Notes on the large time of Euler equations and inviscid damping

Consider the classical incompressible Euler equations in two dimension; namely written in the vorticity formulation, the transport equation for the unknown scalar vorticity ${\omega = \omega(t,x,y)}$ ,

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega =0$

posed on a spatial domain ${\Omega \subset \mathbb{R}^2}$ , where the velocity field ${u\in \mathbb{R}^2}$ is obtained through the Biot-Savart law ${u = \nabla^\perp \Delta^{-1} \omega}$ (with convention $(a_1,a_2)^\perp = (a_2,-a_1)$ ). By construction, the velocity field ${u}$ is incompressible: ${\nabla \cdot u=0}$ . When dealing with domains with a boundary, ${\Delta^{-1} }$ is defined together with a Dirichlet boundary condition that corresponds to the no-penetration condition of fluids ${u\cdot n =0}$ on the boundary.

The global well-posedness theory of 2D Euler is classical: (1) smooth initial data give rise to solutions that remain smooth for all times (e.g. the Beale-Kato-Majda criterium holds, as vorticity is uniformly bounded for all times; in fact, being transported along the volume-preserving flow, all ${L^p}$ norms of vorticity are conserved), and (2) weak solutions with bounded vorticity are unique (see Yudovich ’63).

The major open problem is to understand the large time behavior of solutions to the 2D Euler equations. This is notoriously difficult, being time-reversible Hamiltonian system and having conserved energy and many invariant Casimirs

$\displaystyle \mathcal{E}[u] = \int_\Omega \frac{|u|^2}{2} \;dx , \qquad \mathcal{C}[\omega] = \int_\Omega \Phi(\omega) \; dx$

that are conserved for all times, for any reasonable function ${\Phi(\cdot)}$ . A great reference that discusses in depth this topics and other questions in fluid dynamics is this lecture notes by V. Sverak.

1. Steady states

It is convenient to introduce the stream function ${\phi}$ solving ${\Delta \phi = \omega}$ (with a Dirichlet boundary condition when the spatial boundary is not empty). The velocity is thus defined by ${u = \nabla^\perp \phi}$ . Steady states to 2D Euler are those that satisfy ${\nabla^\perp \phi \cdot \nabla \omega =0}$ . That is, vorticity is constant along the streamlines of the flow. There is a large class of such steady state solutions. Namely, ${\omega = F(\phi)}$ , or equivalently for any given function ${F(\cdot)}$ , solutions ${\phi_*}$ to the elliptic problem

$\displaystyle \Delta \phi_* = F(\phi_*)$

give a steady state of 2D Euler, with velocity ${u_* = \nabla^\perp \phi_*}$ and vorticity ${\omega_* = F(\phi_*)}$ . A major open problem is to determine which ${F(\cdot)}$ would attract the large time dynamics of Euler.

A special class of steady states to Euler are shear flows of the form

$\displaystyle u_{sh} = \begin{pmatrix} U(y) \\ 0 \end{pmatrix}$

which is an exact solution to Euler equations for any function ${U(y)}$ . The vorticity is computed by ${\omega_{sh} = U'(y)}$ . Again, it is not known which shear flows are the asymptotic states of Euler for generic initial data.

2. Arnold stability

Arnold in around 1960s made an attempt to address the question on which function ${F(\cdot)}$ is relevant at the large time dynamics. More precisely, he studies the functional

$\displaystyle I[\omega] = \int_\Omega \frac{|u|^2}{2} \;dx + \int_\Omega \Phi(\omega) \; dx$

namely, the sum of energy and Casimirs, which are invariant in time. The functional is now designed so that the steady states are a critical point, namely the first variation vanishes, leading to

