+1 and width 2 A_q+1 as well as 2 rectangles of length L_q+1 and width A_q+1. By including these two rectangles in larger rectangles and adding to this collection the collection of rectangles given by { F_R, R^2 }_R∈ℛ_q+1(R) we obtain a collection denoted by ℱ_q+1(R) which is of cardinality #ℱ_q+1(R) = 4 n_q+1 + 1 ≤ 6 n_q+1. This ends the proof of <ref> and Lemma <ref>.

Before stating the last lemma of this subsection we will define the sets { D_q }_q ≥ 1.  Let q ≥ 5 be arbitrary and define for each R ∈𝒢_q the set

    D_q,R = [ 0 , L_q/3] ×[ - A_q/2 - B_q/50 , - A_q/2 - B_q/100] ⊆ R

in the coordinates of R.
Then, we define D_q (see Figure <ref>) as

    D_q = ⋃_R ∈𝒢_q D_q,R.

 
We can now prove the following lemma.

 Let D_q ⊂^2 constructed above and  𝒢_q be the collection defined in (<ref>). Then,
  the following properties hold true
 
  
  * inf_q ≥ 1^2(D_q) > 0; 
  
  * for all R ∈𝒢_q, we have b_q+2^(1)_L^∞(D_q,R)≤ v_q+2 and in the coordinates of R we have 
  
    b_q+2(x,y) = b_q+2(x + A_q+2 + B_q+2, y)   ∀ (x,y) ∈ D_q,R[ B_q150].

 


Point <ref> follows from the fact that

    ^2 (D_q) = #𝒢_q ·B_q/100·L_q/3(<ref>)≥ 2^-11 N_q ·B_q L_q/300≥ c > 0

for some constant c depending only on a_0. Point <ref> is a direct consequence of the construction.

We end the subsection by defining some sets which will be useful for the proof of Theorem <ref>. Let q ≥ 2 be arbitrary. For any R ∈ℰ_q-1, up to a rotation and a translation

    R = [0, L_q-1] ×[ - A_q-1 + B_q-1/2, A_q-1 + B_q-1/2].

In these coordinates, we define the sets

    C_q-1, R = { 0 }×[ - A_q-1/4, A_q-1/4] ⊆ R,   Γ_q-1, R = { 0 }×[ - A_q-1/2 + A_q, A_q-1/2 - A_q ] ⊆ R.

and

    S_q-1, R = [0, L_q-1] ×[- A_q-1/2 - ℓ_q-1, - A_q-1/2 + ℓ_q-1] ∪ [0, L_q-1] ×[A_q-1/2 - ℓ_q-1, A_q-1/2 + ℓ_q-1] ⊆ R

where ℓ_q-1 is a mollification parameter which will be explicitly defined at beginning of Subsection <ref>. Note that this corresponds to the set marked in orange in Figure <ref>.
Then, we define

    C_q-1 = ⋃_R ∈ℰ_q-1 C_q-1, R,   Γ_q-1 = ⋃_R ∈ℰ_q-1Γ_q-1, R  and   S_q-1 = ⋃_R ∈ℰ_q-1 S_q-1, R.




 §.§ Construction of C^α velocity fields u_q

The purpose of this subsection is to construct a sequence of C^α-velocity fields { u_q }_q ≥ 1 whose limit u ∈ C^α(^2, ^2) will be the velocity field in Theorem <ref>. We start by defining a sequence of mollification parameters {ℓ_q }_q ≥ 1 as

    ℓ_q = a_q^v_q+1v_qA_q+2.

Then we define { w_q }_q ≥ 1 as the sequence of velocity fields given by

    w_q =
     
      b_q - b_q-4    if q ∈ 4;
    
      0    otherwise.

The sequence { u_q }_q ≥ 0 is then given by

    u_q = b_0 ⋆φ_ℓ_0 + ∑_j = 1^q w_j ⋆φ_ℓ_j.

Of course, one could have selected w_q to be b_q - b_q-1 for each q instead of (<ref>). This choice however would destroy useful geometric features which simplify the proof of Theorem <ref> greatly. We now prove several lemmas regarding the properties of the sequence { u_q }_q ≥ 0.

  
Let { u_q }_q ≥ 0⊂ C^∞(^2; ^2) be the sequence of velocity fields defined above in (<ref>), then there exist a constant C>0 and >0 independent on q ∈ such that the following inequalities hold for all q ≥ 1
 
    u_q+1 - u_q _C^α(^2; ^2)≤ C a_q^;  
     u_q ·∇ u_q - u_q-1·∇ u_q-1_C^α≤ C a_q^ε̃;   
    κ_q Δ u_q+2_C^α≤ C q a_q^ε̃.
 
 



 By (<ref>), w_q _L^∞(^2; ^2)≲ v_q-4 so that
  
    w_q ⋆φ_ℓ_q_C^0(^2; ^2)   ≲ v_q-4 , 
    ∇ ( w_q ⋆φ_ℓ_q ) _C^0(^2; ^2)   ≲ v_q-4ℓ_q^-1.

