title: Homogenization and hypocoercivity for Fokker-Planck equations driven by weakly compressible shear flows
authors: Michele Coti Zelati Grigorios A. Pavliotis
text: 
  
  
  
  We study the long-time dynamics of two-dimensional linear Fokker-Planck equations driven by a drift that can be decomposed in the sum of a large
  shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive
  scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential equation, we 
  give explicit homogenization rates in terms of a family of time-scales depending on the parameter measuring the strength of the incompressible perturbation. This is achieved by exploiting an auxiliary Poisson problem, and by computing the related effective diffusion coefficients. Regarding
  the long-time behaviour of the solution of the Fokker-Planck equation, we provide explicit decay rates to the unique invariant measure by employing
  a quantitative version of the classical hypocoercivity scheme. From a fluid mechanics perspective, this turns out to be equivalent to quantifying the phenomenon of enhanced diffusion for slightly compressible shear flows.
  
  
  
  
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  § INTRODUCTION
  
  
  
  
  Scalar transport is an important problem with many applications to, e.g. atmosphere/ocean science and engineering CKRZ08, Golden_al_2020, kramer. The evolution of the density of a passive tracer is governed by the advection-diffusion (Fokker-Planck) equation
  
      _t ρ= - ∇· (ρ) + κΔρ,
  
  with an initial density ρ(0,)= ρ_0(), where =() denotes the (fluid) velocity field and κ>0 stands for the molecular diffusion coefficient. The stochastic differential equation corresponding to the advection-diffusion equation is
  
      (t) = ((t))  t + √(2κ) (t).
  
  One is usually interested in the long-time, large-scale behaviour of the dynamics of (<ref>) or, equivalently, (<ref>). It is well-known lions, PavlSt08 that, for periodic or random velocity fields, the dynamics of the passive scalar at large scales becomes diffusive and can be quantified by the effective diffusion tensor D Pap95,PavlSt08,kramer. The dependence of the diffusion tensor on the properties of the velocity field , in particular in the asymptotic regime of small molecular diffusion, is a problem that has attracted a lot of attention in recent decades Golden_al_2020, kramer. More precisely, let  be a smooth periodic vector field (which is the case that we will consider in this paper). If e is an arbitrary unit vector in ^d,  the rescaled process X^e_(t):=e ·(t/^2) converges weakly in C([0,T];  ) to a Brownian motion with diffusion coefficient D^e, that is
  
      X^e_(t) →√(2 D^e) W(t),    as →0,
  
  where D^e := e · D e, D being the diffusion matrix. In the above we have assumed that the vector field  is centered with respect to the invariant measure of the process (t) when restricted to the torus ^d=[0,2π)^d, see Equation (<ref>) below. This process is an ergodic Markov process with generator
  
      = ·∇ + κΔ.
  
  This is a partial differential operator on ^d, equipped with periodic boundary conditions. The calculation of the diffusion coefficient along the e-direction  D^e requires the solution of two PDEs on ^d, the stationary Fokker-Planck equation and an appropriate Poisson equation, together with the calculation of an integral over the unit torus <cit.>*Ch. 13. The stationary Fokker-Planck equation reads
  
      ^* ρ_∞ =0,
  
  where ^*, the Fokker-Planck operator appearing in (<ref>), is the L^2(^d)–adjoint of ; the Poisson equation is
  
      - ϕ^e = b^e :=· e.
  
  The diffusion coefficient is given by the formula
  
      D^e := κe + ∇ϕ^e ^2_L^2_ρ_∞ = κ∫_^d |e + ∇ϕ^e|^2 ρ_∞()  .
  
  The PDEs (<ref>) and (<ref>) are equipped with periodic boundary conditions and we have assumed the centering condition
  
      ∫_^d() ρ_∞()   = 0.
  
  The effect of a nonzero mean flow is studied in thesis, MajMcL93 and it will not be considered in this paper. Convergence theorems of the form (<ref>) can be proved using either PDE <cit.> or probabilistic techniques pardoux, bhatta.
  
  A question that has attracted a lot of interest, both from a mathematical and a computational perspective, is the calculation of the diffusion tensor for different types of vector fields . Since it can be calculated analytically only in very few cases, e.g. for shear flows or for gradient flows in one dimension <cit.>*Sec. 13.6, in most cases the best one can hope for is the derivation of estimates on the diffusion tensor and on its dependence on the parameters of the problem such as the molecular diffusivity κ. This problem has been studied in detail for two particular types of vector fields  in (<ref>), namely gradient flows = -∇ V or divergence–free flows, for which ∇· = 0. In the former case, where the SDE (<ref>) becomes
  
      (t) = -∇ V((t))   t  +√(2 κ)(t),
  
  it is well known that the diffusion is always depleted <cit.>*Ch. 13, namely
  
      D^e ≤κ,
  
  for all directions e. In fact, when κ≪ 1, the diffusion coefficient becomes exponentially small in κ <cit.>:
  
      D^e ∼ C_1 ^- C_2/ κ,   κ≪ 1.
  
  On the other hand, when the vector field is divergence–free, then diffusion is always enhanced <cit.>, <cit.>*Ch. 13. Furthermore, both lower and upper bounds for the diffusion coefficient are known:
  
      κ≤ D^e ≤κ + 1/κ,   κ∈ (0, +∞).
  
  The asymptotic behavior of the diffusion coefficient in the limit as κ→ 0 depends on the detailed properties of the vector field  and can be quite different in different directions of ^d <cit.>. This is also reflected on the scaling of the relevant time scales, e.g. the diffusive time scale; this scaling depends crucially on the divergence-free vector field, i.e on whether it has open or closed streamlines. A detailed study of this is presented in <cit.>.
  
  Much less is known about the diffusion coefficient for flows that are neither gradient nor divergence–free. Homogenization problems for compressible flows have been studied in a few papers McLaughlinForest99, vergassola. However, the problem of the derivation of rigorous estimates on the diffusion coefficient for periodic vector fields that are neither gradient nor divergence-free has not been addressed yet. This is precisely the problem that we address in this paper; in particular, we obtain quantitative information on the effect of compressible perturbations of divergence-free flows on the effective diffusion coefficient, and we also study enhanced dissipation rates, for compressible perturbations of shear flows. Our analysis is based on recently developed techniques BW13,BCZ15,CZ19, including the theory of hypocoercivity <cit.>. 
  
  Before discussing in detail the framework that we will consider in this paper we present our main results, we mention a couple of related problems. 
  
  
  
  
  
  ♢ Homogenization for Inertial Particles.
  
  The problem of homogenization and enhanced dissipation for velocity fields that are not divergence-free arises naturally in the study of inertial particles PavlStuZyg07, PavlStZyg09, PavlStBan06, PavSt05b. The equations of motion for inertial particles, written in non-dimensional form, read <cit.>
  
  
      = ( (t,) - ) +β D_t   (t,) + √(2  Pe^-1),
  
  
  where  denotes the Strouhal number, β the fluid density and D_t = ∂_t + ·∇ the material derivative. Rigorous homogenization results for dynamics of the form (<ref>) were obtained in HP04, PavlStuZyg07, PavSt05b. In recent work <cit.> it was shown that in the small inertia (small Strouhal number) limit, the dynamics (<ref>) reduces to a passive tracer equation of the form (<ref>) in a modified velocity field that is no longer incompressible. The velocity field that the inertial particles experiences is 
  
      _e =  - (1 - β) D_t .
  
