Erratum: “Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows” [J. Math. Phys. 53, 115611 (2012)]

Lunasin, Evelyn; Lin, Zhi; Novikov, Alexei; Mazzucato, Anna; Doering, Charles R.

doi:10.1063/1.4816334

We noted in the acknowledgement to our paper that we updated Sec. III A using the Jacobian estimate established in Wirosoetisno.⁸ In particular, we stated in that section that if the flow is constrained to have constant palenstrophy then the H⁻¹ mix-norm decays at most

$\sim e^{-\frac{C}{2\sqrt{\pi }}L\mathcal {P}^{1/2}t}$

\sim e^{- \frac{C}{2 \sqrt{π}} L P^{1 / 2} t}

⁠. The bilinear estimate in Wirosoetisno⁸ led to the exponential lower bound for the finite palenstrophy case improving the lower bound that we initially obtained which is only the Gaussian exponential

$e^{-Ct^2}$

e^{- C t^{2}}

⁠.

We were recently informed (thanks to Gautam Iyer for informing us and shedding light on this) that there is an unsettled issue in one of the references (namely, Kim's paper⁴), used in Ref. 8 when proving the Jacobian estimate (12) p. 115611-6 of our paper. Thus, this estimate will not necessarily hold. The main issue that was overlooked in Ref. 4 is that, in order to apply Coifman and Meyer's result to prove Lemma 1 there (p. 1656 in Ref. 4), one needs the bilinear symbol in ξ and η and its derivatives to decay jointly in ξ and η in a suitable way (see Eq. (2.12) in Ref. 8) . This condition is not satisfied by the choice of symbols in Kim's proof, which are constant in η.

Here we present an alternative proof of an explicit lower bound for the decay of the mix-norm, which in particular rules out finite-time perfect mixing for fixed palenstrophy flows, without using the unsupported lemma in Wirosoetisno.⁸ We use a similar approach, first introduced and utilized in Larios et al.,⁵ by defining an auxiliary “stream-function” associated with the scalar field, analogous to the stream-function associated with the vorticity in the 2D incompressible Euler equations. Then following the proofs in Ref. 5, we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.

We establish below that if the flow is constrained to have constant palenstrophy, then the H⁻¹ mix-norm decays at most

$\sim e^{-Ct^2}$

\sim e^{- C t^{2}}

⁠. Finite-time perfect mixing is thus still certainly ruled out when too much cost is incurred by small scale structures in the stirring.

Starting from the advection equation for the periodic mean-zero scalar field θ, multiplying by −Δ⁻¹θ, integrating over the spatial domain, and integrating by parts, recalling that u is mean zero, periodic and divergence free, we get the following series of estimates (using the notation ∇⁻¹ = ∇Δ⁻¹):

\begin{eqnarray*}\frac{d}{dt}\Vert \nabla ^{-1} \theta \Vert _{L^2}^2 &=& \frac{d}{dt}\Vert \theta \Vert ^2_{H^{-1}}\\&=& -2\int _{\mathbb {T}^2} \nabla ^{-1} \theta \,\nabla \mathbf {u}\, \nabla ^{-1} \theta \,d\mathbf {x}\\&\ge & - 2\Vert \nabla ^{-1}\theta \Vert _{\infty }^{2\epsilon }\int _{\mathbb {T}^2} |\nabla ^{-1} \theta |^{2-2\epsilon }\,|\nabla \mathbf {u}|\,d\mathbf {x},\\&\ge & - 2\Vert \nabla ^{-1}\theta \Vert _{\infty } ^{2\epsilon }\Vert \nabla \mathbf {u}\Vert _{L^{1/ \epsilon }} \Vert \,|\nabla ^{-1} \theta |^{2-2\epsilon }\Vert _{L^{1/(1-\epsilon )}}\quad (\rm{H\ddot{o}lder{\text{'}}s})\\&=& - 2\Vert \nabla ^{-1}\theta \Vert _{\infty } ^{2\epsilon }\Vert \nabla \mathbf {u}\Vert _{L^{1/ \epsilon }} \Vert \nabla ^{-1} \theta \Vert ^{2\,(1-\epsilon )}_{L^2}\\&\ge & \frac{-2L^{2\epsilon }\Vert \nabla ^{-1}\theta \Vert _{\infty } ^{2\epsilon }}{\sqrt{\epsilon }} \Vert \Delta \mathbf {u}\Vert _{L^2}\left(\Vert \nabla ^{-1} \theta \Vert _{L^2}^2\right)^{1-\epsilon }, \quad (\text{Sobolev embedding})\end{eqnarray*}