$\displaystyle \begin{aligned}0 = I'[\omega_*] \omega &= \int_\Omega \Big[u_* \cdot u + \Phi'(\omega_*) \omega \Big] \;dx \\ &= \int_\Omega \Big[\nabla^\perp\phi_* \cdot \nabla^\perp \phi + \Phi'(\omega_*) \omega \Big] \; dx \\ &= \int_\Omega \Big[\phi_* + \Phi'(F(\phi_*)) \Big] \omega \; dx \end{aligned}$

recalling ${\omega = \Delta \phi }$ and ${\omega_* = F(\phi_*)}$ . The choice of Casimirs is clear: namely, take ${\Phi(\cdot)}$ so that ${\phi_* + \Phi' (F(\phi_*)) = 0}$ , or equivalently

$\displaystyle \Phi' := - F^{-1}$

the inverse map of ${F(\cdot)}$ . Now observe that monotonicity of ${F(\cdot)}$ gives convexity of ${\Phi(\cdot)}$ and hence, convexity of the functional ${I[\omega]}$ , which is sufficient for the nonlinear stability in the sense of Lyapunov. Precisely,

Theorem 1 (Arnold ’60s) Let ${\Omega\subset \mathbb{R}^2}$ be bounded and simply connected. The steady state ${u_* = \nabla^\perp \phi_*}$ , with ${\Delta \phi_* = F(\phi_*)}$ and ${\phi_* = 0}$ on the boundary, is nonlinearly stable in ${H^1(\Omega)}$ , provided that ${F'(\cdot) <0}$ .

In the particular case of shear flows ${u_{sh} = [U(y),0]}$ . The vanishing of the first variation now reads ${U + \Phi'' (U') U'' = 0}$ , leading to ${\Phi'' = -\frac{U}{U''}}$ . That is, the Lyapunov’s functional in case of shear flows reads

$\displaystyle I[\omega] = \int_\Omega \Big[ \frac{|u|^2}{2} - \frac{U(y)}{U''(y)} |\omega|^2\Big] \; dx .$

The stability follows for strictly convex shear flows or more precisely for those which satisfy ${\frac{U(y)}{U''(y)} <0}$ (namely, in the presence of an inflection point, ${U(y)}$ is also required to vanish at that point; else, possible unstable modes are known to exist).

However, it is clear that Arnold stability gives no information on the asymptotic behavior of solutions to Euler starting from data that are sufficiently near these steady states. For instance, it’s not even clear whether velocity remains bounded for all times, not to mention any decay!

3. Rayleigh stability

The first step is to understand mode stability, which dates back to classical works by Rayleigh whose study the linearized Euler evolution around a shear flow ${U(y)}$ , which reads

$\displaystyle (\partial_t + U\partial_x) \omega - \partial_x \phi U'' = 0$

where ${\omega = \Delta \phi}$ . Taking Fourier in ${x}$ with variable ${k = i\partial_x}$ and Laplace in ${t}$ with variable ${\lambda = \partial_t}$ , one immediately arrives at the resolvent equation

$\displaystyle Ray_k(\phi): = (\partial_y^2 - k^2) \phi - \frac{U''}{ U - c} \phi = \frac{\omega_0}{ik(U-c)}$

upon setting ${c = -\lambda / ik}$ for sake of convenience. This is the classical Rayleigh equation which is accompanied with the Dirichlet boundary conditions ${ ik \phi = 0}$ . Note that ${\Re \lambda = k \Im c}$ , that is, unstable growth modes exist iff homogenous solutions to the Rayleigh equation exist for some unstable eigenvalue ${c}$ with ${k\Im c >0}$ . By symmetry, it suffices to consider the case when ${k>0}$ .

Theorem 2 (Rayleigh’s inflexion-point criterium, 1880) A necessary condition for instability is that the basic velocity profile must have an inflection point.

