aa r X i v : . [ m a t h . A P ] A ug A NEW APPROACH TO BOUNDS ON MIXING
FLAVIEN LÉGER
Abstract.
We consider mixing by incompressible flows. In 2003, Bres-san stated a conjecture concerning a bound on the mixing achieved bythe flow in terms of an L norm of the velocity field. Existing results inthe literature use an L p norm with p > . In this paper we introducea new approach to prove such results. It recovers most of the exist-ing results and offers new perspective on the problem. Our approachmakes use of a recent harmonic analysis estimate from Seeger, Smartand Street. Introduction
Consider a passive scalar θ which is advected by a time-dependent anddivergence-free velocity field u on the whole space R d . θ might assign labelsto the fluid particles, or represent the concentration of some scalar quantity. θ and u then satisfy the equations ∂ t θ + div( uθ ) = 0div( u ) = 0 on [0 , + ∞ ) × R d θ (0 , · ) = θ on R d (1)In [3], Bressan stated a conjecture relating a bound on the mixing achievedby the flow in terms of an L norm of the velocity field: Conjecture (Bressan, [3]) . For the geometric mixing scale ε ( t ) of θ ( t, · ) (seedef. 2 in Section 4.3) there exists a constant C > depending on θ suchthat ε ( t ) ≥ C − exp (cid:18) − C Z t k∇ u ( t ′ , · ) k dt ′ (cid:19) To the best of our knowledge this conjecture is still open. However, start-ing from work by Crippa and De Lellis [5], there have been several relatedresults bounding mixing in terms of the L p norm in space R t k∇ u ( t ′ , · ) k p dt ′ with p > .This paper develops a new approach to proving bounds on mixing. Itrecovers many of the known results, and (being different from the previousproofs) offers new perspective on the problem. Unfortunately, this methoddoesn’t seem to be effective to deal with the L case.To describe our results, we begin by introducing the following functional(this is different from, but somewhat analogous to, the functionals discussed in [5],[2],[12],[8]) V ( f ) = Z R d log | ξ | | ˆ f ( ξ ) | dξ where ˆ f denotes the Fourier transform of f . V ( f ) captures a logarithm of aderivative of f .Our main result (Theorem 1) says, roughly speaking, that if the velocityfield u is bounded in ˙ W ,p uniformly in time for some p > , then V (cid:0) θ ( t, · ) (cid:1) grows at most linearly(2) V (cid:0) θ ( t, · ) (cid:1) ≤ C (1 + t ) We also offer an additional, related result (Theorem 2). Consider thefunctional W ( f ) = Z R d (log | ξ | ) | ˆ f ( ξ ) | dξ We show that if the velocity field u is bounded in ˙ W ,q uniformly in time forsome q ≥ , then W (cid:0) θ ( t, · ) (cid:1) grows at most quadratically W (cid:0) θ ( t, · ) (cid:1) ≤ C (1 + t ) Our results rely crucially on a harmonic analysis estimate recently provedby Seeger, Smart and Street [12]. It is clear from [12] that the results thereare related to and motivated by Bressan’s conjecture. However [12] does notinclude much information about these connections.Proving linear growth of V (cid:0) θ ( t, · ) (cid:1) recovers most of the existing resultsin the literature. We discuss this in detail in Section 4, giving just a briefsummary here. A popular choice to measure mixing is the ˙ H − norm of θ ( t, · ) or more generally any ˙ H − s norm for s > (see for instance [10] andthe review article [14]). It was shown in [9] and [13] that a uniform in timecontrol of k∇ u ( t, · ) k p implies an exponential lower bound on the ˙ H − normof θ ( t, · ) k θ ( t, · ) k ˙ H − ≥ C − exp( − Ct ) This exponential decay can be recovered by our main theorem coupled tothe simple convexity inequality (see Prop. 1 in Sect. 4.4) k f k ˙ H − s / k f k L ≥ exp (cid:0) − s V ( f ) / k f k L (cid:1) Note that this gives us access to any ˙ H − s norm for s > .To give a more precise statement of our main result we turn to a moreprecise description of the problem.2. Statement of main result
Consider the functional V , defined on functions f in the Schwartz class by V ( f ) = Z R d log | ξ | | ˆ f ( ξ ) | dξ NEW APPROACH TO BOUNDS ON MIXING 3 where ˆ f denotes the Fourier transform of f . This can be written in physicalspace (see Lemma 1) V ( f ) = α d Z Z | x − y |≤ | f ( x ) − f ( y ) | | x − y | d dx dy − Z Z | x − y | > f ( x ) f ( y ) | x − y | d dx dy ! + β d k f k L where α d , β d are constants and α d > .Let θ be a scalar quantity passively advected by a smooth divergence-freetime-dependent velocity field u (equation (1)). A careful computation (seeLemma 2) shows that the time-derivative of V (cid:0) θ ( t, · ) (cid:1) is(3) ddt V (cid:0) θ ( t, · ) (cid:1) = c d PV Z Z θ ( t, x ) θ ( t, y ) (cid:0) u ( t, x ) − u ( t, y ) (cid:1) · x − y | x − y | d +2 dx dy where the principal value means taking the limit ε → of the integral overthe domain | x − y | > ε in R d × R d .The time-derivative of V (cid:0) θ ( t, · ) (cid:1) can be written as a trilinear form in θ ( t, · ) , θ ( t, · ) and ∇ u ( t, · ) . Using the harmonic analysis estimate from Seegeret al. [12], we deduce that the right hand side of (3) can be bounded by C ( d, p ) k θ ( t, · ) k ∞ k θ ( t, · ) k p ′ k∇ u ( t, · ) k p with p > and /p + 1 /p ′ = 1 . We now state our main theorem. Theorem 1.
