Abstract
We consider the 2D Navier–Stokes equation on \(\mathbb T \times \mathbb R\), with initial datum that is \(\varepsilon \)-close in \(H^N\) to a shear flow (U(y), 0), where \(\Vert U(y) - y\Vert _{H^{N+4}} \ll 1\) and \(N>1\). We prove that if \(\varepsilon \ll \nu ^{1/2}\), where \(\nu \) denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains \(\varepsilon \)-close in \(H^1\) to \((e^{t \nu \partial _{yy}}U(y),0)\) for all \(t>0\). Moreover, the solution converges to a decaying shear flow for times \(t \gg \nu ^{-1/3}\) by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than \(\nu ^{1/2}\) for 2D shear flows close to the Couette flow.
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Acknowledgments
The work of J.B. was partially supported by NSF Grant DMS-1462029, and by an Alfred P. Sloan Research Fellowship. The work of V.V. was partially supported by NSF Grant DMS-1514771 and by an Alfred P. Sloan Research Fellowship. The work of F.W. was partially supported by NSF Grant DMS-1514771.
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Communicated by Alex Kiselev.
Appendix: Construction and Properties of the Multiplier M
Appendix: Construction and Properties of the Multiplier M
In this section, we recall some of the technical tools regarding the Fourier multiplier M from Bedrossian et al. (2015d) and adapt them to our simpler setting.
Lemma 4.1
There exists a multiplier M such that the conditions (16a)–(16e) hold for some constant \(0<c<1\).
Proof of Lemma 4.1
We consider a multiplier of the form \(M=M_1M_2\) such that both \(M_1\) and \(M_2\) satisfy (16a). We choose \(M_1\) such that for \(k\ne 0\) it is determined by the ODE
Similar multipliers appeared in Zillinger (2014), Bedrossian et al. (2015b, (2015c, (2015d). The above multiplier clearly satisfies (16c). Notice that for \(k \ne 0\), there holds
and hence (16f) holds for \(M_1\). A direct computation shows that
which implies (16b) holds for \(M_1\). Taking the derivative of \(M_1\) with respect to \(\xi \) gives
which proves that (16d) holds for \(M_1\). Note here that \(k \ne 0\) implies \(|k|\ge 1\).
Next, we define \(M_2\) by the differential equation (for \(k \ne 0\)),
This multiplier was introduced in Bedrossian et al. (2015d). Similarly to \(M_1\), we deduce that (16b), (16d), and (16f) all hold for \(M_2\) (and hence also for \(M = M_1 M_2\)). Since for \(k\ne 0\) we have that
and
inequality (16e) holds for \(M_2\). Therefore, the multiplier M we constructed satisfies conditions (16a)–(16e), completing the proof. \(\square \)
Remark 4.2
In condition (16e), the power \(\nu ^{-1/6}\) in front of the multiplier is sharp in the sense that 1 / 6 is the smallest sacrifice we need to make to bound the expression on the right side from below by a constant. In fact, if we make \(M_2\) to be positive constants independent of \(\nu \) when \(\left| t-\xi /k\right| \ge \nu ^{-1/3}\), then the size of \(\dot{M}_2\) should be approximately \(\nu ^{1/3}\). Hence, if we did not need (16f) or (16d), one could construct \(M_2\) to satisfy (16e) with
We need the following Lemma to commute \(\sqrt{-\dot{M}M}\) with \(\Delta _L\Delta _t^{-1}\), the latter of which is not a Fourier multiplier.
Lemma 4.3
Let \(f\in H^N\) and \(N > 1\). Then the following estimate holds for \(\delta \) sufficiently small,
Proof of Lemma 4.3
Using the equality
we have
By (16f), \(N>1\), and (42), we deduce
and similarly
Since \(\delta \ll 1\), the result follows from (50) immediately. \(\square \)
The following lemma is proved in the same manner as Lemma 4.3, although slightly simpler. In particular, this lemma shows that \(\Delta _L \Delta _t^{-1}\) can be approximately treated as the identity for \(\delta \) sufficiently small.
Lemma 4.4
Let \(f\in H^N\) and \(N > 1\). Then the following estimate holds for \(\delta \) sufficiently small,
The following estimate applies to the zero mode of the velocity field.
Lemma 4.5
Let f be such that \(f_0\in H^N\), \(N>1\), and \(\partial _{v}\Delta _t^{-1}f_0\in L^2\). Then for \(\delta \) sufficiently small, there holds
Proof of Lemma 4.5
Since
we have by \(N>1\) and (42) (interpolating \(H^N\) between \(L^2\) and \(H^{N+1}\)),
For \(\delta \ll 1\), we may absorb the second term on the right side and we obtain the desired result. \(\square \)
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Bedrossian, J., Vicol, V. & Wang, F. The Sobolev Stability Threshold for 2D Shear Flows Near Couette. J Nonlinear Sci 28, 2051–2075 (2018). https://doi.org/10.1007/s00332-016-9330-9
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DOI: https://doi.org/10.1007/s00332-016-9330-9