Proof: Indeed, multiplying the homogenous Rayleigh equation by ${\bar \phi/(U-c)}$ and integrating by parts yield

$\displaystyle \int_0^\infty (|\partial_y \phi|^2 + k^2 |\phi|^2) \;dy+ \int_0^\infty \frac{U''}{U-c}|\phi|^2 \;dy=0, \ \ \ \ \ (1)$

whose imaginary part reads

$\displaystyle \Im c \int_0^\infty \frac{U''}{|U-c|^2}|\phi|^2 \;dy =0. \ \ \ \ \ (2)$

That is, the instability condition ${\Im c>0}$ (if exists) must imply that ${U''}$ changes its sign. $\Box$

As a consequence of the Rayleigh criterium, shear flows ${U(y)}$ without an inflection point are spectrally stable for the linearized Euler equations. There are also stable shear flows with inflection points. For instance,

Lemma 3 (Fjortoft criterium, 1950) A necessary condition for instability is that ${U'' (U - U(y_c))<0}$ somewhere in the flow, where ${y_c}$ is a point at which ${U''(y_c) =0}$ .

Proof: Indeed, taking the real part of (1) and using (2) with ${\Im c>0}$ yield

$\displaystyle \int_0^\infty (|\partial_y \phi|^2 + k^2 |\phi|^2) \;dy+ \int_0^\infty \frac{U'' (U-U(y_c))}{|U-c|^2}|\phi|^2 \;dy=0$

which immediately gives the Fjortoft’s criterium. $\Box$

Lin ’03 identifies a large class of shear flows that are spectrally unstable (necessarily have an inflection point ${y = y_c}$ ). Roughly speaking, the assertion is that the growth mode exists, if there is a positive eigenvalue of the Rayleigh or Schrodinger operator ${Ray_k(\cdot)}$ , defined above, for ${c = U(y_c)}$ and for sufficiently small ${k}$ .

In case of the whole space ${\mathbb{R}^2}$ , all shear flows with different constant asymptotic states ${U_\pm}$ are always unstable (again due to the presence of very low frequency ${|k|\ll1}$ ). On the half-space or in an infinite long channel, all shear flows, but Couette, are unstable at a sufficiently high Reynolds number, including those that are stable to Euler equations. This is a classical Heisenberg’s viscous destabilization, whose complete mathematical proof is given in my recent work, here and here, with Y. Guo and E. Grenier.

4. Inviscid damping

After the breakthrough on nonlinear Landau damping for plasmas by Mouhot-Villani ’11 (see this previous blog of mine on the topics), there are active efforts to establish similar damping for fluids, notably the work by Bedrossian-Masmoudi ’15 for Euler near plane Couette ${U(y)=y}$ on ${\mathbb{T}\times \mathbb{R}}$ , and recent works by Ionescu-Jia ’20 and Masmoudi-Zhao ’20 both for Euler near monotone shear flows in finite channel ${\mathbb{T}\times [0,1]}$ . Like Landau damping by Mouhot-Villani, the nonlinear results are obtained precisely for analytic or some Gevrey data.

The fundamental damping mechanism is phase mixing on torus ${\mathbb{T}\times \mathbb{R}}$ . Namely, the linearized Euler equations near ${U(y)}$ read

$\displaystyle (\partial_t + U\partial_x) \omega - U'' \partial_x \Delta^{-1}\omega = 0$

which can be put in a symmetric form ${ (\partial_t + H\partial_x) \omega =0}$ where ${H = U(y) + K}$ for some compact self-adjoint operator ${K}$ (e.g. see Section 2 in this paper of mine or this previous blog post of mine). Ignoring ${K}$ for a moment, as time increases, solutions are shearing on torus, according to ${\omega(t,x,y) \sim \omega_0(x+U(y)t,y)}$ , which eventually do not see the difference in location in ${x}$ . As a consequence, average in ${y}$ decays in time in accordance to regularity in ${y}$ , which is known as phase mixing. Equivalently, solving in Fourier in ${x}$ ,

$\displaystyle \widehat\omega_k(t,y) \sim e^{-ikU(y)t }\widehat \omega^0_k(y) .$

The decay in time follows: namely, only “local” decay in ${L^2}$ for vorticity is expected (recalling ${L^2}$ norm does not decay)