Let < p ≤ ∞ and p ′ the dual Hölder exponent p/ ( p − .There exists a constant C > depending only on p and the dimension d suchthat a) (cid:12)(cid:12) d/dt V (cid:0) θ ( t, · ) (cid:1)(cid:12)(cid:12) ≤ C k θ k ∞ k θ k p ′ k∇ u ( t, · ) k p where k θ k ∞ = k θ k L ∞ , etc. As consequences:b) If θ ∈ L ∩ L ∞ and u is bounded in ˙ W ,p uniformly in time, then V (cid:0) θ ( t, · ) (cid:1) grows at most linearly;c) More generally, V (cid:0) θ ( t, · ) (cid:1) −V (cid:0) θ (cid:1) ≤ C k θ k ∞ k θ k p ′ Z t k∇ u ( t ′ , · ) k p dt ′ We consider now another functional W ( f ) = Z R d (log | ξ | ) | ˆ f ( ξ ) | dξ which provides slightly stronger control of the high frequencies. We have thefollowing bounds for W : Theorem 2.
Let ≤ q ≤ ∞ and ≤ ˜ q ≤ ∞ be such that /q + 1 / ˜ q = 1 / .There exists a constant C > depending only on the dimension d such thata) (cid:12)(cid:12)(cid:12) d/dt q W (cid:0) θ ( t, · ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ C k θ k ˜ q k∇ u ( t, · ) k q where k θ k ˜ q = k θ k L ˜ q , etc. As a consequence:b) If θ ∈ L ∩ L ∞ and u is bounded in ˙ W ,q uniformly in time, then W (cid:0) θ ( t, · ) (cid:1) grows at most quadratically. FLAVIEN LÉGER
Remark . In Theorem 2, the best we can do is obtaining a bound in termsof the L norm in space k∇ u ( t, · ) k , whereas in Theorem 1 we can go all theway down to k∇ u ( t, · ) k p for p > .Further discussion of our results and some corollaries are given in Sec-tion 4. 3. Proofs
In this section we prove Theorems 1 and 2. We start by expressing V ( f ) in physical space. Lemma 1.
For any f in the Schwartz class, V ( f ) can be written in physicalspace V ( f ) = α d Z Z | x − y |≤ | f ( x ) − f ( y ) | | x − y | d dx dy − Z Z | x − y | > f ( x ) f ( y ) | x − y | d dx dy ! + β d k f k L where α d and β d are two constants and α d > .Proof. Define the following tempered distribution T h T, φ i = Z | h |≤ φ ( h ) − φ (0) | h | d dh + Z | h | > φ ( h ) | h | d dh Then (see the appendix) the Fourier transform of T is an L loc function and h ˆ T , ψ i = Z (cid:0) ζ d − σ d − log | ξ | (cid:1) ψ ( ξ ) dξ for any ψ in the Schwartz class S ( R d ) , where σ d − is the surface area ofthe unit sphere in R d and ζ d is a constant. Consequently if we define thetempered distribution S by S = ζ d /σ d − δ − /σ d − T then for all ψ ∈ S ( R d ) we have h ˆ S, ψ i = Z log | ξ | ψ ( ξ ) dξ NEW APPROACH TO BOUNDS ON MIXING 5
We deduce that for all f ∈ S ( R d ) , writing ˜ f ( x ) = f ( − x ) we have V ( f ) = h ˆ S, ˆ f ¯ˆ f i = h ˆ S, [ f ⋆ ˜ f i = h ˆˆ S, f ⋆ ˜ f i = ζ d /σ d − ( f ⋆ ˜ f )(0) − /σ d − Z | h |≤ Z (cid:0) f ( x − h ) − f ( x ) (cid:1) f ( x ) | h | d dx dh + Z | h | > Z f ( x − h ) f ( x ) | h | d dx dh ! = ζ d /σ d − k f k L − /σ d − − Z Z | x − y |≤ | f ( x ) − f ( y ) | | x − y | d dx dy + Z Z | x − y | > f ( y ) f ( x ) | x − y | d dx dy ! which concludes the proof. (cid:3) We now give the expression of the time-derivative of V (cid:0) θ ( t, · ) (cid:1) . Lemma 2.
