Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary Conditions
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2024
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| author | Bedrossian, Jacob He, Siming Iyer, Sameer Wang, Fei |
| author_facet | Bedrossian, Jacob He, Siming Iyer, Sameer Wang, Fei |
| contents | We consider the 2D, incompressible Navier-Stokes equations near the Couette flow, $ω^{(NS)} = 1 + εω$, set on the channel $\mathbb{T} \times [-1, 1]$, supplemented with Navier boundary conditions on the perturbation, $ω|_{y = \pm 1} = 0$. We are simultaneously interested in two asymptotic regimes that are classical in hydrodynamic stability: the long time, $t \rightarrow \infty$, stability of background shear flows, and the inviscid limit, $ν\rightarrow 0$ in the presence of boundaries. Given small ($ε\ll 1$, but independent of $ν$) Gevrey 2- datum, $ω_0^{(ν)}(x, y)$, that is supported away from the boundaries $y = \pm 1$, we prove the following results: \begin{align*} & \|ω^{(ν)}(t) - \frac{1}{2π}\int ω^{(ν)}(t) dx \|_{L^2} \lesssim εe^{-δν^{1/3} t}, & \text{(Enhanced Dissipation)} \\ & \langle t \rangle \|u_1^{(ν)}(t) - \frac{1}{2π} \int u_1^{(ν)}(t) dx\|_{L^2} + \langle t \rangle^2 \|u_2^{(ν)}(t)\|_{L^2} \lesssim εe^{-δν^{1/3} t}, & \text{(Inviscid Damping)} \\ &\| ω^{(ν)} - ω^{(0)} \|_{L^\infty} \lesssim ενt^{3+η}, \quad\quad t \lesssim ν^{-1/(3+η)} & \text{(Long-time Inviscid Limit)} \end{align*} This is the first nonlinear asymptotic stability result of its type, which combines three important physical phenomena at the nonlinear level: inviscid damping, enhanced dissipation, and long-time inviscid limit in the presence of boundaries. The techniques we develop represent a major departure from prior works on nonlinear inviscid damping as physical space techniques necessarily play a central role. In this paper, we focus on the primary nonlinear result, while tools for handling the linearized parabolic and elliptic equations are developed in our separate, companion work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19249 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary Conditions Bedrossian, Jacob He, Siming Iyer, Sameer Wang, Fei Analysis of PDEs We consider the 2D, incompressible Navier-Stokes equations near the Couette flow, $ω^{(NS)} = 1 + εω$, set on the channel $\mathbb{T} \times [-1, 1]$, supplemented with Navier boundary conditions on the perturbation, $ω|_{y = \pm 1} = 0$. We are simultaneously interested in two asymptotic regimes that are classical in hydrodynamic stability: the long time, $t \rightarrow \infty$, stability of background shear flows, and the inviscid limit, $ν\rightarrow 0$ in the presence of boundaries. Given small ($ε\ll 1$, but independent of $ν$) Gevrey 2- datum, $ω_0^{(ν)}(x, y)$, that is supported away from the boundaries $y = \pm 1$, we prove the following results: \begin{align*} & \|ω^{(ν)}(t) - \frac{1}{2π}\int ω^{(ν)}(t) dx \|_{L^2} \lesssim εe^{-δν^{1/3} t}, & \text{(Enhanced Dissipation)} \\ & \langle t \rangle \|u_1^{(ν)}(t) - \frac{1}{2π} \int u_1^{(ν)}(t) dx\|_{L^2} + \langle t \rangle^2 \|u_2^{(ν)}(t)\|_{L^2} \lesssim εe^{-δν^{1/3} t}, & \text{(Inviscid Damping)} \\ &\| ω^{(ν)} - ω^{(0)} \|_{L^\infty} \lesssim ενt^{3+η}, \quad\quad t \lesssim ν^{-1/(3+η)} & \text{(Long-time Inviscid Limit)} \end{align*} This is the first nonlinear asymptotic stability result of its type, which combines three important physical phenomena at the nonlinear level: inviscid damping, enhanced dissipation, and long-time inviscid limit in the presence of boundaries. The techniques we develop represent a major departure from prior works on nonlinear inviscid damping as physical space techniques necessarily play a central role. In this paper, we focus on the primary nonlinear result, while tools for handling the linearized parabolic and elliptic equations are developed in our separate, companion work. |
| title | Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary Conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.19249 |