Moreover, Equations (<ref>) imply that there exists a > 0 such that

    12+δ -  - α (1 + δ)^4 ( 3  + 3 δ + (1+δ)^2/2 + δ) > (1 + δ)^3 
    
    1 - (2 + δ)  - (1 + α) (1 + δ)^5 ( 3  + 3 δ + (1+δ)^2/2 + δ) > (1 + δ)^5 
    1 + 2(1 + δ)^52 + δ -  - (2 + α) (1 + δ)^4 ( 3  + 3 δ + (1+δ)^2/2 + δ) > (1 + δ)^4

From the definition of ℓ_q, we find ℓ_q ≳ a_q^3  + 3 δ + (1 + δ)^2/2 + δ
and therefore
 
    w_q ⋆φ_ℓ_q_C^α(^2; ^2)   ≲ v_q-4ℓ_q^- α≲ a_q-4^-  + 1/2+ δ - α (1 + δ)^4 ( 3  + 3 δ + (1 + δ)^2/2 + δ)≲ a_q -1 ^.  ,

 Hence,
 
    u_q+1 - u_q _C^α(^2; ^2)≤ w_q+1⋆φ_ℓ_q+1_C^α(^2; ^2)≲ a_q^.

This proves (<ref>). 
 To prove (<ref>) we observe that (<ref>) implies
 
    u_q ·∇ u_q  =  ∑_i, j =0^q  w_i ⋆φ_ℓ_i·∇ (w_j ⋆φ_ℓ_j )      =   ∑_i =0^q  w_i ⋆φ_ℓ_i·∇ (w_i ⋆φ_ℓ_i ) +   ∑_i =0^q-1w_i ⋆φ_ℓ_i·∇ (w_i+1⋆φ_ℓ_i+1 )
     
          +  ∑_i =0^q-1w_i+1⋆φ_ℓ_i+1·∇ (w_i⋆φ_ℓ_i )
  
 where we use the shorthand notation w_0  b_0  and for the last equality we used that 
 
    (w_i ⋆φ_ℓ_i ) ∩ (   w_j ⋆φ_ℓ_j ) = ∅  for any    |i-j| > 1

 which follows from the construction.
 Therefore,
 
    (u_q ·∇) u_q - (u_q-1·∇) u_q-1   = ((w_q⋆φ_ℓ_q) ·∇) (w_q⋆φ_ℓ_q) + ((w_q-1⋆φ_ℓ_q-1) ·∇) (w_q⋆φ_ℓ_q) 
            + ((w_q⋆φ_ℓ_q) ·∇) (w_q-1⋆φ_ℓ_q-1)

 so that
 
    (u_q ·∇) u_q - (u_q-1·∇) u_q-1_C^k   ≤ w_q ⋆φ_ℓ_q_C^k+1 w_q ⋆φ_ℓ_q_C^0 +  w_q ⋆φ_ℓ_q_C^k+1 w_q-1⋆φ_ℓ_q-1_C^0
          +  w_q-1⋆φ_ℓ_q-1_C^k+1 w_q⋆φ_ℓ_q_C^0
       ≲ v_q-4^2 ℓ_q^- k-1 + v_q-4 v_q-5ℓ_q^-k-1 +  v_q-4 v_q-5ℓ_q-1^-k-1
       ≲ v_q-4 v_q-5ℓ_q^- k -1 ,
  
where the last holds from the choice of the parameters (<ref>) and (<ref>). Thanks to interpolation inequalities, we have 

    u_q ·∇ u_q - u_q-1·∇ u_q-1_C^α≲ v_q-4 v_q-5ℓ_q^-1 - α≲ a_q-5^1 - (2 + δ) - (1 + α)(1 + δ)^5 ( 3  + 3 δ + (1 + δ)^2/2 + δ)≲ a_q^.

This proves (<ref>). 
Finally, we prove (<ref>). 
For all q ≥ 1 and all k ≥ 1, we have

    u_q _C^k≤∑_j = 0^q w_j ⋆φ_ℓ_j_C^k≲∑_j = 0^qℓ_j^-k v_j-4≲ q ℓ_q^-k v_q-4

since q ↦ℓ_q^-k v_q-4 is increasing.
Hence by interpolation

    κ_q Δ u_q _C^α≲κ_q q ℓ_q^-2 - α v_q-4≲ q a_q-4^- (2 + α)(1 + δ)^4 ( 3  + 3 δ + (1+δ)^2/2 + δ) -  + 1 + 2 (1 + δ)^5/2 + δ≲ q a_q^

where we used (<ref>) and (<ref>).


Thanks to this lemma, it is clear that { u_q }_q ≥ 0 is a Cauchy sequence in C^α(^2; ^2) and hence we may define

    u = lim_q →∞ u_q ∈ C^α(^2; ^2).

In particular, a direct implication of this definition is that for any q ∈, we have

    u_q+4 - u_q _L^∞≲ v_q  .


From Lemma <ref> and definition of u_q as in (<ref>) the following lemma follows.

 The sequence of velocity fields { u_q }_q ≥ 0 constructed above satisfies the following properties for all q ∈{ 2 + 4 k : k }:
 
  
  *  
  For each R ∈ℛ_q-1 and each R∈ℛ_q(R) it holds that
  
    u_q+2(x,y) = 
       [ v_q;   0;     ]  ∀ (x,y) ∈[0, L_q/2] ×[- A_q/5, A_q/5] ⊆R

  in the coordinates of R.
  