For smooth divergence-free velocity fields u decaying fast enoughat infinity, the time-derivative of V (cid:0) θ ( t, · ) (cid:1) can be written ddt V (cid:0) θ ( t, · ) (cid:1) = c d PV Z Z θ ( t, x ) θ ( t, y ) (cid:0) u ( t, x ) − u ( t, y ) (cid:1) · x − y | x − y | d +2 dx dy where c d is a positive constant.Proof. The derivation has to be done a bit carefully otherwise non integrableterms appear. Recall that the expression of V in physical space is (Lemma 1) V (cid:0) θ ( t, · ) (cid:1) = α d Z Z | x − y |≤ | θ ( t, x ) − θ ( t, y ) | | x − y | d dx dy − Z Z | x − y | > θ ( t, x ) θ ( t, y ) | x − y | d dx dy ! + β d k θ ( t, · ) k L Taking time-derivatives, the L norm term on the right-hand side disappearssince the flow is incompressible. Dropping the t for readability we get ddt V ( θ ) = α d Z Z | x − y |≤ (cid:0) θ ( x ) − θ ( y ) (cid:1)(cid:0) ∂ t θ ( x ) − ∂ t θ ( y ) (cid:1) | x − y | d dx dy − Z Z | x − y | > ∂ t θ ( x ) θ ( y ) + θ ( x ) ∂ t θ ( y ) | x − y | d dx dy ! = α d (cid:0) A − B (cid:1) Note that since θ and ∂ t θ are smooth and decay fast at infinity, there is noissue in differentiating under the integral sign. FLAVIEN LÉGER
Set V ( h ) = 1 | h | d We first simplify A . By symmetry in the integral in x and y we only keepthe term with ∂ t θ ( x ) and write A Z Z | x − y |≤ V ( x − y ) (cid:0) θ ( x ) − θ ( y ) (cid:1)(cid:0) − div( uθ )( x ) (cid:1) dx dy Next we want to integrate by part but doing so directly yields a term whichis not integrable. Thus we proceed in the following way: A ε → Z y Z ε< | x − y |≤ V ( x − y ) (cid:0) θ ( x ) − θ ( y ) (cid:1)(cid:0) − div( uθ )( x ) (cid:1) dx dy = lim ε → Z y Z ε< | x − y |≤ (cid:16) ∇ V ( x − y ) (cid:0) θ ( x ) − θ ( y ) (cid:1) + V ( x − y ) ∇ θ ( x ) (cid:17) · u ( x ) θ ( x ) dx dy − Z y Z x ∈ S ( y, V ( x − y ) (cid:0) θ ( x ) − θ ( y ) (cid:1) θ ( x ) u ( x ) · x − y | x − y | dσ ( x ) dy − Z y Z x ∈ S ( y,ε ) V ( x − y ) (cid:0) θ ( x ) − θ ( y ) (cid:1) θ ( x ) u ( x ) · y − x | y − x | dσ ( x ) dy ! = lim ε → (cid:0) A ( ε ) + A + A ( ε ) (cid:1) where S ( y, denotes the sphere of center y and of radius , etc.We now fix ε > and compute each of the terms A ( ε ) , A , A ( ε ) . Thefirst one A ( ε ) = Z y Z ε< | x − y |≤ (cid:16) ∇ V ( x − y ) (cid:0) θ ( x ) − θ ( y ) (cid:1) + V ( x − y ) ∇ θ ( x ) (cid:17) · u ( x ) θ ( x ) dx dy can be split into three terms (we swap integrals in x and y ) A ( ε ) = Z x θ ( x ) u ( x ) · Z ε< | y − x |≤ ∇ V ( x − y ) dy dx − Z Z ε< | x − y |≤ ∇ V ( x − y ) · u ( x ) θ ( x ) θ ( y ) dx dy + Z x ∇ (cid:0) θ (cid:1) ( x ) · u ( x ) Z ε< | y − x |≤ V ( x − y ) dy dx The first term cancels since the integral in y is zero, and the last term cancelssince the integral in y doesn’t depend on x and u is divergence-free. Thuswe are left with (after symmetrizing in x and y in the second term) A ( ε ) = − Z Z ε< | x − y |≤ θ ( x ) θ ( y ) (cid:0) u ( x ) − u ( y ) (cid:1) · ∇ V ( x − y ) dx dy NEW APPROACH TO BOUNDS ON MIXING 7
We now turn our attention to A (which doesn’t depend on ε ) A = − Z y Z x ∈ S ( y, V ( x − y ) (cid:0) θ ( x ) − θ ( y ) (cid:1) θ ( x ) u ( x ) · x − y | x − y | dσ ( x ) dy We split the integral in two and swap the integrals in x and y in the firstterm A = − Z x θ ( x ) u ( x ) · Z y ∈ S ( x, V ( x − y ) x − y | x − y | dσ ( y ) dx + Z y Z x ∈ S ( y, V ( x − y ) θ ( y ) θ ( x ) u ( x ) · x − y | x − y | dσ ( x ) dy The first term cancels since the integral in y is zero. Thus we are left with A = Z y Z x ∈ S ( y, V ( x − y ) θ ( y ) θ ( x ) u ( x ) · x − y | x − y | dσ ( x ) dy Now we simplify the third term A ( ε ) = Z y Z x ∈ S ( y,ε ) V ( x − y ) (cid:0) θ ( y ) − θ ( x ) (cid:1) θ ( x ) u ( x ) · y − x | y − x | dσ ( x ) dy and show that lim ε → A ( ε ) = 0 We use the following Taylor’s formula with integral remainder to estimatethe quantity θ ( y ) − θ ( x ) θ ( y ) − θ ( x ) = ∇ θ ( x ) · ( y − x ) + Z (1 − s ) D θ (cid:0) x + s ( y − x ) (cid:1) ( y − x, y − x ) ds and plug it in the expression of A ( ε ) . A ( ε ) decomposes into two terms A ′ ( ε ) = Z y Z x ∈ S ( y,ε ) V ( x − y ) ∇ θ ( x ) · ( y − x ) θ ( x ) u ( x ) · y − x | y − x | dσ ( x ) dy and A ′′ ( ε ) = Z y Z x ∈ S ( y,ε ) V ( x − y ) Z (1 − s ) D θ (cid:0) x + s ( y − x ) (cid:1) ( y − x, y − x ) ds θ ( x ) u ( x ) · y − x | y − x | dσ ( x ) dy Swapping the integrals in x and y and doing a change of variables h = y − x yields A ′ ( ε ) = Z x Z h ∈ S (0 ,ε ) V ( h ) ∇ (cid:0) θ ( x ) / (cid:1) · h u ( x ) · h | h | dσ ( h ) dx Rearranging terms, we get A ′ ( ε ) = Z x u i ( x ) ∂ j (cid:0) θ / (cid:1) ( x ) Z h ∈ S (0 ,ε ) V ( h ) h j h i | h | dσ ( h ) dx FLAVIEN LÉGER where the summation over indices i and j is implied. In the integral in h ,note that | h | = ε and V ( h ) = ε − d . Furthermore, by rotationnal symmetryof the sphere S (0 , ε ) it is easy to see that Z h ∈ S (0 ,ε ) h i h j dσ ( h ) = ( if i = jd − σ d − ε d +1 if i = j Consequently A ′ ( ε ) = d − σ d − Z u i ( x ) ∂ i (cid:0) θ / (cid:1) ( x ) dx = 0 since u is divergence-free.Finally, we can crudely bound the term A ′′ ( ε ) (cid:12)(cid:12) A ′′ ( ε ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x Z y ∈ S ( x,ε ) V ( x − y ) Z (1 − s ) D θ (cid:0) x + s ( y − x ) (cid:1) ( y − x, y − x ) dsθ ( x ) u ( x ) · y − x | y − x | dσ ( y ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z x Z y ∈ S ( x,ε ) V ( x − y ) k D θ k L ∞ | y − x | | θ ( x ) | | u ( x ) | dσ ( y ) dx ≤ Z x σ d − ε d − ε − d k D θ k L ∞ ε | θ ( x ) | | u ( x ) | dx ≤ ε σ d − k u k L k θ k L ∞ k D θ k L ∞ This shows that A ′′ ( ε ) → as ε → and thus A ( ε ) = A ′ ( ε ) + A ′′ ( ε ) → as ε → We now deal with the B term. By symmetry in x and y we have B Z Z | x − y | > V ( x − y ) ∂ t θ ( x ) θ ( y ) dx dy Here we can substitute ∂ t θ by − div( uθ ) and directly do an integration bypart B Z y θ ( y ) Z | x − y | > V ( x − y ) (cid:0) − div( uθ ) (cid:1) ( x ) dx dy = Z Z | x − y | > θ ( y ) ∇ V ( x − y ) · u ( x ) θ ( x ) dx dy − Z y θ ( y ) Z x ∈ S ( y, V ( x − y ) θ ( x ) u ( x ) · y − x | y − x | dσ ( x ) dy = B + B NEW APPROACH TO BOUNDS ON MIXING 9
Note that B = A . Moreover, after symmetrizing in x and y , B can bewritten B = 12 Z Z | x − y | > θ ( x ) θ ( y ) (cid:0) u ( x ) − u ( y ) (cid:1) · ∇ V ( x − y ) dx dy In conclusion, grouping everything together we get A − B ε → (cid:0) A ( ε ) + A + A ( ε ) (cid:1) − ( B + B )= lim ε → A ( ε ) − B = lim ε → − Z Z ε< | x − y |≤ θ ( x ) θ ( y ) (cid:0) u ( x ) − u ( y ) (cid:1) · ∇ V ( x − y ) dx dy − Z Z | x − y | > θ ( x ) θ ( y ) (cid:0) u ( x ) − u ( y ) (cid:1) · ∇ V ( x − y ) dx dy Hence A − B = lim ε → − Z Z | x − y | >ε θ ( x ) θ ( y ) (cid:0) u ( x ) − u ( y ) (cid:1) · ∇ V ( x − y ) dx dy which concludes the proof since ∇ V ( h ) = − d h/ | h | d +2 . (cid:3) We are now able to prove Theorem 1.
Proof of Theorem 1.
With some notation we can write the time derivative ddt V (cid:0) θ ( t, · ) (cid:1) = c d PV Z Z θ ( t, x ) θ ( t, y ) (cid:16) u ( t, x ) − u ( t, y ) (cid:17) · x − y | x − y | d +2 dx dy as the type of multilinear singular integral studied in [12]. Define K ( h ) = c d h ⊗ h − /d | h | I | h | d +2 for h ∈ R d \ { } , where h ⊗ h is the d × d matrix defined by ( h ⊗ h ) i,j = h i h j and I is the d × d identity matrix.It is easy to see that K is a matrix-valued Calderón-Zygmund kernel (i.e.each entry K i,j is a Calderón-Zygmund kernel). Let m x,y L denote the aver-age of a function L between x and ym x,y L = Z L (cid:0) (1 − s ) x + sy (cid:1) ds and let M : N denote the contraction M : N = P i,j M i,j N i,j . Thanks to the divergence-free condition ∇ u : I = 0 , where (cid:0) ∇ u (cid:1) i,j = ∂ i u j ,we can write (summing over repeated indices) (cid:0) u j ( t, x ) − u j ( t, y ) (cid:1) x j − y j | x − y | d +2 = Z ∇ u j (cid:0) t, (1 − s ) x + sy (cid:1) · ( x − y ) ds x j − y j | x − y | d +2 = m x,y (cid:0) ∇ u ( t, · ) (cid:1) : ( x − y ) ⊗ ( x − y ) − /d | x − y | I | x − y | d +2 and thus ddt V (cid:0) θ ( t, · ) (cid:1) = Z Z θ ( t, x ) θ ( t, y ) m x,y (cid:0) ∇ u ( t, · ) (cid:1) : K ( x − y ) dx dy Using the terminology from [12], this is a first order d -commutator. Themain result of [12] is that Hölder estimates are valid on this types of trilinearform. More precisely we have the bound Z Z θ ( t, x ) θ ( t, y ) m x,y (cid:0) ∇ u ( t, · ) (cid:1) : K ( x − y ) dx dy ≤ C ( d, p ) k θ ( t, · ) k L ∞ k θ ( t, · ) k L p ′ k∇ u ( t, · ) k L p for any p > .The L p ′ and L ∞ norms of θ ( t, · ) are conserved quantities since the flow isincompressible, and this concludes the proof of Theorem 1. (cid:3) To prove Theorem 2 we start with the following lemma