\begin{matrix} \frac{d}{d t} {‖ \nabla^{- 1} θ ‖}_{L^{2}}^{2} & = & \frac{d}{d t} {‖ θ ‖}_{H^{- 1}}^{2} \\ = & - 2 \int_{T^{2}} \nabla^{- 1} θ \nabla u \nabla^{- 1} θ d x \\ \geq & - 2 ‖ \nabla^{- 1} {θ ‖}_{\infty}^{2 ε} \int_{T^{2}} | \nabla^{- 1} {θ |}^{2 - 2 ε} | \nabla u | d x, \\ \geq & - 2 ‖ \nabla^{- 1} {θ ‖}_{\infty}^{2 ε} {‖ \nabla u ‖}_{L^{1 / ε}} ‖ | \nabla^{- 1} θ {|^{2 - 2 ε} ‖}_{L^{1 / (1 - ε)}} (H \ddot{o} lder' s) \\ = & - 2 ‖ \nabla^{- 1} {θ ‖}_{\infty}^{2 ε} {‖ \nabla u ‖}_{L^{1 / ε}} {‖ \nabla^{- 1} θ ‖}_{L^{2}}^{2 (1 - ε)} \\ \geq & \frac{- 2 L^{2 ε} {‖ \nabla^{- 1} θ ‖}_{\infty}^{2 ε}}{\sqrt{ε}} {‖ Δ u ‖}_{L^{2}} {(‖, \nabla^{- 1}, {θ ‖}_{L^{2}}^{2})}^{1 - ε}, (Sobolev embedding) \end{matrix}

where 0 < ε ⩽ 1/2.

Above, we used Hölder's inequality and the embedding of the Sobolev space

$H^1(\mathbb {T}^2)$

H^{1} (T^{2})

into every

$L^p(\mathbb {T}^2)$

L^{p} (T^{2})

⁠, for p ⩾ 2. More precisely, in two dimensions we have that for any w ∈ H¹ with average zero and p ∈ [2, ∞),

\begin{equation}\Vert \mathbf {w}\Vert _{L^p}\le L^{2/p}\sqrt{p-1}\Vert \nabla \mathbf {w}\Vert _{L^2},\end{equation}

{‖ w ‖}_{L^{p}} \leq L^{2 / p} \sqrt{p - 1} {‖ \nabla w ‖}_{L^{2}},

(1)

where we recall that L is the size of the box. This estimate can be established for L = 1 via a Fourier series representation and the Hausdorff-Young inequality, using that the first Fourier coefficient vanishes (see, for example, Danchin and Paicu,² Lieb and Loss,⁶ and Robinson⁷ for the case of

$\mathbb {R}^2$

R^{2}

⁠). Then, the estimate for L ≠ 1 follows by rescaling.

Denoting

$z = \Vert \nabla ^{-1} \theta \Vert _{L^2}^2$

z = ‖ \nabla^{- 1} {θ ‖}_{L^{2}}^{2}

⁠, we can rewrite the above equation as

\begin{equation}\frac{d}{dt} z \ge -\frac{2L^{2\epsilon } \Vert \nabla ^{-1}\theta \Vert _{\infty }^{2\epsilon }}{\sqrt{\epsilon }} \Vert \Delta \mathbf {u}\Vert _{L^2}\, z^{1-\epsilon }\, .\end{equation}

\frac{d}{d t} z \geq - \frac{2 L^{2 ε} {‖ \nabla^{- 1} θ ‖}_{\infty}^{2 ε}}{\sqrt{ε}} {‖ Δ u ‖}_{L^{2}} z^{1 - ε} .

(2)

Dividing both sides by z^{1 − ε} we get

\begin{equation}\frac{1}{z^{1-\epsilon }} \frac{dz}{dt} = \frac{1}{\epsilon } \frac{d}{dt} z^{\epsilon } \ge - 2L^{2\epsilon } \Vert \nabla ^{-1}\theta \Vert _{\infty }^{2\epsilon }(\sqrt{\epsilon})^{-1} \Vert \Delta \mathbf {u}\Vert _{L^2}.\end{equation}

\frac{1}{z^{1 - ε}} \frac{d z}{d t} = \frac{1}{ε} \frac{d}{d t} z^{ε} \geq - 2 L^{2 ε} ‖ \nabla^{- 1} {θ ‖}_{\infty}^{2 ε} {(\sqrt{ε})}^{- 1} {‖ Δ u ‖}_{L^{2}} .