  *  
  For each R ∈ℛ_q, the collection of rectangles ℱ_q(R) given by Item <ref> of Lemma <ref> has the property that
  
    ( u_q+2^(2)) ∩ R ⊆⋃_R∈ℱ_q(R)R

  and
  
    u_q+2^(2)_L^∞(R)≤ v_q+1  ∀R∈ℱ_q(R).

 






 §.§ Stability results

In this subsection, we will establish stability results. 
Precisely, we will conclude that stochastic backward trajectories starting from C_q-1 remain inside the pipe structure and eventually reach Γ_1. The proof is decomposed into two lemmas. Firstly, we show that any stochastic backward trajectory starting from a point in Γ_p reaches Γ_p-1 for all p ≤ q - 4. 
Secondly, we prove that any backward stochastic trajectory starting from C_q-1 reaches Γ_q-4. The main ingredient in these proofs is Lemma <ref>. An extra technical point comes from how we constructed in the velocity fields { u_q }_q ≥ 1. Indeed, recall that b_q+1 is constructed from b_q, by replacing straight pipes by branching-merging pipes. When constructing the velocity fields { u_q }_q ≥ 1 we mollify b_j - b_j-4 with different mollification parameters ℓ_j. Since the mollification parameters ℓ_q go to zero as q →∞, a rectangle R ∈ℛ_j-1 with (j-1) ∈ 4 does not simply contain a mollified branching-merging pipe but also shear flows in the thin neighbourhoods given by S_j-1, R (see Figure <ref> where this set is marked in orange). Hence, we have to take into account that these shear flows with velocity bounded by v_j-1 intersect the pipes of width A_j perpendicularly. We refer to this as the intersection problem which is something that needs to be taken into account in the stability results.
Before stating the two lemmas, we introduce a subset Ω_T ⊆Ω defined by

    Ω_T = {ω∈Ω : sup_t ∈ [0, T] | W_t - W_T| ≤ K }

where T and K are numbers still to be determined (it will be chosen at the end of Section <ref>).

 Let q ∈ 2 + 4 and X_t,s^κ_q be the backward stochastic flow with respect to u_q+2 and noise parameter √(2 κ_q) starting from (x,y) ∈Γ_p for some p ≤ q-4. Let T_0 Ω→ [0,T] and consider the reverse stopping time τΩ→ [0, T_0] defined by
 
    τ(ω) = sup{ s ∈ (0, T_0] : X_T_0,s^κ_q(x, y, ω) ∈Γ_p-1}∨ 0.

 Then there exist three constants Q, C and C^' depending only on a_0 such that if q ≥ Q then
 
    C a_p-1^3 δ≤ T_0(ω) - τ(ω) ≤ C^' a_p-1^  ∀ω∈Ω_T.




 We fix an arbitrary ω∈Ω_T and write T_0 = T_0(ω).
 Let t_min be the smallest time such that 
 
    X_T_0, t_min(x,y) ∈Γ_p - 1( A_p+1A_p/3 A_p).

 and t_max be the largest time with the same property.
 It is clear from the construction that
 
    a_p-1^3 δ≲B_pv_p-1 - A_p+1A_p/3 A_p v_p-1≤ T_0 - t_max≤ T_0 - t_min≤L_p + A_p-1v_p + L_p-1v_p-1 + A_p+1A_p/3 A_p v_p-1≲ a_p-1^.

 We have to consider two distinct cases: p ∈ 1 + 4 or p ∉1 + 4. 

 Case 1:  (p ∉1 + 4).
 By definition of the parameters, (see (<ref>), (<ref>))
 
    A_p/A_pℓ_p ≤A_p+1A_p/3 A_p

 Therefore, Lemma <ref> gives
 
    ∫_t_min^T_0∇ u_q+2_L^∞( B_A_p+1A_p/3 A_p(X_T_0,t(x,y)) )  dt ≤ C

 where C is a universal constant given by Lemma <ref>.
 Define the reverse stopping time τΩ→ [t_min, T_0] by
 
    τ(ω) = sup{ t ∈ [t_min, T_0] : |X_T_0,t^κ_q(x,y,ω) - X_T_0,t(x,y)| > A_p+1A_p/3 A_p}∨ t_min.

 Then, thanks to Lemma <ref>, for all ω∈Ω_T and all t ∈ [τ(ω), T_0]
 
    |X_T_0, t^κ_q(x,y,ω) - X_T_0, t(x,y)|    ≤ 2 B_q K exp( ∫_τ(ω)^T_0∇ u_q+2_L^∞( B_A_p+1A_p/3 A_p(X_T_0, s(x,y)) )  ds ) 
       ≤ 2 B_q K exp( ∫_t_min^T_0∇ u_q+2_L^∞( B_A_p+1A_p/3 A_p(X_T_0, s(x,y)) )  ds ) (<ref>)≤ 2 ^C B_q K.