Lemma 3 (Time-derivative of W (cid:0) θ ( t, · ) (cid:1) ) . For smooth divergence-free veloc-ity fields u decaying fast enough at infinity, the time-derivative of W (cid:0) θ ( t, · ) (cid:1) can be written ddt W (cid:0) θ ( t, · ) (cid:1) = c d PV Z Z φ ( t, x ) θ ( t, y ) (cid:0) u ( t, x ) − u ( t, y ) (cid:1) · x − y | x − y | d +2 dx dy where we denote ˆ φ ( t, ξ ) = log( | ξ | ) ˆ θ ( t, ξ ) and c d is the same constant as in Lemma 2.Proof. Let f = f ( x ) be any functions (regular enough, e.g. in the Schwartzclass) and u = u ( t, x ) be any divergence-free velocity field (also regularenough). Let θ be the solution of ( ∂ t θ + div( uθ ) = 0 on ( − , × R d θ (0 , · ) = f on R d and denote v ( x ) = u (0 , x ) Then Lemma 2 shows that(4) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 V (cid:0) θ ( t, · ) (cid:1) = c d PV Z Z f ( x ) f ( y ) (cid:0) v ( x ) − v ( y ) (cid:1) · x − y | x − y | d +2 dx dy NEW APPROACH TO BOUNDS ON MIXING 11
On the other hand, computing this time-derivative in Fourier space gives us ddt V (cid:0) θ ( t, · ) (cid:1) = ddt Z log | ξ | | ˆ θ ( t, ξ ) | dξ = 2 ℜ Z log | ξ | ∂ t ˆ θ ( t, ξ )ˆ θ ( t, ξ ) dξ = 2 ℜ Z log | ξ | ( − i ) ξ · (ˆ u ⋆ ˆ θ ( t, · ))ˆ θ ( t, − ξ ) dξ = 2 ℜ Z Z log | ξ | ( − i ) ξ · ˆ u ( t, ξ + η )ˆ θ ( t, − η )ˆ θ ( t, − ξ ) dξ dη where ℜ denotes the real part. Writing this last expression at time t = 0 and using (4), we get an equivalence formula in physical and Fourier space: c d PV Z Z f ( x ) f ( y ) (cid:0) v ( x ) − v ( y ) (cid:1) · x − y | x − y | d +2 dx dy = 2 ℜ Z Z log | ξ | ( − i ) ξ · ˆ v ( ξ + η ) ˆ f ( − ξ ) ˆ f ( − η ) dξ dη true for any f and divergence-free v ; thus if we polarize the following equalityis true(5) c d PV Z Z f ( x ) g ( y ) (cid:0) v ( x ) − v ( y ) (cid:1) · x − y | x − y | d +2 dx dy =2 ℜ Z Z log | ξ | ( − i ) ξ · ˆ v ( ξ + η ) ˆ f ( − ξ ) ˆ g ( − η ) dξ dη for any (smooth, fast-decaying) f , g and divergence-free v .Furthermore, in Fourier space the time-derivative of W (cid:0) θ ( t, · ) (cid:1) is (com-putations are similar to the previous ones) ddt W (cid:0) θ ( t, · ) (cid:1) = 2 ℜ Z Z (cid:0) log | ξ | (cid:1) ( − i ) ξ · ˆ u ( t, ξ + η )ˆ θ ( t, − η )ˆ θ ( t, − ξ ) dξ dη = 2 ℜ Z Z log | ξ | ( − i ) ξ · ˆ u ( t, ξ + η )ˆ θ ( t, − η ) ˆ φ ( t, − ξ ) dξ dη where ˆ φ ( t, ξ ) = log( | ξ | ) ˆ θ ( t, ξ ) Using equality (5) then yields the desired result. (cid:3)
Proof of Theorem 2.
Lemma 3 just showed that ddt W (cid:0) θ ( t, · ) (cid:1) = c d PV Z Z φ ( t, x ) θ ( t, y ) (cid:16) u ( t, x ) − u ( t, y ) (cid:17) · x − y | x − y | d +2 dx dy where ˆ φ ( t, ξ ) = log( | ξ | ) ˆ θ ( t, ξ ) . This is the same first order d -commutatorconsidered previously (see the proof of Theorem 1), but this time in φ ( t, · ) , θ ( t, · ) and ∇ u ( t, · ) . The results in [12] imply Z Z φ ( t, x ) θ ( t, y ) (cid:16) u ( t, x ) − u ( t, y ) (cid:17) · x − y | x − y | d +2 dx dy ≤ C ( d ) k φ ( t, · ) k L k θ ( t, · ) k L ˜ q k∇ u ( t, · ) k L q for any ≤ q ≤ ∞ , where ≤ ˜ q ≤ ∞ is such that / /q + 1 / ˜ q = 1 .Note that the constant C ( d ) doesn’t depend on q . It is immediate to see inFourier space that k φ ( t, · ) k L = q W (cid:0) θ ( t, · ) (cid:1) which is enough to conclude to proof. (cid:3) Discussion and corollaries
The functionals V and W . Let us give some background on the func-tionals V and W introduced in this work. A regularized version of V wasconsidered in [2], on a problem that shares similarities with the one consid-ered in this paper. In [5], the idea of controlling the log of a derivative of θ ( t, · ) is also present. Additionally, the usual homogeneous Sobolev semi-norm k f k ˙ H s can be defined in Fourier space by k f k H s = Z R d | ξ | s | ˆ f ( ξ ) | dξ This can be written in physical space for < s < k f k H s = c d,s Z Z R d × R d | f ( x ) − f ( y ) | | x − y | d +2 s dx dy where the constant c d,s blows up as s → (see for instance [11] for moreinformation). The functionals V and W involve taking a limit s → . Moreprecisely, note that formally for small s | ξ | s = 1 + 2 s log | ξ | + 2 s (log | ξ | ) + O ( s ) Thus V ( f ) and W ( f ) are the first terms of the expansion of k f k H s as s → k f k H s = k f k L + 2 s V ( f ) + 2 s W ( f ) + O ( s ) See Section 4.5 for additional insight.4.2.