  In particular, for time-independent flows we have that ∇·_e = - (1 - β) ∇·((·∇) ). Even though this quantity vanishes for shear flows, the flows studied in this paper, we believe that the connection between the study of inertial particles and the problem of homogenization and enhanced dissipation for compressible flows <cit.> is an interesting one and we plan to investigate this further in future work. Some preliminary numerical experiments are presented in Section <ref>.
  
  
  
  
  
  
  ♢ Nonreversible Langevin Samplers. 
  
  A fundamental problem in statistics and in computational statistical mechanics is that of sampling from a probability measure π() = 1/Z^-V()^2 that is known up to the normalization constant. A standard approach to sampling from π() is to consider dynamics that is ergodic with respect to this measure. The natural choice is that of the overdamped Langevin dynamics
  
      (t) = - ∇ V((t))   t + √(2) (t).
   
  The rate of convergence of (<ref>) to the target distribution π() = 1/Z^-V() is given by the Poincaré (spectral gap) and logarithmic Sobolev inequalities <cit.>, and it depends only on the properties of the potential function V. In order to speed up convergence to equilibrium and to reduce the asymptotic variance, a natural approach is to perturb the dynamics (<ref>) by adding a divergence-free perturbation 1/ν () with 
  
      ∇·( e^-V) = 0
  
  that does not change the invariant measure:
  
      ^ν(t) = (-∇ V(^ν(t)) + 1/ν(^ν(t)) )   t + √(2) (t).
  
  It is indeed possible to prove that the divergence-free perturbation accelerates convergence to the target distribution Hwang_al1993,Hwang_al2005,LelievreNierPavliotis2013 and, in addition, that it reduces the asymptotic variance <cit.>; the asymptotic variance plays a role analogous to that of the effective diffusion coefficient, and given by the same formula, in terms of an appropriate Poisson equation, or equivalently, of the Green-Kubo formula <cit.>. It is worth noting that the set-up considered in this paper is in essence the opposite to this, i.e. we are interested in analyzing the effect of reversible perturbations to incompressible (divergence-free) flows on the long time behaviour of advection-diffusion equations.  We also mention that, in order for the 
  divergence-free condition (<ref>) to be satisfied, it is sufficient for the velocity field  to be divergence-free and orthogonal to ∇ V. In this paper we will consider the SDE (<ref>) for such velocity fields:
  
      (t) =((t)) t + √(2) (t),
  
  with suitable initial conditions, with
  
      ()=1/ν()-∇ V(),     ·∇ V=0,
   
  and where ν>0 and :^d→^d is divergence-free and satisfies the condition
  
      ·∇ V=0.
   
  Our goal is to investigate the long-time behavior of its solutions in the limit ν→ 0. 
  
  
  
  
   §.§ Enhanced diffusion
  
  The probability density  ρ:[0,∞)×^d → of the solution (t) of  (<ref>) satisfies the Fokker-Planck equation 
  
      _t ρ+1/ν·∇ρ= ∇·(ρ∇ V +∇ρ),     ∫_^dρ(t,)=1.
  
  As mentioned above, the unique invariant density ρ_∞  of the dynamics (<ref>) is given by  
  the Gibbs measure 
  
      ρ_∞()=1/Z^- V(),      Z=∫_^d^-V().
  
  In order to study the convergence rates to ρ_∞ of solutions to the Fokker-Planck equation, it is convenient to normalize with respect to the invariant distribution 
  ρ_∞ and consider the unknown h defined by
  
      h(t,)=ρ(t,)/ρ_∞()-1,     ∈^d.
  
  Indeed, h satisfies the backward Kolmogorov equation
  
      _t h+1/ν·∇ h=Δ h-∇ V·∇ h,     h(0,)=h^in(),
  
  where h^in is defined in terms of the initial distribution function of the process (t) as 
  
      h^in()= ρ(0,)/ρ_∞()-1.
  
  Note that since (<ref>) conserves mass, the same is true for (<ref>) for the weighted mass
  
      ∫_^dh(t,)ρ_∞()=0,     ∀ t≥ 0.
  
  Define the L^2-weighted space 
  
      L^2_ρ_∞={ f:^d→,   ∫_^d |f()|^2ρ_∞()<∞,   ∫_^df()ρ_∞()=0 },
  
  endowed with the natural norm and scalar product
  
      ł f,g=̊∫_^d f() g()ρ_∞(),     f^2=∫_^d |f()|^2ρ_∞().
  
  It is straightforward to check that the operator Δ -∇ V·∇ is symmetric in L^2_ρ_∞ and
  
      łΔ f -∇ V·∇ f,g=̊-ł∇ f,∇ g,̊
  
  while, thanks to (<ref>),
  we have
  
      ł·∇ f,g=̊-ł  f,·∇ g,̊
  
  for sufficiently smooth functions f,g. 
  
  In advection-diffusion equations, when we take  V=0 in (<ref>), the enhancement of diffusive mixing
  in passive tracers  by a fast incompressible flow was studied in great generality in CKRZ08, Zlatos2010 from a qualitative standpoint, and quantitatively in the more recent works BW13, BCZ15, CZDE18,CZD19,CZ19,CZDri19,IXZ19,FI19,WEI18. 
  In particular, a necessary and sufficient condition for diffusion enhancement is that
  the operator ·∇ has no eigenfunctions in the homogeneous Sobolev space Ḣ^1.
  
  The main result of <cit.> on qualitative enhanced diffusion still holds with the presence of the potential V, thanks to the Hilbert space setting illustrated above, providing a framework for slightly compressible perturbations of incompressible flows. In this paper, we address the issue of quantitative estimates, when the velocity field  is a two-dimensional shear flow with simple critical points, and V is a potential depending on one of the two variables only. To put it in terms of the advection-diffusion equation (<ref>), we study a slightly compressible velocity field , in which the main incompressible part  in (<ref>) is a shear flow.
  We devise enhanced diffusion estimates via hypocoercivity methods, and relate them to the diffusive time-scales of the process (t). The corresponding result for the so-called Kolmogorov flow was proven in <cit.>, and later generalized in <cit.> to all  incompressible shear flows with a finite number of critical points. 
  
  
  
  
  
  
   §.§ A general abstract framework
  
  The setting described in Section <ref> can be put in a more general abstract fashion as follows, see <cit.>*Chapter 2. Suppose that we are given a Gibbs measure (<ref>). Any smooth vector field on ^d admits the decomposition 
  
      = 1/ν+∇lnρ_∞,
  
  where ∇· (ρ_∞ ) = 0 and ρ_∞ is the solution of the stationary Fokker-Planck equation
  
      - ∇· (ρ_∞) +  Δρ_∞ =0.
  
  In writing (<ref>) we have already normalized the various vector fields and we have introduced a parameter ν>0 which measures the strength of the deviation from the reversible dynamics. We remark that the stationary Fokker-Planck equation (<ref>) plays precisely the role of the Poisson equation - Δ V = ∇· in the Helmholtz decomposition in a flat L^2 space. Using now the decomposition (<ref>) we can decompose the generator  of the Markov process  on ^d into a symmetric and an antisymmetric part in L^2_ρ_∞ as defined in (<ref>), representing the reversible and irreversible parts of the dynamics, respectively:
  
      =1/ν+  ,
  
  where = (∇lnρ_∞)·∇ +  Δ and = ·∇. With this abstract setting at hand, the results of <cit.> can  be rephrased in terms of weighted L^2 spaces, giving a characterization of relaxation enhancement in terms of eigenfunctions of 
  in the (operator) domain of ^1/2. In this paper, we prove a quantitive version of this result in a special case, described in the next section. Estimates on the diffusion coefficient as a function of strength of the nonreversible perturbation, in the abstract setting considered in this subsection, are presented in <cit.>.
  