 If 2 ^C B_q K ≤A_p+1A_p/3 A_p then it immediately follows from the inequalities above that τ(ω) = t_min for all ω∈Ω̃_T. Hence 
 
    |X_T_0, t^κ_q(x,y,ω) - X_T_0, t(x,y)| ≤ 2 ^C B_q K   ∀ω∈Ω_T,  ∀ t ∈ [t_min, T_0].

 From this, it immediately follows that (<ref>) holds for the stopping time τ. To finish the proof in this case, all we need to show is 2 ^C B_q K ≤A_p+1A_p/3 A_p, which thanks to (<ref>) and (<ref>) reduces to a_q^1+δ/2 + δ≪ a_p^δ a_p-1^ + δ + 1 + δ/2 + δ to reabsorb constants depending on a_0, C and K. This holds since p ≤ q-4 and (<ref>). 

 Case 2:  (p ∈ 1 + 4).
 In this case, we have to consider the so-called intersection problem. 
 We note that p ≤ q-5. 
 We start by studying the backward Lagrangian flow.
We will work in the coordinates of an arbitrary rectangle R ∈ℰ_p-1, see Figure <ref>. 
 Let t∈ [0, T_0] be the largest time such that X_T_0, t(x,y) ∉S_p-1, R(2 ℓ_p-1) for all t ∈ [0, t]. 
 We refer to (<ref>) for the definition of the set S_p-1, R as well as to Figure <ref>.
 Define the reverse stopping time τ̃Ω→ [0,T] as
 
    τ̃(ω) = sup{ t ∈ [t,T_0] : |X_T_0,t^κ_q(x,y,ω) - X_T_0,t(x,y)| > A_p+1/10}.

 It is clear that u_q+2^(2)(X_T_0,t^κ_q(x,y,ω)) = u_q+2^(2)(X_T_0,t(x,y)) for all [τ̃(ω), T_0] and all ω∈Ω_T. For this we conclude that
 
    |X_T_0, t^κ_q   (2)(x,y,ω) - X_T_0, t^(2)(x,y)| ≤ B_q K

 for all [τ̃(ω), T_0] and all ω∈Ω_T.
 For the first component, we observe that for all [τ̃(ω), T_0] and all ω∈Ω_T
 
    |X_T_0, t^κ_q   (1)(x,y,ω) - X_T_0, t^(1)(x,y)|    ≤∫_τ̃(ω)^T_0 |u_q+2^(1)(X_T_0, s^κ_q(x,y,ω) )|   ds + ∫_τ̃(ω)^T_0 |u_q+2^(1)(X_T_0, s(x,y) )| + B_q 
       ≤6 ℓ_p-1/v_p v_p-1 + 2 ℓ_p-1/v_p v_p-1 + B_q K ≤A_p+1/20

 where we used (<ref>) and a_0 sufficiently small so that a_0^≤ K. Hence τ̃(ω) = t.
 At this point we can finish the proof in the same way as in the first case.
 
 
Let q ∈ 2 + 4 and X_t,s^κ_q be the backward stochastic flow with respect to u_q+2 and noise parameter √(2 κ_q) starting from (x,y) ∈ C_q-1. Let T_0 Ω→ [0,T] and consider the stopping time τΩ→ [0, T_0] defined by
 
    τ(ω) = sup{ s ∈ (0, T_0] : X_T_0,s^κ_q(x,y,ω) ∈Γ_q-4}

 Then there exist three constants Q, C and C^' independent of q such that for all q ≥ Q
 
    C a_q-4^3 δ≤ T_0(ω) - τ(ω) ≤ C^' a_q-4^  ∀ω∈Ω_T.

 
 
  We fix an arbitrary ω∈Ω_T and write T_0 = T_0(ω).
The first challenge to overcome is the so-called intersection problem. By arguments similar to Case 2 in the proof of the previous lemma, we overcome the intersection problem because
  
    B_q K v_q-2/v_q-1≤A_q-1/100.

  for q large enough.
  The remainder is based on Lemma <ref> and Lemma <ref> in the same way as the proof of Proposition <ref>.
  Let t_min be the smallest time so that 
  
    X_T_0, t_min(x,y) ∈Γ_q-4( A_q-1/100)
 
  and let t_max be the largest time such that the same property holds.
  In a similar way as in the previous proof, we have
  
    a_q-3^3 δ≲ T_0 - t_max≤ T_0 - t_min≲ a_q-3^.

  Since A_q-1/A_q-1ℓ_q-1≤A_q-1/100 for q large enough, Lemma <ref> applies and gives
  
    ∫_t_min^T_0∇ u_q+2_L^∞( B_A_q-1/100(X_T_0,t(x,y)) )  dt ≤ 3C.

  where C is the universal constant coming from Lemma <ref>.
  Define the stopping time τ̃Ω→ [t_min, T_0] by
 
    τ̃(ω) = sup{ t ∈ [0, T_0] : |X_T_0,t^κ_q(x,y,ω) - X_T_0,t(x,y)| > A_q-1/100}∨ t_min.