$\displaystyle \|\langle \partial_y \rangle^{-N} \widehat\omega_k(t,\cdot)\|_{L^2_y(K)} \lesssim \langle kt \rangle^{-N}$

for any ${N\ge 0}$ , provided that ${U'>0}$ on a compact subset ${K}$ . Near critical points of ${U(y)}$ , decay is slower: e.g., only at rate of order ${\langle kt\rangle^{-N/2}}$ near non-degenerate critical points (such as ${U(y) = y^2}$ ). That is, for monotone shear flows, the horizontal component of velocity decays at rate ${t^{-1}}$ , while the vertical component decays at rate ${t^{-2}}$ .

In case of the Couette flow ${U(y) = y}$ , the linearized Euler becomes ${(\partial_t + y \partial_x)\omega =0}$ , and decay can be calculated more explicitly in Fourier in both ${x}$ and ${y}$ , giving

$\displaystyle \widehat \omega_{k,\eta}(t) = \widehat \omega^0_{k,\eta + kt}$

where ${k = i\partial_x}$ and ${\eta = i \partial_y}$ . That is, if vorticity is initially located around ${(k_0,\eta_0)}$ , then it will be located around ${(k_0, \eta_0 - k_0 t)}$ at all the later times: polynomially localized for Sobolev data and exponentially localized for analytic data. Decay for velocity is also explicit:

$\displaystyle \widehat u^1_{k,\eta} = \frac{-i\eta}{k^2 + \eta^2} \widehat \omega^0_{k,\eta + kt} \sim \langle kt\rangle^{-1}, \qquad \widehat u^2_{k,\eta} = \frac{ik}{k^2 + \eta^2} \widehat \omega^0_{k,\eta + kt}\sim |k|\langle kt\rangle^{-2},\ \ \ \ \ (3)$

since ${\eta}$ is localized near ${\eta_0 - k_0t}$ (for initial vorticity located around ${(k_0,\eta_0)}$ in Fourier frequency space).

5. Nonlinear damping

In the remaining part of the post, I give a sketch of the proof of nonlinear inviscid damping near Couette obtained by Bedrossian and Masmoudi. That is, the inviscid damping (3) holds for the nonlinear equations under small analytic and Gevrey initial perturbations. I focus only the case of analytic data, as the result for Gevrey data is similar, upon making use of integration by parts to avoid the apparent loss of one derivatives in the transport equation. The first step is to factor out the dynamics of the free transport by introducing

$\displaystyle g(t,x,y) = \omega(t,x+yt,y) , \qquad \phi(t,x,y) = \theta(t,x+yt,y)$

leading to

$\displaystyle \partial_t g = - \partial_y \phi \partial_x g + \partial_x \phi \partial_y g , \qquad \Delta_t \phi = g, \ \ \ \ \ (4)$

where we have used the notation ${\Delta_t = \partial_x^2 + (\partial_y - t\partial_x )^2}$ . Observe that the transport equation (4) presents a loss of one derivatives, and a convenient way to fight this loss is to use the classical Cauchy-Kovalevskaya theory, which allows one to view it as an ODE.

To follow the analytic regularity of solutions, following what I did with Emmanuel Grenier and Igor Rodnianski on Landau damping, we may introduce the generator functions

$\displaystyle Gen[g](z) :=\sum_{k\in \mathbb{Z}} \int_{\mathbb{R}} e^{z\langle k,\eta\rangle } |\widehat{g}_{k,\eta}| \; d\eta$

for analyticity radius ${z\ge 0}$ , where ${\widehat{g}_{k,\eta}}$ denotes the Fourier transform of ${g(t,x,y)}$ in ${x,y}$ , respectively. Note that inviscid damping (3) follows from the boundedness of ${Gen[g](z)}$ for some ${z>0}$ . The generator functions are non negative, and all their derivatives are non negative and non decreasing in z. Moreover, they commute with the product, the sum and the differentiation, making their use very versatile not only to provide existence of analytic solutions, including at the large time such as Landau damping as done in my recent work, but also to capture instabilities or physics that may not be seen for analytic data (plasma echoes, boundary layers,…). See also this previous blog post of mine. For instance, we have