  
  
  Alternatively, one could ask what the effect of a small reversible perturbation of the divergence-free dynamics is on the diffusion coefficient. The two approaches are equivalent, and in this paper, we prefer to consider the influence of a large incompressible flow on the dynamics, as it is clear from the way we write (<ref>). As we shall see, a simple time rescaling makes the problem equivalent to a small compressible perturbation, with noise strength equal to √(2ν). 
  
  
  
  
  
  
   §.§ Setting and main results
  
  Let u,v∈ C^1() be two given functions, and define 
  
      (x,y)=[ u(y);    0 ],
           V(y)=-∫_0^yv(y') y'.
  
  Throughout the paper, we will assume a zero-mean condition on v, namely
  
      ∫_ v(y) y=0,
  
  and a centering condition for u, that is,
  
      ∫_ u(y)^-V(y) y=0.
  
  Notice that thanks to (<ref>), V is a periodic potential.
  Writing (<ref>) explicitly for the two-component process (t)=(X(t),Y(t)), we obtain the system 
  
      X(t)=1/νu(Y(t))  dt+√(2)  W_1(t),
       Y(t)= v(Y(t))   dt +√(2)  W_2(t).
  
  Note that (<ref>) is automatically satisfied, independently of the choice of u and v. 
  It turns out that the noise driving the process {X(t)}_t≥ 0 is not essential, so that we consider the stochastic differential equations
  
      X(t)=1/νu(Y(t))   dt,
       Y(t)= v(Y(t))   dt +√(2)  W(t),
  
  with initial conditions 
  
      X(0)=X_0,      Y(0)=Y_0.
  
  Our first main result is a homogenization theorem for a suitable rescaling of the solution of  (<ref>), with explicit rates of convergence.
  
  Assume u,v∈ C^1() are given functions such that (<ref>) and (<ref>) hold. Consider the solution  (X(t),Y(t)) of (<ref>), and for β>0 define the one-parameter family of rescaled processes
  
      X^ν(t)=ν^1+β X(t/ν^2β),      Y^ν(t)=ν^β Y(t/ν^2β).
  
  Let α∈(0,1/2), and p∈[1/2α,∞), and assume that the initial conditions satisfy
  
      |X_0|^p<∞,      |Y_0|^p<∞.
  
  Then, for any T>0,
  there holds the convergence estimate
  
      sup_t∈[0,T] |X^ν(t)- √(2D_u)  W_x(t)|^p+ sup_t∈[0,T] |Y^ν(t)- √(2D_v)  W_y(t)|^p ≲ν^αβ p,
   
  for two independent one-dimensional Brownian motions W_x(t),   W_y(t), where
  
      D_u= _yχ_u^2,      D_v= 1+_yχ_v^2,
  
  and χ_u,χ_v:→ are the unique solutions to the one-dimensional periodic Poisson problems
  
      v _y χ_u+_yyχ_u=-u,     ∫_χ_u(y)ρ_∞(y) y=0,
         v _y χ_v+_yyχ_v=-v,     ∫_χ_v(y)ρ_∞(y) y=0.
  
  
  
  
  It will be clear from the proof that deducing a convergence estimate for the fully diffusive problem (<ref>) amounts to changing the diffusion coefficient D_u in (<ref>) to 1+_yχ_u^2.
  
  
  
  
  Due to the degenerate noise considered for (<ref>), the unique invariant measure for (X(t),Y(t)) is the measure HP04, HP08
  
  
      ρ_∞(y)= 1/Z^- V(y),      Z=2π∫_^-V(y) y,
  
  while (<ref>) becomes 
  
      _t h+1/ν u_x h=_yy h-v_y h,     h(0,x,y)=h^in(x,y).
  
  In what follows, we will be consistent with the notation introduced in (<ref>) and (<ref>), so that
  
      L^2_ρ_∞={ f:^2→,   ∫_^2 |f(x,y)|^2ρ_∞(y) x  y<∞,   ∫_^2f(x,y)ρ_∞(y) x y=0 },
  
  and
  
      ł f,g=̊∫_^2 f(x,y) g(x,y)ρ_∞(y) x  y,     f^2=∫_^2 |f(x,y)|^2ρ_∞(y) x  y.
  
  An important feature of (<ref>) is that it decouples in the x-Fourier modes. By expanding the solution h as a Fourier series in the  x variable, namely
  
      h(t,x,y)=∑_ℓ∈𝔥_ℓ(t,y)^iℓ x,     𝔥_ℓ(t,y)=1/2π∫_0^2πh(t,x,y)^-iℓ x y.
   
  for any integer ℓ we have from (<ref>) that
  
      _t 𝔥_ℓ+iℓ/ν u 𝔥_ℓ=_yy𝔥_ℓ-v_y 𝔥_ℓ,    𝔥_ℓ(0,y)=𝔥_ℓ^in(y).
  
  However, in order not to deal with  complex-valued function and heavier notation, it is more convenient to deal with functions that 
  are localized on a single band ±ℓ. Thus, for k∈_0 we set
  
      h_k(t,x,y):=∑_|ℓ|=k𝔥_ℓ(t,y)^iℓ x.
  
  This way we may write 
  
      h(t,x,y)=∑_k∈_0h_k(t,x,y),
   
  as a sum of real-valued functions h_k that are localized in x-frequency on a single band ± k, k∈_0. In particular,
  for the x-average of the function h corresponds to h_0=𝔥_0.
  When norms and scalar products are applied to Fourier modes, it is understood that we will consider the complex one-dimensional version of (<ref>), as no confusion will arise.
  
  
  
  
  Our second main result consists of explicit rates of convergence to 0 for h_k. 
  
  
  
  Assume u,v∈ C^2() are given functions such that (<ref>) and (<ref>) hold, and further assume that
  u has a finite number of critical points such that u”(y_crit)≠0.
  Then there exist constants ν_0,_0∈ (0,1)  such that the following holds:
  there exist positive numbers å_0,_0,_0 only depending on _0
  for each integer k∈ and ν>0 with
  ν k^-1≤ν_0
   the energy functional
  
      Ψ_k=1/2[h_k^2 + ν^1/2å_0/k^1/2_y h_k^2+2_0/kł u' _x h_k, _y h_k +̊_0/ν^1/2k^3/2 u'_x h_k^2 ]
  
  satisfies the differential inequality 
  
      Ψ_k+_0k^1/2/ν^1/2Ψ_k+å_0ν^1/2/2k^1/2_yyh_k-v_y h_k^2+_0/2ν^1/2k^3/2 u'_xy h_k^2 ≤ 0.
  
  for all t≥0.
  In particular
  
      Ψ_k(t) ≤^-_0k^1/2/ν^1/2tΨ_k(0),     ∀ t≥ 0.
  