 Then, thanks to Lemma <ref>, for all ω∈Ω_T and all t ∈ [τ̃(ω), T_0]
 
    |X_T_0,t^κ_q(x,y,ω) - X_T_0, t(x,y)|    ≤ 2 B_q K exp( ∫_τ̃(ω)^T_0∇ u_q+2_L^∞( B_A_q-1/100(X_T_0, s(x,y)) )  ds ) 
       ≤ 2 B_q K exp( ∫_t_min^T_0∇ u_q+2_L^∞( B_A_q-1/100(X_T_0, s(x,y)) )  ds ) ≤ 2 ^3 C B_q K.

 If 2 ^3 C B_q K ≤A_q-1/100 then it immediately follows from the inequalities above that τ^β(ω) = t_min for all ω∈Ω_T. Hence 
 
    |X_T_0, t^κ_q(x,y,ω) - X_T_0, t(x,y)| ≤ 2 ^3 C B_q K   ∀ω∈Ω_T,  ∀ t ∈ [t_min, T_0].

 From this, it immediately follows that (<ref>) holds for the stopping time defined in the statement of the proposition. To finish the proof in this case, all we need to show is that 2 ^3 C B_q K ≤A_q-1/100 for q large enough which holds thanks to (<ref>).
 
 The following corollary is an immediate consequence of the two previous lemmas.
 
 Let q ∈ 2 + 4 with q ≥ 10, let j < q and X_t,s^κ_q be the backward stochastic flow with respect to u_q+2 and noise parameter √(2 κ_q) starting from (x,y) ∈ C_q-1. Let T_0 Ω→ [0,T] and consider the stopping time τΩ→ [0,T_0] defined by
 
    τ(ω) = sup{ s ∈ [0,T_0] : X_T_0,s^κ_q(x,y,ω) ∈Γ_j}

 Then there exist three positive constants Q, C and C^' independent of q such that for all q ≥ Q
 
    C a_j^3 δ≤ T_0(ω) - τ(ω) ≤ C^' a_j^  ∀ω∈Ω_T.

 In particular, for all ω∈Ω_T, (X_T_0,0^κ_q(x,y,ω), (x,y) ) ≥ A_0.
 
 



§ PROOF OF THEOREM <REF>

In this section we prove that all the assumptions of Proposition <ref> are satisfied for the sequence of velocity fields { u_q+2}_q ∈ 2+4 and the sequence of diffusivity parameters {κ_q }_q ∈ 2+4.
Recall the definition of the dissipative sets D_q in Item <ref> in Lemma <ref>. We stress again that

    inf_q ≥ 1^2(D_q) >0.

We consider X_t,s the backward flow of u_q+2 with noise parameter √(2 κ_q) defined in Section <ref>. Firstly, we notice that   

    u - u_q+2_L^2^2/κ_q ≲v_q+2^2/κ_q≲ a_q+1^- 2  (1 + δ) + 2 δ/2 + δ≲ a_q+1^→ 0  ,

thanks to Equation (<ref>), Equations (<ref>) and (<ref>) in Lemma <ref> and the fact that 0< ≪δ < 1.

Before stating the main claim, we take T=1, but we could have considered any fixed positive time.




Our goal is now to prove the following: 

Main Claim:
There exist two constants c_1, c_2, Q > 0 depending only on a_0 such that for any q ≥ Q and any x ∈ D_q, there exist two sets Ω_q,1,x, Ω_q,2,x⊆Ω for which:

 
  * min{ (Ω_q,1,x),  (Ω_q,2,x) }≥c_1 for all q ≥ Q;
 
  * for all q ≥ Q, we have
 
    X_T,0(x, ω) ∈ B_c_2(x)   ∀ω∈Ω_q,1,x;

 and
 
    X_T,0(x, ω) ∉B_2 c_2(x)   ∀ω∈Ω_q,2,x.


Assuming that this last claim holds, the proof follows from Proposition <ref>. Indeed, the main claim implies that

    max( inf_ω∈Ω_q,1,x |X_T,0(x, ω) -  [X_T,0(x, ·)]| , inf_ω∈Ω_q,2,x |X_T,0(x, ω) -  [X_T,0(x, ·)]| ) ≥c_2/2  ∀ x ∈ D_q

which in turn leads to

    inf_x ∈ D_q[ |X_T,0(x, ω) -  [X_T,0(x, ·)]|^2 ] ≥c_1 c_2^2/4.

Since this last inequality holds for all q ≥ Q, all the assumptions of Proposition <ref> are satisfied and the proof of Theorem <ref> follows. It only remains to prove the main claim above.




Proof of Main Claim: The proof is divided into two parts. In the first part, we prove that there exists a set Ω_q,1,x⊂Ω such that (<ref>) holds. In the second part, we show that for any integer q ≥ 1, there exists a set Ω_q,2⊂Ω such that (<ref>) holds. 

Part 1: (Proof of (<ref>)) 
We now fix x ∈ D_q, then x ∈ D_q,R for some R ∈𝒢_q (see (<ref>)). We define the stopping time  (recalling that we denote as A(·) the enlarged set of A⊂^2)

    τ (ω) = sup{ s ∈ [0,T]: X_T,s (x, ω) ∉ D_q,R (B_q/100 ) } .