Brief comments on the harmonic analysis estimate.
For our the-orems 1 and 2 we rely crucially on a hard harmonic analysis estimate; moreprecisely we need Hölder-type bounds on a bilinear singular integral. With-out going in too much depth, let us point out that usual Calderón-Zygmundtheory does not apply here as the kernel is too singular (if d ≥ ) and eventhe multilinear singular integral framework developed by Grafakos and Tor-res [7] is not adapted for this problem.In [4] Christ and Journé studied the type of multilinear singular integraloperators we consider in this paper, however applying their results wouldonly give us a bound on d/dt V (cid:0) θ ( t, · ) (cid:1) in terms of the L ∞ norm of ∇ u ( t, · ) .This is not useful here as controlling k∇ u ( t, · ) k ∞ makes everything obvious;thus only the very recent work [12] contains the needed estimates. We referto [12] for more information on these types of multilinear singular integraloperators. NEW APPROACH TO BOUNDS ON MIXING 13
Two ways to quantify mixing.
Two measures of mixing have beenmainly considered in the literature. We follow here the naming of [1].
Definition 1.
The functional mixing scale of θ ( t, · ) is k θ ( t, · ) k ˙ H − .The second notion has traditionally been considered for functions withvalues ± , but it can be extended naturally to functions in L ∞ Definition 2.
Given < κ < , the geometric mixing scale of θ ( t, · ) is theinfimum ε ( t ) of all ε > such that k θ ( t, · ) ⋆ χ ε k L ∞ k θ ( t, · ) k L ∞ ≤ − κ where ⋆ denotes the convolution, χ is the indicator function of the unit ballin R d modified to have total mass : χ ( x ) = | B (0) | B (0) ( x ) and χ ε ( x ) = ε − d χ ( x/ε ) Relations between our results and previous ones.
A consequenceof the main result in [5] is an exponential lower bound on the geometricmixing scale ε ( t ) ≥ C − exp (cid:18) − C Z t k∇ u ( t ′ , · ) dt ′ k L p (cid:19) for p > , while the works [9],[13] showed an exponential lower bound on thefunctional mixing scale k θ ( t, · ) k ˙ H − ≥ C − exp (cid:18) − C Z t k∇ u ( t ′ , · ) dt ′ k L p (cid:19) We recover both results. Indeed upper bounds on V ( f ) imply lower boundson the functional and geometric mixing scales. More precisely, for the func-tional mixing scale we have the following proposition Proposition 1 (Exponential decay of the functional mixing scale) . For all s > we have the following convexity inequalitya) For all non zero f in the Schwarz class Sk f k ˙ H − s / k f k L ≥ exp (cid:0) − s V ( f ) / k f k L (cid:1) As a consequence:b) Fix any s > . There is a constant C depending only on p and thedimension d such that k θ ( t, · ) k ˙ H − s ≥ k θ k L exp (cid:0) − s V ( θ ) / k θ k L (cid:1) exp (cid:18) − C k θ k L ∞ k θ k L p ′ k θ k L Z t k∇ u ( t ′ , · ) k L p dt ′ (cid:19) where p ′ = p/ ( p − , for any smooth,fast-decaying solution θ of (1) .Remark . The decay rate we obtain this way is C ( d, p ) k θ k L ∞ k θ k L p ′ k θ k L For p = 2 for instance we get decay rate C ( d, p ) k θ k L ∞ k θ k L which is a slight improvement from [9], where the authors find the followingdecay rate (in the L case p = 2 ) c ( d, p ) | A λ | / where A λ is the set { x | θ ( x ) / k θ k L ∞ > λ } and | A λ | denotes its Lebesguemeasure. The parameter λ is a fixed number in (0 , . Note that a weak L estimate yields | A λ | / ≤ λ − k θ k L k θ k L ∞ so that c ( d, p ) | A λ | / ≥ c ( d, p ) λ k θ k L ∞ k θ k L In any case, both decay rates are larger when the support of θ is smaller,which matches the intuition, see the related discussion in [9, Sect. 1].For the geometric mixing scale we have the following proposition Proposition 2 (Exponential decay of the geometric mixing scale) . a) There exists a constant A > depending on the dimension d and on κ (see def. 2) such that for any f ∈ S with V ( f ) > ε < A − exp (cid:0) − A V ( f ) / k f k L (cid:1) ⇒ k f ⋆ χ ε k L ∞ k f k L ∞ > (1 − κ ) k f k L k f k L k f k L ∞ As a consequence:b) Consider the system (1) on the flat torus T d . Assume that θ onlyhas values ± . Then there exist constants A ( d, κ ) and C ( d, p ) suchthat the geometric mixing scale ε ( t ) satisfies ε ( t ) ≥ A − exp( − A V ( θ )) exp (cid:18) − A C Z t k∇ u ( t ′ , · ) k L p dt ′ (cid:19) Remark . For part b) of Prop. 2 we consider the dynamic on the torus T d .To be rigorous we need to define a version e V of the functional V which actson (smooth) functions defined on the torus e V ( f ) = X k ∈ Z d \{ } log | k | | ˆ f ( k ) | where ˆ f ( k ) = Z T d e − iπk · x f ( x ) dx NEW APPROACH TO BOUNDS ON MIXING 15
We have the same bounds on e V as for V in Theorem 1, since we can computethe time-derivative similarly to Lemma 2 (which only uses integrations bypart) and the harmonic analysis estimate from [12] is also valid for e V .We now prove both propositions. Proof of Prop. 1.