  
  The above result confirms that the gradient perturbation given by the potential V does not affect the enhanced diffusion time-scales of the 
  backward Kolmogorov equations. From a fluid dynamics perspective, scalars advected by a shear flow ν^-1(u(y),0) (as studied in <cit.>) or the slightly compressible perturbation of a shear ν^-1(u(y),ν v(y)) have the same decay properties.
  
  While the dependence on ν and k of the functional Ψ_k may be cumbersome for interpreting 
  the real decay properties of h_k, the following simple consequence
  entails a clearer result that simply requires that the initial condition be in L^2_ρ_∞. The assumptions are the same as in Theorem <ref>.
  
  There exist constants ν_0,_0∈ (0,1) and c_0>1 such that the following holds:
  for each integer k∈ and ν>0 with
  ν k^-1≤ν_0
  there holds the estimate
  
      h_k(t)^2 ≤  c_0 h^in_k^2 ^-_0k^1/2ν^-1/2/1+|lnν|+ln kt  ,     ∀ t≥ 0.
  
  Moreover,
  
      h_0(t)^2 ≤h^in_0^2 ^-_0 t  ,     ∀ t≥ 0.
  
  
  Estimate (<ref>) contains two very important pieces of information. On the one hand, it quantifies precisely the influence of a large drift 
  in the Kolmogorov equation, which allows the convergence mode-by-mode. On the other hand, it shows how the drift has an instantaneous regularization effect in the x-variable, in which diffusion is not present: from L^2 initial data, (<ref>) shows that Fourier coefficients decay exponentially fast, giving rise to Gevrey-type regularization effects. It is worth mentioning that, due to the results in <cit.>, the  decay rate is 
  optimal up to the logarithmic correction.
  
  Since the x-average of h is not influenced by the drift (see (<ref>) for k=0),  the results of Corollary <ref> can be stated
  for the solution of the advection diffusion equation (<ref>) as follows. 
  
  There exist constants ν_0∈ (0,1) and c_0>1 such that the following holds:
  for each ν∈(0,ν_0] with
  there holds the estimate
  
      h(t)-∫_ h(t,x,y) x^2 ≤  c_0 h^in-∫_ h^in(x,y) x^2 ^-_0ν^-1/2/1+|lnν|t  ,     ∀ t≥ 0,
  
  for any h^in∈ L^2_ρ_∞.
  
  
  The result of Corollary <ref> is easily explained by looking at Figure <ref>, in which various snapshots of the solution 
  of (<ref>) with u(y)=-3cos (3y) and v(y)≡ 0 are plotted. 
  
  
  
  For time-scales faster than O(ν^1/2(1+|lnν|)),  the dominant behavior is mixing by incompressible velocities: (<ref>) says that if ν is chosen small enough, h is very close to its x-average, and hence tends to become x-independent. This coincides with the appearance of “horizontal” stripes. At this point, only diffusion is relevant, and the solution slowly decays to zero at a ν-independent rate, as prescribed by (<ref>).
  
  
  
  § RATES OF CONVERGENCE IN HOMOGENIZATION
  
  
  
  In this section we  prove Theorem <ref>. The main ingredient to prove explicit convergence rate is the use
  of a second Poisson equation and an auxiliary process that we introduce in the next section. The rates follows from the
  Dambis-Dubins-Schwarz theorem (see <cit.>*Thm 3.4.6) on time-change for martingales, together with the
  Hölder continuity properties of Brownian motion.
  
  
  
   §.§ Auxiliary processes and the Poisson equation
  
  In order to prove Theorem <ref>, we derive from (<ref>) the SDEs for the rescaled process
  in (<ref>), which read
  
      X^ν(t)    =1/ν^β u(Y^ν(t)/ν^β) t,
       Y^ν(t)    =1/ν^β v(Y^ν(t)/ν^β) t+√(2)  W(t),
  
  with initial conditions
  
      X^ν(0)=ν^1+β X_0,      Y^ν(0)=ν^β Y_0.
  
  In fact, it is convenient to introduce the auxiliary process 
  
      R^ν(t)=Y^ν(t)/ν^β,
  
  and re-write (<ref>) as an augmented system of the form
  
      X^ν(t)    =1/ν^β u(R^ν(t)) t,
       Y^ν(t)    =1/ν^β v(R^ν(t)) t+√(2)  W(t),
       R^ν(t)    =1/ν^2β v(R^ν(t)) t+√(2)/ν^β  W(t).
  
  In what follows, an important role will be played by the generator   of the process R^1 (i.e. for ν=1), namely the operator
  
      =v _y +_yy.
  
  Given a function ϕ∈ C^1() such that
  
      ∫_ϕ(y)ρ_∞(y) y=0,
  
  it is not hard to verify that the unique solution to the Poisson equation
  
      χ =ϕ,     ∫_χ(y)ρ_∞(y) y=0,
  
  is given by
  
      χ(y)=B+∫_0^y^V(y')[A+ ∫_0^y'ϕ(y̅)^-V(y̅)y̅] y',
  
  where the constant A,B are chosen to enforce periodicity and the weighted mean-zero condition in (<ref>) as
  
      A=-[∫_^V(y') y']^-1∫_∫_0^y'ϕ(y̅)^V(y')-V(y̅)y̅  y'
  
  and
  
      B=-∫_ρ_∞(y)∫_0^y^V(y')[A+ ∫_0^y'ϕ(y̅)^-V(y̅)y̅] y'  y.
  
  Here, the notation is that of (<ref>) and (<ref>). In particular, notice that χ∈ C^1() (at least).
  
  
  
   §.§ Convergence rates for Y^ν
  
  
  We begin to deal with the convergence for the process Y^ν.
  
  
  Let α∈(0,1/2) and p∈[1/2α,∞) be arbitrarily fixed, and assume that
  
      |Y_0|^p<∞.
  
  Then we have the convergence estimate
  
      sup_t∈[0,T] |Y^ν(t)- √(2D_v)  W(t)|^p ≲ν^αβ p,
   
  where
  
      D_v= 1+_yχ_v^2=∫_[1+_yχ_v(y)]^2 ρ_∞(y) y
  
  and χ_v:→ is the unique solution to the elliptic equation
  
      χ_v=-v,     ∫_χ_v(y)ρ_∞(y) y=0.
  
  
  
  
  First of all, notice that in light of (<ref>) and (<ref>), we have
  
      ∫_ v(y)ρ_∞(y) y =-1/Z∫_ V'(y) ^- V(y) y=1/Z∫__y(^- V(y)) y=0.
  
  As a consequence, the unique solution to (<ref>) satisfies χ_v∈ C^1().
  Then, using Ito's formula we find
  
      χ_v (R^ν)   =1/ν^2βχ_v(R^ν)+√(2)/ν^β_yχ_v(R^ν)  W
      =-1/ν^2β  v(R^ν)  t +√(2)/ν^β_yχ_v(R^ν)  W.
  
  In turn,
  
      1/ν^β∫_0^t  v(R^ν(s))  s = -ν^β[χ_v (R^ν(t)) - χ_v (R^ν(0))]+√(2)∫_0^t _yχ_v(R^ν(s))  W(s).
  
  Substituting in the equation for Y^ν, we arrive at
  
      Y^ν(t)=Y^ν(0)   -ν^β[χ_v (R^ν(t)) - χ_v (R^ν(0))]+√(2)∫_0^t [1+_yχ_v(R^ν(s)) ] W(s).
  