We claim that there exists a set Ω_q,1,x⊂Ω with ℙ (Ω_q,1,x) ≥ c >0, with the constant c>0 independent on q, such that τ (ω ) =0 for any ω∈Ω_q,1,x.
 Let 

    Ω_1 = {ω∈Ω : sup_t ∈ [0,T] |W_t - W_T| ≤1200}.

It is clear that (Ω_1) ≥ c >0. We work in the coordinates of R and we denote u_q+2 = (u_q+2^(1) , u_q+2^(2)).
It is clear from property <ref> in Lemma <ref> that u_q+2 is A_q+1 + B_q+1 periodic in the dissipative set D_q,R [B_q 150]. 
Recall the collection of rectangles ℱ_q(R) coming from Item <ref> in Lemma <ref> and note that it follows from Item <ref> of Lemma <ref> that

    ∫_Ω_1∫_τ(ω)^T | u_q+2^(2) ( X_T,s (x , ω))  | ds    dℙ (ω)    ≤∫_0^T v_q+1ℙ[ Ω_1 ∩{ω∈Ω : X_T,s (x , ω) ∈⋃_R∈ℱ_q(R)R}] ds.



An Itǒ-Tanaka trick applied to the first component of the velocity field (which is zero average) together with(<ref>) yields 

    sup_t ∈ [0,T]∫_Ω | ∫_t^Tu_q+2^(1)(X_1,s) ds  | ≲ a_q+1^- εℓ_q+2^- εB_q+1/B_q v_q+2≲ a_q+1^δ/2 + δ - 4  (1 + δ) A_q+1.

Indeed, using the Itǒ formula with f such that 

    Δ f = u_q+2^(1),     ∫ f =0  ,

we deduce 

    | ∫_t^Tu_q+2^(1) (X_s,1) ds  | = 1/κ_q | f(X_t,1) - f(x) + ∫_t^T∇ f (X_s,1) ·u_q+2 (X_s,1) ds + √(2 κ_q)∫_t^T ∇ f (X_s,1) · d W_s   |.

Thanks to Property <ref> in Lemma <ref> and (<ref>) we have 

    ∇^k f _L^∞≲ B_q+1^2-k u_q+2^(1)_C^ε (D_q,R (B_q /100) )≲ℓ_q+2^- ε B_q+1^2-k v_q+2
 for k=0,1. Therefore, using that κ_q = B_q^2 and the bounds on the parameters (<ref>), in particular v_q+2≤ B_q, we have 

    | ∫_t^Tu_q+2^(1) (X_s,1) ds  | ≤ f _L^∞/κ_q + u_q+2_L^∞∇ f _L^∞/κ_q + ∇ f _L^∞/√(κ_q)≲ a_q+1^- εB_q+1/B_q v_q+2 ,

where the norms are taken in the set D_q,R (B_q /100).
This proves (<ref>).
Therefore, by (<ref>) and Markov's inequality we obtain 

    ℙ  ( ω∈Ω : sup_t ∈ [0,T] | ∫_t^Tu_q+2^(1) (X_T,s (x, ω)) ds  | ≥   A_q+1 ) ≤ C a_q+1^δ/2 + δ - 4  (1 + δ)  ,

and we denote this set Ω_u_q+2.
 Using the previous property we conclude from the formula of the integral curves of X_T,s that 

    {ω∈Ω: X_T,s (x, ω ) ∈⋃_R∈ℱ_q(R)R}⊆{ω∈ (Ω_u_q+2)^c  :  √(κ_q) W_s^(1)∈⋃_R∈ℱ_q(R)π_1 ( R (A_q+1) ) }∪Ω_u_q+2 .

From this, it follows that

    ∫_0^T v_q+1ℙ( Ω_1 ∩ X_T,s (x , ω) ∈⋃_R∈ℱ_q(R)R) ds   
        ≤∫_0^T v_q+1ℙ( Ω_1 ∩√(κ_q) W_s^(1)∈⋃_R∈ℱ_q(R)π_1( R (A_q+1) ) ) ds 
      
          + C v_q+1 a_q+1^δ/2 + δ - 4  (1+ δ) ,

and we can bound, C v_q+1 a_q+1^δ/2 + δ - 4  (1+ δ)≤ a_0^2 B_q thanks to (<ref>) for any q sufficiently large thanks to δ≫ε, i.e. (<ref>).
 
Using Item <ref> of Lemma <ref> about the structure of the collection of rectangles ℱ_q(R), we can estimate the first term on the right-hand side above as follows:

    ∫_0^T v_q+1ℙ( Ω_1 ∩√(κ_q) W_s^(1)∈⋃_R∈ℱ_q(R)π_1( R (A_q+1) ) ) ds 
       ≤∫_0^T∑_j=0^n_q+1∫_ j B_q+1 +[ - 2 A_q+1 , 2 A_q+1] ∩  |y | ≤√(κ_q)1/√(4 π s κ_q)exp ( - |y|^2/4 κ_q s ) v_q+1 dy ds 
       ≤∫_0^T∑_j=0^2 B_q  /B_q+1∫_j B_q+1 +[ - 2 A_q+1 , 2 A_q+1]  1/√(4 π s κ_q)exp ( - |y|^2/4 κ_q s ) v_q+1 dy ds 
       ≤ B_q B_q+1A_q+1 v_q+1/√(κ_q)≤ a_0 B_q  ,

where in the last we used the properties of the parameters (<ref>). Then, by Markov for any x ∈ D_q we bound

