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We study the stability of spectrally stable, strictly monotone, and smooth shear flows in the 2D Navier-Stokes equations on $$\mathbb{T} \times \mathbb{R}$$ with small viscosity $$\nu$$. We establish nonlinear stability in $H^s$ for $$s \geq 2$$ with a threshold of size $$\epsilon \nu^{1/3}$$ for time smaller than $$c_*\nu^{-1}$$ with $$\epsilon, c_* \ll 1$$. Additionally, we demonstrate nonlinear inviscid damping and enhanced dissipation.
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Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D
In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $$\mathbb{T} \times [-1,1]$$, supplemented with Navier boundary conditions $$\omega|_{y = \pm 1} = 0$$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $$\nu$$. On the other hand, the nonzero modes are assumed size $$O(\nu^{\frac12})$$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the \textit{inviscid damping}, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework.
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- Award ID(s):
- 2108633
- PAR ID:
- 10597650
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- ISSN:
- 1432-0916
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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