  Thus
  
      Y^ν(t)- √(2D_v)  W(t)   =Y^ν(0)    -ν^β[χ_v (R^ν(t)) - χ_v (R^ν(0))]
           +√(2)∫_0^t [1+_yχ_v(R^ν(s)) -√(D_v)] W(s)
  
  Hence, for any p≥ 1 we have
  
      |Y^ν(t)- √(2D_v)  W(t)|^p    ≲ |Y^ν(0)|^p  + ν^ pβ[|χ_v (R^ν(t))|^p +|χ_v (R^ν(0))|^p]
           + |∫_0^t (1+_yχ_v(R^ν(s)) -√(D_v)) W(s)|^p.
   
  The first two terms are essentially harmless, given the fact that χ_v is at least C^1, the assumption (<ref>) on the initial condition and the rescaling  (<ref>) . To estimate the last term,  first note that  the Dambis-Dubins-Schwarz theorem (see <cit.>*Thm 3.4.6)
  implies that in law we have
  
      ∫_0^t (1+_yχ_v(R^ν(s)) -√(D_v)) W(s) = W( ∫_0^t (1+_yχ_v(R^ν(s)) )^2 s ) -W(D_vt).
  
  Hence, by the Hölder-continuity of Brownian motion and the Cauchy-Schwarz inequality inequality we deduce that
  
      sup_t∈[0,T]|∫_0^t (1+_yχ_v(R^ν(s)) -√(D_v)) W(s)|^p
                 =sup_t∈[0,T]|W( ∫_0^t (1+_yχ_v(R^ν(s)) )^2 s ) -W(D_vt)|^p 
                 ≲[Hol^p_α (W(t)) sup_t∈[0,T]|( ∫_0^t (1+_yχ_v(R^ν(s)) )^2 s ) -D_vt|^α p] 
                 ≲([ sup_t∈[0,T]|( ∫_0^t (1+_yχ_v(R^ν(s)) )^2 s ) -D_vt|^2α p])^1/2
                 ≲([ sup_t∈[0,T]|( ∫_0^t g(R^ν(s)) s )|^2α p])^1/2,
   
  where Hol_α (W(t)) is the Hölder constant of the Brownian motion W(t) and we have conveniently defined
  
      g(y)=(1+_yχ_v(y) )^2-D_v.
  
  In this way,
  
      ∫_ g(y)ρ_∞(y) y=0.
  
  Let now χ_g∈ C^1() be the unique solution to
  
      χ_g=-g,     ∫_χ_g(y)ρ_∞(y) y=0.
  
  As before, using Ito's formula we find
  
      χ_g (R^ν)=-1/ν^2β  g(R^ν)  t +√(2)/ν^β_yχ_g(R^ν)  W.
  
  In turn,
  
      ∫_0^t  g(R^ν(s))  s = -ν^2β[χ_g  (R^ν(t)) - χ_g  (R^ν(0))]+ν^β√(2)∫_0^t _yχ_g (R^ν(s))  W(s).
  
  Using the Burkholder-Davis-Gundy inequality, we then find for any q≥ 1 that
  
      sup_t∈ [0,T]|∫_0^t  g(R^ν(s))  s |^q   ≲ν^2qβ sup_t∈ [0,T][ |χ_g (R^ν(t))|^q + |χ_g (R^ν(0))|^q]
           +ν^qβsup_t∈ [0,T]|∫_0^t _yχ_g(R^ν(s))  W(s)|^q
         ≲ν^2qβ+ν^qβ∫_0^T |_yχ_g(R^ν(s)) |^q s≲ν^qβ.
  
  Going back to (<ref>) and using the above bound (here is where we need 2α p≥ 1),  we find
  
      sup_t∈[0,T]|∫_0^t (1+_yχ_v(R^ν(s)) -√(D_v)) W(s)|^p
      ≲ν^pαβ
   
  From (<ref>) and using the rescaling of the initial data (<ref>), we take the supremum in time and expectations to deduce the desired estimate
  
      sup_t∈[0,T] |Y^ν(t)- √(2D_v)  W(t)|^p
         ≲ |Y^ν(0)|^p+ ν^pβsup_t∈[0,T][|χ_v (R^ν(t))|^p +|χ_v (R^ν(t))|^p]+ν^ pαβ
         ≲ν^pβ |Y_0|^p+ν^ pαβ.
   
  This concludes the proof.
  
  
  
  
   §.§ Convergence rates for X^ν
  
  We now turn to the process X^ν. The proof is somewhat similar, so we will only highlight the main points.
  
  Let α∈(0,1/2) and p∈[1/2α,∞) be arbitrarily fixed, and assume that
  
      |X_0|^p<∞.
  
  Then we have the convergence estimate
  
      sup_t∈[0,T] |X^ν(t)- √(2D_u)  W(t)|^p ≲ν^αβ p,
   
  where
  
      D_u= _yχ_u^2
  
  and χ_u:→ is the unique solution to the elliptic equation
  
      χ_u=-u,     ∫_χ_u(y)ρ_∞(y) y=0.
  
  
  
  
  From (<ref>) and (<ref>), we have that
  
      X^ν(0)=ν^1+β X_0 +1/ν^β∫_0^t u (R^ν(s))  s.
  
  Therefore, using assumption (<ref>), (<ref>)  and the Ito's formula we infer that
  
      χ_u (R^ν)=-1/ν^2β  u(R^ν)  t +√(2)/ν^β_yχ_u(R^ν)  W.
  
  As before,
  
      1/ν^β∫_0^t  u(R^ν(s))  s = -ν^β[χ_u (R^ν(t)) - χ_u (R^ν(0))]+ √(2)∫_0^t _yχ_u(R^ν(s))  W_s,
  
  so that
  
      X^ν(t)-√(2D_u)  W(t)=   ν^1+β X_0 -ν^β[χ_u (R^ν(t)) - χ_u (R^ν(0))]
         + √(2)∫_0^t [_yχ_u(R^ν(s))-√(D_u)] W(s).
  
  By repeating the same proof as in Lemma <ref>, we arrive at
  
      sup_t∈[0,T]|∫_0^t (_yχ_u(R^ν(s)) -√(D_u)) W(s)|^p
      ≲ν^pαβ.
   
  Hence, the convergence estimate follows immediately from
  
      sup_t∈[0,T] |X^ν(t)-√(2D_u)  W(t)|^p ≲ν^p(1+β) |X_0|^p  + ν^ pβsup_t∈[0,T][|χ_u (R^ν(t))|^p +|χ_u (R^ν(0))|^p]+ ν^pαβ,
   
  and the proof is over.
  
  It is clear that Theorem <ref> is simply a combination of Lemma <ref> and Lemma <ref>. 
  
  
  
  
  
  
  § ENHANCED DIFFUSION WITH SLIGHTLY COMPRESSIBLE PERTURBATIONS
  
  
  This section is devoted to the proof of Theorem <ref>. Instead of working with (<ref>), we rescale time by defining
  
      f(t,x,y)=h(ν t,x,y),
  
  so that from (<ref>) we infer that f satisfies
  
      _t f+ u_x f=ν(_yyf-v_y f),     f(0,x,y)=f^in(x,y)=h^in(x,y).
  
  Clearly, the  Fourier-decomposition  in (<ref>) is preserved, and therefore we can write the above as
  
      _t f_k+ u_x f_k=ν(_yyf_k-v_y f_k),     f_k(0,y)=f_k^in(y),
  
  having in mind the localization to band k∈.
  The analogous of Theorem <ref> for f is the following.
  