(3)

Multiplying both sides by ε and integrating we obtain

\begin{equation}z^{\epsilon }(t) \ge \left[z(0)\right]^\epsilon - 2\epsilon ^{1/2} L^{2\epsilon } \Vert \nabla ^{-1}\theta \Vert _{\infty }^{2\epsilon } \int _0^t\Vert \Delta \mathbf {u}\Vert _{L^2}\, ds .\end{equation}

z^{ε} (t) \geq {[z (0)]}^{ε} - 2 ε^{1 / 2} L^{2 ε} ‖ \nabla^{- 1} {θ ‖}_{\infty}^{2 ε} \int_{0}^{t} {‖ Δ u ‖}_{L^{2}} d s .

(4)

We next observe that again the Sobolev embedding gives (see, e.g., p. 56 of Ref. 9)

\begin{equation*}-\Vert \nabla ^{-1}\theta \Vert _{L^\infty } \ge - C\,L\Vert \theta \Vert _{L^\infty }\ge - C\, L \Vert \theta _0\Vert _\infty\end{equation*}

- ‖ \nabla^{- 1} {θ ‖}_{L^{\infty}} \geq - {C L ‖ θ ‖}_{L^{\infty}} \geq - C L {‖ θ_{0} ‖}_{\infty}

since θ is taken initially in L^∞ and all L^p norms of θ are preserved by the velocity field u ∈ H² (see, for example, DiPerna and Lions³). Above, we used again that the average of ∇⁻¹θ is zero, and note that C is a universal, dimensionless constant.

The last step involves taking the εth root of both sides of the inequality:

\begin{eqnarray}\Vert \theta (t)\Vert _{H^{-1}}^2 &\ge & \left[\Vert \theta _0\Vert _{H^{-1}}^{2\epsilon }-2 \epsilon ^{1/2} (C L)^{2\epsilon }L^{2\epsilon }\Vert \theta _0\Vert _{L^\infty }^{2\epsilon } \int _0^t\Vert \Delta \mathbf {u}\Vert _{L^2}\, ds \right]^{1/\epsilon },\nonumber\\&\ge & \Vert \theta _0\Vert _{H^{-1}}^{2} \left[1- 2\epsilon ^{1/2}\left(\frac{C^2 L^2\Vert \theta _0\Vert _{L^\infty }}{\Vert \theta _0\Vert _{H^{-1}}}\right)^{2\epsilon } \int _0^t\Vert \Delta \mathbf {u}\Vert _{L^2}\, ds \right]^{1/\epsilon },\\&\ge & \Vert \theta _0\Vert _{H^{-1}}^{2} \left[1- \epsilon ^{1/2}\beta _{\max } \int _0^t\Vert \Delta \mathbf {u}\Vert _{L^2}\, ds \right]^{1/\epsilon }, \nonumber\end{eqnarray}

\begin{matrix} {‖ θ (t) ‖}_{H^{- 1}}^{2} & \geq & {[‖ θ_{0} ‖_{H^{- 1}}^{2 ε} - 2 ε^{1 / 2} {(C L)}^{2 ε} L^{2 ε} ‖ θ_{0} ‖_{L^{\infty}}^{2 ε} \int_{0}^{t} {‖ Δ u ‖}_{L^{2}} d s]}^{1 / ε}, \\ \geq & ‖ θ_{0} ‖_{H^{- 1}}^{2} {[1 - 2 ε^{1 / 2} {(\frac{C^{2} L^{2} {‖ θ_{0} ‖}_{L^{\infty}}}{‖ θ_{0} ‖_{H^{- 1}}})}^{2 ε} \int_{0}^{t} {‖ Δ u ‖}_{L^{2}} d s]}^{1 / ε}, \\ \geq & ‖ θ_{0} ‖_{H^{- 1}}^{2} {[1 - ε^{1 / 2} β_{\max} \int_{0}^{t} {‖ Δ u ‖}_{L^{2}} d s]}^{1 / ε}, \end{matrix}

(5)

where

$\beta _{\max } = \max \left\lbrace 2, 2\left(\frac{ C^2 L^2\Vert \theta _0\Vert _{L^\infty }}{\Vert \theta _0\Vert _{H^{-1}}}\right)\right\rbrace$

β_{\max} = \max \{2, 2 (\frac{C^{2} L^{2} {‖ θ_{0} ‖}_{L^{\infty}}}{‖ θ_{0} ‖_{H^{- 1}}})\}