$\displaystyle Gen[fg] \le Gen[f] Gen[g], \qquad Gen[\partial_x g ] + Gen[\partial_y g] \le \partial_z Gen[g] .$

The goal is to derive a suitable Hopf-type differential inequality for generators ${Gen[g(t)](z)}$ : for instance,

$\displaystyle \partial_t Gen[g(t)](z) \le C_0 \langle t\rangle^{-1-\epsilon} Gen[g(t)](z) \partial_z Gen[g(t)](z) \ \ \ \ \ (5)$

for ${g(t)}$ being solutions to (4), ${\epsilon>0}$ , and ${z\ge 0}$ . This would imply that the analyticity radius does not shrink to zero in finite time, since ${\int_0^\infty \langle t\rangle^{-1-\epsilon}\; dt < \infty}$ . The existence of analytic or Gevrey solutions, and hence inviscid damping, would follow from a standard bootstrap argument.

To proceed, we take the Fourier transform of (4) in both ${x,y}$ , yielding

$\displaystyle \partial_t \widehat{g}_{k,\eta}(t) = \sum_{k_1\in \mathbb{Z}}\int K_{k_1,\eta_1, k_2, \eta_2}(t) \widehat{g}_{k_1,\eta_1}\widehat{g}_{k_2, \eta_2} \; d\eta_1$

for each ${(k,\eta) \in \mathbb{Z}\times \mathbb{R}}$ , where ${k_2 = k - k_1}$ , ${\eta_2 = \eta -\eta_1}$ , and the interaction kernel ${K_{k_1,\eta_1, k_2, \eta_2}(t)}$ is given by

$\displaystyle K_{k_1,\eta_1, k_2, \eta_2}(t): =\frac{\eta_1 k_2 - \eta_2 k_1}{k_1^2 + |\eta_1 - k_1 t|^2} .$

Note that in the case when ${k_1 =0}$ , the interaction kernel does not decay in time, and we thus need to mode out the contribution of ${\widehat{g}_{0,\eta_1}}$ , namely the average of ${g(t,x,y)}$ in ${x\in \mathbb{T}}$ , which is a standard practice. On the other hand, the case when ${k_2=0}$ is simpler, since the kernel presents no loss of derivatives (for bounded ${\eta_2}$ ).

I focus the discussion on the case when both ${k_1\not =0}$ and ${k_2\not =0}$ . Observe that the interaction kernel ${K_{k_1,\eta_1, k_2, \eta_2}(t)}$ is integrable in time, however the time decay is not uniform near the critical time ${t\sim \eta_1 / k_1}$ , which complicates the use of the classical Cauchy-Kovalevskaya theory. For this reason, introduce the weighted vorticity

$\displaystyle \widehat{G}_{k,\eta}(t) = a_{k,\eta}(t)\widehat{g}_{k,\eta}(t),$

where ${a_{k,\eta}(t)}$ is some Fourier multiplier to be determined, leading to

$\displaystyle \begin{aligned} \partial_t \widehat{G}_{k,\eta} - \frac{\dot a_{k,\eta}(t)}{a_{k,\eta}(t)} \widehat{G}_{k,\eta} = \sum_{k_1 \in \mathbb{Z}}\int \mathcal{K}_{k_1,\eta_1, k_2, \eta_2}(t)\widehat{G}_{k_1,\eta_1}\widehat{G}_{k_2, \eta_2} \; d\eta_1 \end{aligned} \ \ \ \ \ (6)$

with the new interaction kernel

$\displaystyle \mathcal{K}_{k_1,\eta_1, k_2, \eta_2}(t): =\frac{\eta_1 k_2 - \eta_2 k_1}{k_1^2 + |\eta_1 - k_1 t|^2} \frac{a_{k,\eta}(t)}{a_{k_1,\eta_1}(t) a_{k_2,\eta_2}(t)} .$

The goal is now to derive a similar Hopf-type differential inequality (5) for the analytic norm of ${G(t)}$ . Several observations are to follow.