  
  There exist constants ν_0,_0∈ (0,1)  such that the following holds:
  there exist positive numbers å_0,_0,_0 only depending on _0
  such that for each integer k≠ 0 and ν>0 with
  ν k^-1≤ν_0
   the energy functional
  
      Φ_k=1/2[f_k^2 + ν^1/2å_0/k^1/2_y f_k^2+2_0/kł  u' _x f_k, _y f_k +̊_0/ν^1/2k^3/2  u' _x f_k^2 ]
  
  satisfies the differential inequality 
  
      Φ_k+_0ν^1/2k^1/2Φ_k+å_0ν^1/2/2k^1/2_yyf_k-v_y f_k^2+_0/2ν^1/2k^3/2  u'_xy f_k^2 ≤ 0.
  
  for all t≥0.
  In particular
  
      Φ_k(t) ≤^-_0ν^1/2k^1/2tΦ_k(0),     ∀ t≥ 0.
  
  
  It is clear that Theorem <ref> is a straightforward consequence of the above result, provided we rescale time according to (<ref>). Notice that finiteness of Φ_k(0)  requires f^in_k,_yf^in_k∈ L^2_ρ_∞ for each k∈. 
  At the cost of a logarithmic  loss on the rate, it is possible to relax this requirement. 
  
  
  There exist constants ν_0,_0∈ (0,1) and c_0>1 such that the following holds:
  for each integer k≠ 0 and ν>0 with
  ν k^-1≤ν_0
  there holds the estimate
  
      f_k(t)^2 ≤  c_0 f^in_k^2 ^-_0ν^1/2k^1/2/1+|lnν|+ln kt  ,     ∀ t≥ 0.
  
  
  The above result is a semigroup estimate for the solution operator of (<ref>), and it implies the result of Corollary <ref> for k≠ 0. The proof of this is postponed in Section <ref>.
  
  
  The k=0 mode (or the x-average of f in real variables)  satisfies the equation
  
      _t f_0= ν(_yyf_0 -v_y f_0).
  
  In view of the fact that
  
      ∫_^2 f(t,x,y)ρ_∞(y)=0,     ∀ t≥0,
  
  we have that 
  
      ∫_ f_0(t,y)ρ_∞(y)=0,     ∀ t≥0.
  
  Note that the above is only true for f_0, and it cannot in general be imposed on any other Fourier modes.
  A simple energy estimate performed on (<ref>) implies that
  
      1/2 f_0^2+ν_yf_0^2=0,
  
  so, in view of (<ref>), we are in the position of applying the Poincaré inequality and obtain
  
      f_0(t)≤ f^in_0^-_0ν t,     ∀ t≥ 0,
  
  for some _0>0. In turn,
  
      h_0(t)≤ h^in_0^-_0 t,     ∀ t≥ 0,
  
  as stated in  Corollary <ref>. In other words, the x-average of h does not see the effect of the drift u.
  
  
  
  
   §.§ Some energy estimates
  
  In this section, we perform energy estimates on (<ref>).  In what follows,
  we will tacitly make use of the antisymmetry properties
  
      ł u_x g, g=̊-ł g, u_x g,̊    ł u_x g,g=̊0.
  
  Also, we define the operator
  
      L= _yy -v_y
  
  which satisfies the symmetry properties
  
      ł L g, g=̊-ł_y g, _y g,̊    ł L g, g=̊-_y g^2.
  
  Testing (<ref>) with f in L^2_ρ_∞ we have 
  
      1/2f^2  =-ν_y f^2.
  
  Analogously, taking _y of (<ref>) and testing with _y f we obtain
  
      1/2_y f^2   =νł_y Lf, _y f  -̊ł_y (u_x f),_y f =̊ -νL f^2 -ł u'_x f,_y f .̊
  
  We now turn to the cross term ł u'_x f,_y f $̊. Using  (<ref>), we have
  
      ł u'_x f,_y f =̊ν[ł u'_x Lf,_y  f +̊ł u'_x f,_y Lf ]-[ł u'_x (u_x f),_y f +̊ł u'_x f,_y (u_x f)].
  
  Now, recalling that_y ρ_∞=v ρ_∞, we have
  
      ł u'_x f,_y Lf =̊∫_^2u'_x f _y Lf ρ_∞ x  y=-ł u'_xy f,Lf -̊ł u”_x f, Lf -̊ł v u'_x f, Lf .̊
  
  and therefore
  
      ł u'_x Lf,_y  f +̊ł u'_x f,_y Lf =̊-2ł u'_xy f,Lf -̊ł u”_x f, Lf -̊ł v u'_x f, Lf .̊
  
  On the other hand,
  
      ł u'_x (u_x f),_y f +̊ł u'_x f,_y (u_x f)=̊ u'_x f^2,
  
   and therefore
  
      ł u'_x f,_y f =̊-ν[2ł u'_xy f,Lf +̊ł u”_x f, Lf +̊ł v u'_x f, Lf ]- u'_x f^2.
  
  We are left with one more energy estimate, namely
  
      1/2 u'_x f^2   = νł u'_x f,u'_x Lf -̊ł u'_x f,u'_x (u_x f) =̊- νł (u')^2_xx f, Lf 
         = νł_y[(u')^2_xx f], _yf =̊νł (u')^2_xxy f, _yf +̊2νł u' u”_xx f, _yf 
         =- ν u'_xy f^2 +2νł u' u”_xx f, _yf ,̊
  
  which we rewrite for further reference as
  
      1/2 u'_x f^2=- ν u'_xy f^2 +2νł  u”_x f, u'_xyf .̊
  
  We now combine the above estimates in a precise way in order to derive a good differential inequality.
  
  
  
  
  
   §.§ The hypocoercivity scheme
  
  Forå,,>0to be determined, define the functional
  
      Φ=1/2[f^2 + å_y f^2+2ł u'_x f, _y f +̊ u'_x f^2 ].
  
  Notice that, up to rescaling of the various coefficients,Φhas exactly the form (<ref>), as long as we assume thatfis concentrated in one single frequencyk.
  Since
  
      2 |ł u'_x f, _y f |̊≤å/2_y f^2+ 2^2/å u'_x f^2,
  
  if we assume that
  
      ^2/å≤1/4,
  
  we have that
  
      1/2[f^2 + å/2_y f^2+/2 u'_x f^2 ]≤Φ≤1/2[f^2 + 3å/2_y f^2+3/2 u'_x f^2 ] .
  
  Collecting (<ref>),  (<ref>),  (<ref>), and  (<ref>), we find that
  
      Φ+ν_y f^2+åνL f^2 + u'_x f^2+ν u'_xy f^2
             =-åł u'_x f,_y f -̊ν[2ł u'_xy f,Lf +̊ł u”_x f, Lf +̊ł v u'_x f, Lf ]+2νł  u”_x f, u'_xyf .̊
  
  We now estimates the error terms one by one, possibly adding further constraints onå,,, and then verify that a suitable choice is possible. In what follows,C_0≥2is a constant that depends onu,vand that can change from line to line, but it is crucially independent ofå,,,νand thex-Fourier modek. 
  For the term containingå, we have
  
      å|ł u'_x f,_y f |̊≤/2u'_x f^2+ C_0 å^2/_y f^2.
  
  while the first-error terms can be estimated as
  
      2ν|ł u'_xy f,Lf |̊≤ν/4u'_xy f^2 +C_0 ^2ν/L f^2.
  