⁠. The last inequality follows from the fact that the maximum of

$\left(\frac{ C^2 L^2\Vert \theta _0\Vert _{L^\infty }}{\Vert \theta _0\Vert _{H^{-1}}}\right)^{2\epsilon }$

{(\frac{C^{2} L^{2} {‖ θ_{0} ‖}_{L^{\infty}}}{‖ θ_{0} ‖_{H^{- 1}}})}^{2 ε}

is attained either at ε = 0 or ε = 1/2 when 0 < ε ⩽ 1/2. Then denote

$A(t) = \beta _{\max }\int _0^t \Vert \Delta \mathbf {u}\Vert _2 \,ds$

A (t) = β_{\max} \int_{0}^{t} {‖ Δ u ‖}_{2} d s

⁠. We next choose ε small enough, so that for a given t > 0, we have

\begin{equation}\epsilon ^{1/2} A(t) = 1/2,\end{equation}

ε^{1 / 2} A (t) = 1 / 2,

(6)

we obtain

\begin{equation}\Vert \theta (t)\Vert _{H^{-1}}^2 \ge \Vert \theta _0\Vert _{H^{-1}}^{2} \left[1/2 \right]^{4A(t)^2} \ge \Vert \theta _0\Vert _{H^{-1}}^{2} e^{-(\ln 16)\cdot A(t)^2}.\end{equation}

{‖ θ (t) ‖}_{H^{- 1}}^{2} \geq ‖ θ_{0} ‖_{H^{- 1}}^{2} {[1 / 2]}^{4 A {(t)}^{2}} \geq {‖ θ_{0} ‖}_{H^{- 1}}^{2} e^{- (\ln 16) \cdot A {(t)}^{2}} .

(7)

Observe that since A(t) ⩽ β_maxt^1/2B(t)^1/2, where

$B(t) = \int _0^t\Vert \Delta \mathbf {u}\Vert ^2_{L^2}\,ds$

B (t) = \int_{0}^{t} {‖ Δ u ‖}_{L^{2}}^{2} d s

⁠, we also have

\begin{equation}\Vert \theta (t)\Vert _{H^{-1}}^2 \ge \Vert \theta _0\Vert _{H^{-1}}^{2} \left[1/2 \right]^{4A(t)^2} \ge \Vert \theta _0\Vert _{H^{-1}}^{2}e^{-(\ln 16)\beta ^2_{\max }\cdot t B(t)}.\end{equation}

{‖ θ (t) ‖}_{H^{- 1}}^{2} \geq ‖ θ_{0} ‖_{H^{- 1}}^{2} {[1 / 2]}^{4 A {(t)}^{2}} \geq {‖ θ_{0} ‖}_{H^{- 1}}^{2} e^{- (\ln 16) β_{\max}^{2} \cdot t B (t)} .

(8)

For constant palenstrophy stirring,

$B(t)=\mathcal {P}t$

B (t) = P t

increases linearly in time so

\begin{equation}\Vert \theta (t)\Vert _{H^{-1}}^2 \ge \Vert \theta _0\Vert _{H^{-1}}^{2}e^{-(\ln 16)\beta ^2_{\max }\mathcal {P}\cdot t^2 }.\end{equation}

{‖ θ (t) ‖}_{H^{- 1}}^{2} \geq {‖ θ_{0} ‖}_{H^{- 1}}^{2} e^{- (\ln 16) β_{\max}^{2} P \cdot t^{2}} .

(9)

We finally observe that exponential decay can be obtained for velocities fields that are slightly more regular than H². Indeed, if the velocity field is in the Besov space

$B^{2}_{2,1}$

B_{2, 1}^{2}

⁠, then its gradient is in the Besov space

$B^1_{2,1}\subset L^\infty$

B_{2, 1}^{1} \subset L^{\infty}

(see, e.g., Proposition 2.39 of Ref. 1). Then, Hölder's inequality gives

\begin{equation*}\Vert J(f,g)\Vert _{L^2} \le C\, \Vert \nabla f\Vert _{L^\infty } \Vert \nabla g\Vert _{L^2},\end{equation*}

{‖ J (f, g) ‖}_{L^{2}} \leq {C ‖ \nabla f ‖}_{L^{\infty}} {‖ \nabla g ‖}_{L^{2}},

in place of Eq. (12) in our paper, and the rest of the argument in Sec. III can be carried out.

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Erratum: “Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows” [J. Math. Phys. 53, 115611 (2012)]

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