Let us consider the probability measure dµ ( ξ ) = | ˆ f ( ξ ) | / k f k L dξ Then exp (cid:0) − s V ( f ) / k f k L (cid:1) = exp (cid:18) − s Z log | ξ | dµ ( ξ ) (cid:19) = exp (cid:18)Z log (cid:0) | ξ | − s (cid:1) dµ ( ξ ) (cid:19) (Jensen inequality) ≤ Z | ξ | − s dµ ( ξ ) = k f k H − s k f k L This proves a ) . Combining it with Theorem 1 proves b ) . (cid:3) Proof of Prop. 2.
Let < η < , then there exists a small ρ > such thatfor all ξ ∈ R d (6) | ˆ χ ( ξ ) | ≥ √ η B ρ (0) ( ξ ) where B ρ (0) is the ball of radius ρ centered at . Indeed ˆ χ (0) = R χ ( x ) dx = 1 (see Appendix for the definition of the Fourier transform) and ˆ χ is continuousat (in fact ˆ χ ( ξ ) = | B (0) | J d/ (2 πξ ) / | ξ | d/ where J ν is the classical Besselfunction of order ν , see for instance [6, Sec. B.4]).Let B > , consider a Schwartz function f such that V ( f ) > and let < ε < ρ exp (cid:0) − B V ( f ) / k f k L (cid:1) . We have k f ⋆ χ ε k L ≤ k f ⋆ χ ε k L ∞ k f k L using Hölder’s inequality followed by Young’s inequality, since k χ ε k L = k χ k L = 1 . On the other hand k f ⋆ χ ε k L = R | ˆ χ ( εξ ) | | ˆ f ( ξ ) | dξ and usingthe previous lower bound (6) on ˆ χ we have k f ⋆ χ ε k L ≥ η Z | ξ |≤ ε − ρ | ˆ f ( ξ ) | dξ ≥ η k f k L − Z | ξ | >ε − ρ | ˆ f ( ξ ) | dξ ! Because of the way ε was chosen, (cid:8) | ξ | > ε − ρ (cid:9) ⊂ (cid:8) | ξ | > exp (cid:0) B V ( f ) / k f k L (cid:1) (cid:9) .Consequently we can bound the total mass of the high frequencies Z | ξ | >ε − ρ | ˆ f ( ξ ) | dξ ≤ Z | ξ | > exp( B V ( f ) / k f k L ) | ˆ f ( ξ ) | dξ ≤ Z | ξ | > exp( B V ( f ) / k f k L ) log | ξ | B V ( f ) / k f k L | ˆ f ( ξ ) | dξ ≤ k f k L B Note that we used V ( f ) > . We deduce that k f ⋆ χ ε k L ≥ η (1 − /B ) k f k L which implies k f ⋆ χ ε k L ∞ k f k L ∞ ≥ η (1 − /B ) k f k L k f k L k f k L ∞ Choosing at the beginning η > − κ and B big enough such that η (1 − /B ) > − κ and finally A = max { B, ρ − } proves part a ).Let us now prove part b ). On the torus T d , if θ only has values ± then all the L q norms of θ ( t, · ) are equal to . Thus part a ) implies that if ε < A − exp (cid:0) − A V ( θ ( t, · )) (cid:1) then k θ ( t, · ) ⋆ χ ε k L ∞ k θ ( t, · ) k L ∞ > − κ Thus, the definition of the geometric mixing scale ε ( t ) implies that ε ( t ) ≥ A − exp (cid:0) − A V ( θ ( t, · )) (cid:1) Combining this last bound with Theorem 1 proves b ). (cid:3) Blowup of positive fractional Sobolev norm.
In this section weassume that the velocity field u is bounded in ˙ W , uniformly in time. Byincompressibility the L norm of θ ( t, · ) is constant in time k θ ( t, · ) k L = k θ k L On the other hand, in [1] the authors construct a solution θ ( t, · ) whose ˙ H s norm blows up. More precisely, the initial value θ is in C ∞ c ( R d ) and thesolution θ ( t, · ) does not belong to ˙ H s ( R d ) for any s > and t > .Theorems 1 and 2 give us insight on the blow up of positive fractionalSobolev norms by providing intermediate results between the conservationof the L norm and the (possible) blow up of the ˙ H s norm: NEW APPROACH TO BOUNDS ON MIXING 17 If u is bounded in ˙ W , uniformly in time, there exist various constants C > (denoted by the same letter for readability) such that for all t ≥ Z | ˆ θ ( t, ξ ) | dξ = C (7) Z log | ξ | | ˆ θ ( t, ξ ) | dξ ≤ C (1 + t ) (8) Z (log | ξ | ) | ˆ θ ( t, ξ ) | dξ ≤ C (1 + t ) (9) Z | ξ | s | ˆ θ ( t, ξ ) | dξ can blow up, for any s > (10)We can add one more item to the list by recalling from Section 4.1 thatfor small s , k f k H s = k f k L + 2 s V ( f ) + 2 s W ( f ) + O ( s ) Evidently, the first terms of the expansion of k θ ( t, · ) k H s as s → do notblow up.4.6. Linear growth of V (cid:0) θ ( t, · ) (cid:1) is sharp. In this section we show thatthe linear growth of V (cid:0) θ ( t, · ) (cid:1) in Theorem 1 is sharp, i.e. there is an initialdistribution θ and a velocity field u bounded in ˙ W ,p uniformly in time suchthat V (cid:0) θ ( t, · ) (cid:1) grows linearly. To show this we use the results in [1], wherethe authors prove that the exponential lower bound on the functional mixingscale k θ ( t, · ) k ˙ H − is sharp. We have Proposition 3.