    ℙ ( ω∈Ω_1 :  ∫_0^T | u_q+2^(2) (X_T,s (x, ω)) | ds >  a_0^1/2 B_q  ) ≤∫_Ω_1∫_0^T | u_q+2^(2) (X_T,s (x, ω)) | ds /a_0^1/2 B_q≤ a_0^1/2 ,

where in the last inequality we used the  previous computation (<ref>). Therefore,
 there exists Ω_q,1,x,1⊂Ω_1 with ℙ (Ω_q,1,x,1) ≥ℙ(Ω_1) - a_0^1/2  such that

    ∫_0^T | u_q+2^(2) (X_T,s (x, ω)) | ds ≤ a_0^1/2 B_q

for any  ω∈Ω_q,1,x,1.
 Finally, choosing Ω_q,1,x =  Ω_q,1,x,1∩Ω_u_q+2, satisfying ℙ (Ω_q,1,x) ≥ℙ (Ω_1) - 2 a_0 ≥ c>0 thanks to the fact that a_0 can be taken sufficiently small compared to the universal constant ℙ (Ω_1),  we get 

    |X_T, τ(x, ω) - x| ≤∫_0^T |u_q+2(X_T, s (x, ω))| ds  + √(2 k_q) |W_1 - W_0| ≤  2 a_0 B_q +  B_q/200≤B_q/100    ∀ω∈Ω_q,1,x ,

and  thanks to the fact that a_0 ≤1/200 we get τ (ω) ≡ 0 for any ω∈Ω_q,1,x. Therefore, we conclude the proof of  (<ref>).
 

Part 2: (Proof of (<ref>))
Fix some x ∈ D_q. Let R ∈𝒢_q be such that x ∈ D_q,R.
Recall that the set D_q,R (in the coordinates of R) is given by (see Item <ref> in Lemma <ref>)

    D_q,R = [ 0, L_q/3] ×[ A_q/2 + B_q/100, A_q/2 + B_q/50] ⊆ R.

From now on, we will always work in the coordinates of R. Let τ^(1), τ^(2), τ^(out)Ω→ [0, T] be the reverse stopping times defined by

    τ^()(ω)    = sup{ t ∈ [0, T] : X_T, t(x, ω) ∉ [0, L_q/3] × [ - A_q/2 - B_q/2, A_q/2 + B_q/2]  }. 
    τ^(1)(ω)    = sup{ t ∈ [0, T] : X_T, t(x, ω) ∈[ 0, L_q/3] ×{ 0 }}; 
    τ^(2)(ω)    = sup{ t ∈ [0, T] : X_T, t(x, ω) ∈{ 0 }×[ - A_q/2 - B_q/2, A_q/2 + B_q/2] };

Our first goal is to prove

    [ τ^(2)≥T2] ≥ c > 0

for some constant depending only on a_0.
We will compute

    [ τ^(1)≥3 T4, τ^()≥T2,  τ^(2)≥T2] 
            = [ τ^(2)≥T2|τ^(1)≥3 T4, τ^(out)≥T2] [ τ^(1)≥3 T4, τ^()≥T2]

Thanks to the fact that κ_q = B_q^2 and (<ref>) it holds that [ τ^(1)≥3T4, τ^()≥T2] ≥ c > 0.
Then we notice that

    [ τ^(2)≥T2|τ^(1)≥3 T4, τ^()≥T2] 
           ≥[ ∫_T/2^3 T/4 v_q 1_ [ X_T, t(x, ω) ∈ [0, L_q3] × [- A_q2, A_q2 ]]  dt ≥ L_q |τ^(1)≥34, τ^()≥12] 
           ≥[ ∫_T2^3 T4 v_q 1_ [ √(2 κ_q) W_t ∈  [- A_q2, A_q2 ] ]   dt ≥ L_q ] 
           ≥[ ∫_0^T41_[ √(2 κ_q/A_q^2) W_t ∈  [- 12, 12 ] ]   dt ≥L_qv_q] 
            = [ A_q^22 κ_q∫_0^2 κ_q/4 A_q^2 T1_[ √(2 κ_q/A_q^2) W_A_q^2/2 κ_q s∈  [- 12, 12 ] ]   ds ≥L_qv_q] 
            = [ ∫_0^2 κ_q/4 A_q^2 T 1_ [ W_s∈  [- 12, 12 ] ]   ds ≥2 κ_q L_qA_q^2 v_q] ≥ c_0

for some universal c_0 > 0.
Indeed, 2 κ_q/4 A_q^2≈ a_q^-2 and 2 κ_q L_qA_q^2 v_q≈ a_q^- up to constants depending only on a_0 and therefore the last inequality holds for q sufficiently large thanks to Theorem <ref>.
Hence, (<ref>) holds.
Define a new reverse stopping time, but now in the coordinates of R_i. Precisely, we define τ^(3)Ω→ [0,T] by

    τ^(3)(ω)    = sup{ t ∈ [0, T] : X_T, t(x, ω) ∈{ u_q+2 = v_q-1} [ B_q]  }.