  Using the boundedness ofu”, we also have
  
      ν|ł u”_x f, Lf |̊≤ν_x f^2 +C_0 ^2ν/L f^2,
  
  while using the boundedness ofvwe can derive the bound
  
      ν|ł v u'_x f, Lf |̊≤ν_x f^2 +C_0 ^2ν/L f^2.
  
  Finally, the-error term is estimated as
  
      2ν| ł  u”_x f, u'_xyf |̊≤ν/4u'_xyf^2+ C_0ν_x f^2.
  
  Collecting all the bounds above and assuming the  more stringent constraints
  
      ^2/å≤1/6C_0,
  
  and
  
      å^2/≤ν/2C_0,
  
  we arrive at
  
      Φ+ν/2_y f^2+åν/2L f^2 +/2 u'_x f^2+ν/2 u'_xy f^2≤
      C_0ν_x f^2.
  
  To estimate the right-hand side above, it is convenient to think of_x=i kand rescale the parameters in the following way:
  
      å=ν^1/2/k^1/2å_0,     =1/k_0,     =1/ν^1/2k^3/2_0,
  
  withå_0,_0,_0independent ofν,kand such that
  
      _0^2/å_0_0≤1/6C_0
  
  and
  
      å_0^2/_0≤1/2C_0,
  
  so that  the constraints (<ref>), (<ref>) and (<ref>) are automatically satisfied, provided we
  can chooseå_0,_0,_0as above. Re-writing (<ref>), we end up with
  
      Φ+ν/2_y f^2 +_0/21/k u'_x f^2+å/2L f^2+/2 u'_xy f^2≤
      C_0_0ν^1/2k^1/2f^2.
  
  We now make use of the following inequality, derived in <cit.>*Proposition 2.7, 
  
      σg^2≤ C_0[σ^2 _yg^2+  u'g^2],
  
  valid for any functiong:→inH^1and for any sufficiently smallσ>0. In our case, we will apply it withg=f, identifyingfwith itsk-th Fourier mode, and 
  
      σ^2=1/_0ν/k≤ν_0/_0≪ 1,
  
  provided we chooseν_0≪_0.
  In this way
  
      ν^1/2k^1/2f^2≤C_0/_0^1/2[ν_yf^2+ k u'f^2]=  C_0/_0^1/2[ν_yf^2+ _0/k u'_x f^2].
  
  Thus, from (<ref>), the above inequality and the further constraint 
  
      _0/_0^1/2≤1/2C_0,
  
  we learn that
  
      Φ+ν/4_y f^2+_0/41/k u'_x f^2+å/2L f^2+/2 u'_xy f^2 ≤ 0.
  
  We use (<ref>) once more to deduce that
  
      Φ+_0^1/2/C_0ν^1/2k^1/2f^2+ ν/4_y f^2 +_0/41/k u'_x f^2+å/2L f^2+/2 u'_xy f^2≤ 0.
  
  Equivalently, we can make the decay rate explicit by writing
  
      Φ+ν^1/2k^1/2[_0^1/2/C_0f^2+ å/4å_0_y f^2 +_0/4_0 u'_x f^2]+å/2L f^2+/2 u'_xy f^2 ≤ 0.
  
  We now chooseå_0,_0,_0complying with (<ref>), (<ref>) and (<ref>).
  Let 
  
      δ_0=[1/288 C_0^3]^1/4,
  
  and set
  
      å_0=12 C_0δ_0^3,     _0=δ_0^2,     _0=δ_0/2 C_0.
  
  In this way, 
  
      _0^2/å_0_0= 1/6C_0,     _0/_0^1/2= 1/2C_0,     å_0^2/_0=1/2C_0,
  
  so that  (<ref>), (<ref>) and (<ref>) are automatically satisfied, and since
  
      _0^1/2/C_0= δ_0/C_0,     1/4å_0=1/12C_0δ_0^3,    _0/4_0=C_0/2δ_0,
  
  we deduce from (<ref>) that
  
      Φ+δ_0/C_0ν^1/2k^1/2[  f^2+ å/12δ_0^4_y f^2 +C_0^2/2 u'_x f^2]+å/2L f^2+/2 u'_xy f^2 ≤ 0.
  
  Consequently, sinceC^2_0≥3andδ_0^4≤1/18, we arrive at
  
      Φ+δ_0/C_0ν^1/2k^1/2[  f^2+ 3å/2_y f^2 +3/2 u'_x f^2]+å/2L f^2+/2 u'_xy f^2 ≤ 0.
  
  Defining
  
      _0=2δ_0/C_0,
  
  and using (<ref>), we finally obtain the differential inequality
  
      Φ+_0ν^1/2k^1/2Φ+å/2L f^2+/2 u'_xy f^2 ≤ 0.
  
  This concludes the proof of Theorem <ref>.
  
  
  
  
  
   §.§ Estimates in L^2_ρ_∞
  
  We prove here Corollary <ref>. 
  We first prove that 
  
      f(t)^2 ≤  C_0 f^in^2 ^-_0ν^1/2k^1/2/1+|lnν|+ln kt ,     ∀ t≥ T_ν,k:=1+|lnν|+ln k /_0ν^1/2k^1/2,
  
  where again we neglect the dependence onkoffandC_0≥1is some constant.
  From (<ref>) and the mean value theorem, it is easy to see that there exists 
  
      t_⋆∈(0,1/_0ν^1/2k^1/2)
  
  such that
  
      ν^1/2/k^1/2_yf (t_⋆)^2≤_0/2f^in^2.
  
  Moreover, from (<ref>), we deduce that
  
      Φ(t) ≤^-_0ν^1/2k^1/2tΦ(t_⋆),     ∀ t≥ t_⋆.
  
  Now, using (<ref>) and  (<ref>), we see that
  
      1/2f^2 ≤Φ≤ C_0[f^2 +  ν^1/2/k^1/2_y f^2+k^1/2/ν^1/2f^2 ] .
  
  Therefore,  using (<ref>) and (<ref>) we arrive at
  
      Φ(t_⋆)≤ C_0[f(t_⋆)^2+f^in^2 +k^1/2/ν^1/2f(t_⋆)^2 ]≤ C_0[f^in^2 +k^1/2/ν^1/2f^in^2 ].
  
  By noticing that sinceν_0≪1there holds
  
      k^1/2/ν^1/2^-_0ν^1/2k^1/2t≤^-_0ν^1/2k^1/2/1+|lnν|+ln kt,    ∀ t≥ T_ν,k,
  
  we then find from (<ref>) that for allt≥t_⋆there holds
  
      Φ(t)≤Φ(t_⋆) ^-_0ν^1/2k^1/2(t-t_⋆)   ≤ C_0[f^in^2 +k^1/2/ν^1/2f^in^2 ] ^-_0ν^1/2k^1/2(t-t_⋆)
         = C_0^_0ν^1/2k^1/2t_⋆[f^in^2 +k^1/2/ν^1/2f^in^2 ] ^-_0ν^1/2k^1/2t
         ≤ C_0 [f^in^2 +k^1/2/ν^1/2f^in^2 ] ^-_0ν^1/2k^1/2t
         ≤ C_0 f^in^2 ^-_0ν^1/2k^1/2/1+|lnν|+ln kt,
  
  which in particular implies (<ref>) upon using (<ref>) once more. Now, to extend (<ref>) to allt∈[0,T_ν,k),
  we simply notice that f(t)is decreasing and
  
      min_t∈[0,T_ν,k]C_0 ^-_0ν^1/2k^1/2/1+|lnν|+ln kt=C_0 ^-_0ν^1/2k^1/2/1+|lnν|+ln kT_ν,k= C_0^-1≤ C_0,
  
  concluding the proof of Corollary  <ref>.
  