5.1. Localized interaction.

The convenient fact about working with analytic or Gevrey data is that the nonlinear interaction is essentially exponentially localized. Precisely, take

$\displaystyle a_{k,\eta}(t) = e^{\langle k, \eta\rangle^\sigma}$

for some ${\sigma \in (0,1)}$ , which does not limit further the analytic class of initial data due to the weight ${e^{z\langle k, \eta\rangle}}$ inserted in the generator norm. Conveniently, a direct calculation yields

$\displaystyle C_{k,\eta}: = \frac{a_{k,\eta}}{a_{k_1,\eta_1} a_{k_2,\eta_2}} \le \left \{ \begin{aligned} e^{-\theta_0 \langle k_1,\eta_1\rangle^{\sigma}}, \qquad &\langle k_2,\eta_2\rangle \ge \frac 12 \langle k,\eta\rangle \\ e^{-\theta_0\langle k_2,\eta_2\rangle^{\sigma}}, \qquad &\langle k_1,\eta_1\rangle \ge \frac 12 \langle k,\eta\rangle \end{aligned} \right.$

for any ${k = k_1 + k_2}$ and ${\eta = \eta_1 + \eta_2}$ , and for ${\sigma \in (0,1)}$ . The constant ${\theta_0}$ may depend on ${\sigma<1}$ . Note that the triangle inequality ${\langle k_1,\eta_1\rangle + \langle k_2, \eta_2\rangle \ge \langle k,\eta\rangle }$ implies that one of the two cases in the above expression must occur. This shows that the interaction kernel ${\mathcal{K}_{k_1,\eta_1, k_2, \eta_2}(t)}$ is exponentially localized either in ${\langle k_1,\eta_1\rangle^\sigma}$ or ${\langle k_2, \eta_2\rangle^\sigma}$ near zero.

5.2. Case 1: ${\langle k_1,\eta_1 \rangle \approx 0}$ .

In this case, we may use ${ \langle t - \eta_1/k_1 \rangle \gtrsim \langle t\rangle}$ to bound

$\displaystyle \mathcal{K}_{k_1,\eta_1, k_2, \eta_2}(t) \lesssim \langle k, \eta \rangle \langle t\rangle^{-2} .$

That is, the nonlinear interaction experiences one loss of derivatives (i.e., the factor ${\langle k, \eta\rangle}$ ), but decays in time at an integrable rate: ${\langle t\rangle^{-2}}$ . This proves the differential inequality (5) for ${G(t)}$ in this case, yielding the global analytic solution for such a kernel.

5.3. Case 2: ${\langle k_2,\eta_2\rangle \approx 0}$ .

As ${(k_1,\eta_1)}$ may be very large, the interaction kernel is of order one near the critical time ${t \sim \eta_1/k_1}$ . Recall that the interaction kernel is integrable in time, however the bound is not uniform in the sense of (5). To overcome this issue near critical times, Bedrossian and Masmoudi constructs weight functions ${a_{k,\eta}(t)}$ so that the extra damping term in the equation (6): namely

$\displaystyle \frac{|\dot a_{k,\eta}|}{a_{k,\eta}} \widehat{G}_{k,\eta}$

controls precisely with the interaction kernel near these critical times, provided that ${a_{k,\eta}(t)}$ is nonnegative and decreasing in time. I now precise this point.