Note that

    [ τ^(3)≥T4]    ≥ [ τ^(3)≥T4 , τ^(2)≥T2 ] 
       =  [ τ^(3)≥T4 | τ^(2)≥T2 ]  [ τ^(2)≥T2 ].

By construction of the velocity fields in Section <ref>, we see that

    [ τ^(3)≥T4 | τ^(2)≥T2 ] ≥ c_1  ,

for some constant c_1 >0.
From this and (<ref>) we conclude that

    [ τ^(3)≥T4] ≥ c_1 c >0  .

Now select K sufficiently large so that (Ω_T) > 1 - c_1 c/10. Thus, [ Ω_T ∩{τ^(3)≥T4} ] > c_1 c/2. We then conclude thanks to Corollary <ref>.



§ PROOF OF THEOREM <REF> ABOUT THE 3D FORCED NAVIER–STOKES EQUATIONS

 We use the trick of the (2+ 1/2)-dimensional Navier–Stokes equations that has been already  exploited in other papers, see for instance <cit.>. More precisely, we suppose that all the functions in the system are independent on the last variable x_3 where x = (x_1, x_2, x_3) ∈^3. In this case the first two equations decouple with the last one, i.e.  given the initial data u_, κ : ^2 →^2 and θ_, κ : ^2 → and the force f_ν : ^2 →^2, if (u_ν, p_ν , θ_ν) is a solution of 

    ∂_t u_ν + u_ν·∇ u_ν + ∇ p_ν = νΔ  u_ν + f_ν ,
    
     (u_ν) =0  ,
    
    
    u (0, · ) = u_, ν (·)  ,
    
    ∂_t θ_ν +  u_ν·∇θ_ν = νΔθ_ν ,
    
    θ_ν (0, ·) = θ_, ν (·)  ,

where u_ν : [0,1 ] ×^2 →^2 and p_ν : ^2 → and θ_ν : [0,1] ×^2 →,
then 
v_ν (t, x_1, x_2, x_3) = (u_ν (t, x_1, x_2), θ_ν (t, x_1, x_2)) and P_ν (t, x_1, x_2, x_3) = p_ν (t, x_1, x_2) is a solution of (<ref>) with initial datum v_, ν (x_1, x_2, x_3)= (u_, ν (x_1, x_2) , θ_, ν (x_1, x_2)) and force F_ν (x_1, x_2, x_3) = f_ν (x_1, x_2). 



We aim at finding a solution of (<ref>) which is independent on x_3.
We choose ν_q = κ_q, u_, ν_q (x_1, x_2)= u_q+2, where u_q+2  is the autonomous velocity field defined  in (<ref>), 
F_ν_q = ℒ ( u_q+2·∇ u_q+2 ) - ν_q Δ u_q+2, where ℒ is the Leray projector  into divergence-free velocity fields and θ_, ν = θ_ defined in Proposition <ref>.  Let θ_κ_q : [0,1] ×^2 → be the solution to 

    ∂_t θ_κ_q + u_q+2·∇θ_κ_q = κ_q Δθ_κ_q
    θ_κ_q (0, ·) = θ_ (·)  .

and p_q+2 : ^2 → be  the zero average solution to 

    Δ p_q+2 = -  (u_q+2⊗ u_q+2)  .


It is straightforward to check that (u_ν, p_ν, θ_ν)= (u_q+2, p_q+2, θ_κ_q)
is the unique solution to (<ref>) with the given initial data and force defined above (recall that  u_q+2  is the autonomous velocity field defined  in (<ref>)).
  Finally, for v_ν_q = (u_ν_q, θ_ν_q) we have
 
    lim sup_ν_q → 0ν_q ∫_0^1 ∫_^3 | ∇ v_ν_q (t, x)|^2 dx dt    ≥lim sup_κ_q → 0κ_q ∫_0^1 ∫_^3 | ∇θ_κ_q (t, x)|^2 dx dt >0  ,
 
where for the last we used Proposition <ref>, Proposition  and Lemma <ref> together with u - u_q+2_L^2/√(κ_q)→ 0 as κ_q → 0 thanks to (<ref>).
Thanks to Lemma <ref> it is straightforward to check that 
F_ν_q -  F_0 _C^α +  v_, ν_q - v__C^α→ 0 as ν_q → 0 for some force F_0 ∈ C^α and initial datum v_∈ C^α.

We now prove the last remaining property. Thanks to the fact that u_q+2→ u in C^α and  

    sup_ν_q θ_ν_q_L^∞≤θ__L^∞≤ C

then there exists a converging subsequence of { v_ν_q = (u_q+2, θ_ν_q) }_q in the weak*-L^∞ topology and v_0 = (u, θ_0) where θ_ν_q*-L^∞⇀θ_0 up to subsequences.
Thanks to the fact that all the functions are independent on x_3, it is straightforward to check that v_0 is a solution to the 3D forced Euler equations with initial datum v_∈ C^α and force F_0 ∈ C^α. The Lipschitz property of the energy of v_0 follows from Proposition <ref> since u is independent of time. This concludes the proof.


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