  
  
  
  
  § NUMERICAL SIMULATIONS
  
  
  In this section we illustrate some of the theoretical results obtained in the previous sections by means of some simple numerical experiments. In particular, we confirm the scalings for the diffusion coefficient, as a function of the strength of the gradient perturbation, for a simple two-dimensional shear flow and we also explore this scaling in the presence of closed streamlines, i.e. cat's eye flows and cellular flows.
  
  We consider the two-dimensional Langevin dynamics, consistent with the rescaled SDEs (<ref>),
  
      ^,ν(t) = [ 1/ν∇^⊥ψ_(^,ν(t)) - ∇ψ_(^,ν(t)) ] t + √(2κ) (t),
  
  
  where∇^⊥ = (-_y,_x )and for  the one-parameter family of stream functions  
  
      ψ_()= sin (3x)+ sin (3y) ,     ∈[0,1].
  
  This family of stream functions can be mapped to the corresponding Childress-Soward flow <cit.>, given byψ̃_(x,y) = sin(3x) sin(3y) + cos(3x) cos(3y), via an appropriate rotation/change of coordinates. 
  Typical streamlines for different values ofare shown in Figure <ref>.
  
  
  
   §.§ Effective diffusivity
  
  We calculate the effective diffusion tensor using Monte Carlo simulations McL98, PavlStBan06 and using the Langrangian definition of the effective diffusivity:
  
      D_eff = lim_t → +∞(^,ν(t)-^,ν(t)) ⊗ (^,ν(t)-^,ν(t)) /2 t.
  
  Equations (<ref>)-(<ref>), read, with^,ν = (X(t),   Y(t))and(t) = (W_1(t),   W_2(t)):
  
      X(t)    = -3/νcos(3 Y(t))   t - 3 cos(3 X(t))   t + √(2 κ)  W_1(t),  
      
       Y(t)    = 3/νcos(3 X(t))   t - 3  cos(3 Y(t))   t + √(2 κ)  W_2(t).
  
  We solve the SDEs (<ref>) using the Euler-Maruyama scheme. An alternative is 
  the numerical method developed in <cit.> that is particularly tailored to the calculation of
  the eddy diffusivity for periodic vector fields, and that is computationally more efficient in
  the smallκregime. Given that our primary focus is on the study of the effect of the
  compressible perturbation on the eddy diffusivity, in our numerical experiments we setκ=1. We also consider the regimeν∈(0,1), since for values ofνofO(1)and larger, dynamic is dominated by molecular diffusion.
   
  For=0, the effective diffusion coefficient in theydirection is independent ofνand is given by the standard Lifson-Jackson formula for the diffusion coefficient of a Brownian particle moving in a one dimensional periodic potential <cit.>*Eqn. (13.6.13). In Figure <ref> we plot the effective diffusion coefficient along thexdirection. Using Theorem <ref> and an appropriate rescaling, it is easy to check that the effective diffusion coefficient in thexdirection, and in the absence of noise in thexdirection, isD_xx= D_0ν^-2, whereD_0is given by the Lifson-Jackson formula for the potentialψ_0=sin(3 y).  Thus it is in agreement with the slope in this figure. 
  
  At=0.5, we have both open and closed streamlines, and the diffusion coefficient scales differently in thexandydirections, as a function ofν. In fact, the diffusion coefficient in theydirection,D_yy, depends weakly onand we do not present the plot. The diffusion coefficientD_xxfor=0.5is presented in Figure <ref>. Finally, we plot the diffusion coefficient for=1, the case of closed streamlines. Due to symmetries, the diffusion coefficient is the same in thexandydirections, and we only plotD_xxin Figure <ref>. Based on our numerical experiments for different values of, not presented in this section, and elementary least squares fitting, we conjecture that the diffusion in thexdirection scales withν, forν∈(0,1)asD_xx ∼ν^-2 + , at least for∈[0,1), but not necessarily at=1. We will return to this conjecture in future work.
  
  
  
   §.§ Enhanced diffusion
  
  Turning to the enhanced diffusion problem, when=1/2, the stream function in (<ref>) has both open and closed level sets (see Figure <ref>). From the 
   mixing and relaxation enhancing perspective, the situation is depicted in Figure <ref>. As in (<ref>), the functionh, solution to
   
      _t h+1/ν∇^⊥ψ_·∇ h=Δ h-∇ψ_·∇ h,     h(0,)=h^in(),
  
   concentrates, on very short time-scales, on the streamlines ofψ_.  Hence mixing along streamlines is most efficient on a time scale
   shorter than the natural diffusive one (of order 1 in this case). For longer times instead diffusion takes over and dissipation happens mainly
  across streamlines. Although this is similar to what happen in the case=0(see Figure <ref>), the analysis here is much more complicated due to nontrivial  symmetries of the flow. 
  
  
  
  
  A similar situation appears also for=1(see Figure <ref>), wherehnow 
   follows the streamlines in Figure <ref>, which are closed except for the hyperbolic manifolds. It is not clear how the dependence onis reflected on an estimate of the type (<ref>). Of course, thex-average should be replaced by an average on streamlines. However,
   it is not clear if the decay rate inνwill undergo significant changes. Although mixing is very fast near hyperbolic points, a global rate is
   very likely to be similar to that  in (<ref>), since diffusion will “push” the solutionhinto cells, away from the hyperbolic points.
  
  
  
  
  
  
  § CONCLUSIONS
  
  
  The problem of the effect of compressible perturbations on the long time behaviour of solutions to the advection-diffusion equation was studied in this paper. In particular, for shear flows we characterize homogenization rates (see Theorem <ref>) and the longtime dynamics (see Theorem <ref>) of the solution very precisely in terms of the
  dependence on the parameter (given by1/ν) measuring the size of the incompressible perturbation. In the case of more complicated flows, such as
  the Childress-Soward flow, we exhibit numerical evidence for what the expected behaviour should be.
  
  There are several open questions that we plan to return to in future work. First obtaining similar sharp quantitative estimates for flows with closed streamlines is a challenging and interesting problem. This problem has already been studied for radial flows. Second, understanding in more depth the connection between the scaling of the effective diffusion coefficient with respect to the strength of the perturbation and the long time behaviour of solutions to the advection-diffusion equation. Third, studying the “inverse” problem, namely identifying the optimal nonreversible perturbation that maximizes the rate of convergence to equilibrium. Finally studying similar problems for random velocity fields. 
  
   Acknowledgments
  M. Coti Zelati acknowledges funding from the Royal Society through a University Research Fellowship (URF\ R1\ 191492). The work of G.P. was partially funded by the EPSRC, grant number EP/P031587/1, and by J.P. Morgan Chase&Co. Any views or opinions expressed herein are solely those of the authors listed, and may differ from the views and opinions expressed by J.P. Morgan Chase&Co. or its affiliates. This material is not a product of the Research Department of J.P. Morgan Securities LLC. This material does not constitute a solicitation or offer in any jurisdiction.
  
  
  
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