On the two-dimensional torus T , for any p > there existsa velocity field u bounded in ˙ W ,p uniformly in time and a solution θ to (1) such thatFor n ≤ t < n + 1 , V (cid:0) θ ( t, · ) (cid:1) = V (cid:0) θ ( t − n, · ) (cid:1) + n log( λ − ) k θ − ¯ θ k L where ¯ θ = R θ ( x ) dx is the average of θ . Thus V (cid:0) θ ( t, · ) (cid:1) grows linearly V (cid:0) θ ( t, · ) (cid:1) ≥ m + ( t −
1) log( λ − ) k θ − ¯ θ k L with m = inf ≤ t ≤ V (cid:0) θ ( t, · ) (cid:1) Remark . Here we consider the dynamic on the torus T , like in Prop. 2.To be rigorous we then need to consider the functional e V and not V (seeRemark 3). Proof of Prop. 3.
We use the following result from [1]
Fact (Alberti, Crippa, Mazzucato ’14 [1, Sect .1]) . On the two-dimensionaltorus T , there exists a velocity field u bounded in ˙ W ,p ( T ) and a solution θ to (1) such that for all integers n ≥ ,If n ≤ t < n + 1 , θ ( t, x ) = θ (cid:16) t − n, xλ n (cid:17) where /λ is a positive integer. Then for n ≤ t < n + 1 and k ∈ Z d we have ˆ θ ( t, k ) = ( ˆ θ ( t − n, λ n k ) if k ∈ λ − n Z otherwisethus, defining e V like in Remark 3: e V ( f ) = X k ∈ Z d \{ } log | k | | ˆ f ( k ) | we have for n ≤ t < n + 1 e V (cid:0) θ ( t, · ) (cid:1) = e V (cid:0) θ ( t − n, · ) (cid:1) + n log( λ − ) Z T θ ( t − n, x ) dx − (cid:18)Z T θ ( t − n, x ) dx (cid:19) ! which concludes the proof. (cid:3) Acknowledgements
The author would like to thank Nader Masmoudi for suggesting the prob-lem, Pierre Germain for offering helpful comments and Robert Kohn forhis invaluable help in organizing the present work. Support is gratefullyacknowledged from NSF grants DMS-1211806 and DMS-1311833.5.
Appendix
Fourier Transform of log . We use the following Fourier transform:for functions f in the Schwarz class S ( R d ) the Fourier transform F ( f ) or ˆ f of f is ˆ f ( ξ ) = Z R d e − iπξ · x f ( x ) dx Let us now define the tempered distribution T ∈ S ′ ( R d ) by h T, f i = Z | x |≤ f ( x ) − f (0) | x | d dx + Z | x | > f ( x ) | x | d dx for any f ∈ S ( R d ) . Proposition 4.
The Fourier transform of T is h ˆ T , ψ i = Z (cid:0) ζ d − σ d − log | ξ | (cid:1) ψ ( ξ ) dξ for any ψ ∈ S ( R d ) , where σ d − is the surface area of the unit sphere in R d S d − and ζ d is a constant. NEW APPROACH TO BOUNDS ON MIXING 19
Proof.
We adapt the proof of [15, §9,8.(d)]. Let ψ be a test function in S .Then h ˆ T , ψ i = h T, ˆ ψ i = Z | x |≤ | x | d (cid:0) ˆ ψ ( x ) − ˆ ψ (0) (cid:1) dx + Z | x | > | x | d ˆ ψ ( x ) dx = Z | x |≤ | x | d Z ξ (cid:0) e − iπx · ξ − (cid:1) ψ ( ξ ) dξ dx + Z | x | > | x | d Z ξ e − iπx · ξ ψ ( ξ ) dξ dx = A + B To compute further we write the x integral in spherical coordinates x = rω ,with r > and ω ∈ S d − the unit sphere in R d . Then the first term can bewritten A = Z r =0 Z ω ∈ S d − r d Z ξ (cid:0) e − iπx · ξ − (cid:1) ψ ( ξ ) dξ dσ ( ω ) r d − dr = Z r =0 r Z ξ Z ω (cid:0) e − iπξ · rω − (cid:1) ψ ( ξ ) dσ ( ω ) dξ dr where we have swapped the integrals in ω and ξ . We have (see [6, Sec. B.4.]) Z S d − e − iπξ · rω dσ ( ω ) = (2 π ) d/ ˜ J d − (2 πr | ξ | ) where ˜ J ν ( s ) = s − ν J ν ( s ) and J ν is the classical Bessel function of order ν .Continuing the computation, A = Z r Z ψ ( ξ ) (cid:16) (2 π ) d/ ˜ J d − (2 πr | ξ | ) − σ d − (cid:17) dξ dr = Z ψ ( ξ ) Z | ξ | (cid:16) (2 π ) d/ ˜ J d − (2 πs ) − σ d − (cid:17) dss dξ where we swapped the integrals in r and ξ and did a change of variables inthe r integral s = r | ξ | .Similarly for the second term B = Z ψ ( ξ ) Z ∞| ξ | (2 π ) d/ ˜ J d − (2 πs ) dss dξ Putting together A and B we get A + B = Z ψ ( ξ ) (cid:0) ζ d − σ d − log | ξ | (cid:1) dξ with ζ d = Z (cid:16) (2 π ) d/ ˜ J d − (2 πs ) − σ d − (cid:17) dss + Z ∞ (2 π ) d/ ˜ J d − (2 πs ) dss (cid:3) References [1] Giovanni Alberti, Gianluca Crippa, and Anna L Mazzucato. Exponential self-similarmixing and loss of regularity for continuity equations.
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