Let ${I_{k,\eta}}$ be the critical time interval:

$\displaystyle I_{k,\eta} =\Big [\frac{\eta}{k} - \frac{|\eta|}{2k^2},\frac{\eta}{k} + \frac{|\eta|}{2k^2} \Big]$

if ${\eta k >0}$ and ${|\eta| \gg k^2}$ . Otherwise, set ${I_{k,\eta} = \emptyset}$ . One may first “symmetrize” the interaction kernel in ${(k_1,\eta_1)}$ and ${(k,\eta)}$ by introducing the Ansatz

$\displaystyle a_{k,\eta}(t) = \left \{ \begin{aligned} \frac{|\eta / k^2|}{\langle t - \eta / k\rangle} \widetilde{a}(t,\eta),\qquad & t \in I_{k,\eta} , \\ \widetilde{a}(t,\eta),\qquad & t\not \in I_{k,\eta}, \end{aligned}\right. \ \ \ \ \ (7)$

for some new weight function ${\widetilde{a}(t,\eta)}$ that is independent of ${k}$ , leading to the following bound on the interaction kernel

$\displaystyle \mathcal{K}_{k_1,\eta_1, k_2, \eta_2}(t) \lesssim \frac{|\eta / k^2|}{ \langle t - \eta_1 / k_1\rangle \langle t - \eta/ k\rangle}$

for ${k_1 \not =0}$ and ${k_1\not = k}$ (which is essentially symmetric in ${(k_1,\eta_1)}$ and ${(k,\eta)}$ ).

If ${t \not \in (I_{k_1,\eta_1} \cup I_{k,\eta})}$ , the interaction kernel ${\mathcal{K}_{k_1,\eta_1, k_2, \eta_2}(t) }$ is again bounded by ${\langle \eta \rangle \langle t\rangle^{-2}}$ , which is integrable and (5) holds. If ${t \in I_{k_1,\eta_1}}$ , then since ${k_1 \not = k}$ , ${t \not \in I_{k,\eta_1}}$ . Thus, the interaction kernel is bounded by ${\langle t - \eta_1 / k_1\rangle ^{-1}}$ , which can be controlled by the left hand side of (6), provided that

$\displaystyle \frac{\dot {\widetilde a}(t,\eta_1)}{\widetilde{a} (t,\eta_1)} \approx \frac{-1}{\langle t - \eta_1 / k_1\rangle} , \ \ \ \ \ (8)$

when ${t \in I_{k_1,\eta_1}}$ . Namely, the weight function ${a_{k,\eta}(t)}$ was introduced precisely to control the ${1/t}$ decay near critical times. Such a weight function can be easily constructed by integrating in time the above identity.

Finally, if ${t \in I_{k,\eta}}$ , then again since ${k_1 \not = k}$ , ${t\not \in I_{k_1,\eta}}$ , and the interaction kernel is now of order ${\langle t - \eta / k\rangle ^{-1}}$ . In order to absorb this into the left, we again need

$\displaystyle \frac{1}{\langle t - \eta / k\rangle} \approx \frac{|\dot{\widetilde{a}} (t,\eta)|}{\widetilde{a} (t,\eta)} \approx \frac{|\dot{\widetilde{a}} (t,\eta_1)|}{\widetilde{a} (t,\eta_1)}$

for ${t\in I_{k,\eta}}$ and for any ${k_1 \not =0, k}$ , in which the first approximation is due to construction (8) and the last approximation holds, up to some growth in ${\eta_2 = \eta - \eta_1}$ , which is easily controlled thanks to the exponential localization in ${\eta_2}$ near zero.

To conclude, the non-decaying, but integrable, interaction kernel near critical times is treated or absorbed to the left, precisely thanks to the construction of weight functions ${a_{k,\eta}(t)}$ satisfying (7) and (8) and the damping term ${\frac{|\dot a_{k,\eta}(t)|}{a_{k,\eta}(t)} \widehat{G}_{k,\eta}}$ in the left of (6), yielding the Hopf-type differential inequality for generators of the weighted vorticity function, from which the existence of solutions easily follows. The weight function also plays a crucial role in the recent important advances for monotone shear flows by Ionescu-Jia ’20 and Masmoudi-Zhao ’20.

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Notes on the large time of Euler equations and inviscid damping

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