skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Stationary Structures Near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations
Abstract We study the behavior of solutions to the incompressible 2dEuler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier–Stokes problems. We exhibit a large family of new, non-trivial stationary states that are arbitrarily close to the Kolmogorov flow on the square torus$$\mathbb {T}^2$$ T 2 in analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the long-time behavior is not possible for solutions near the Kolmogorov flow on$$\mathbb {T}^2$$ T 2 . Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on$$\mathbb {T}^2$$ T 2 , and we also show a lack of correspondence between the linearized description of the set of steady states and its true nonlinear structure. Both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different. We show that the only stationary states near them must indeed be shears, even in relatively low regularity. In addition, we show that this behavior is mirrored closely in the related Navier–Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on$${\mathbb T}^2$$ T 2 .  more » « less
Award ID(s):
2043024
PAR ID:
10394858
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Archive for Rational Mechanics and Analysis
Volume:
247
Issue:
1
ISSN:
0003-9527
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, SciPost Physics)
    It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D\mathbb{Z}_2 ℤ 2 gauge theory with an emergent fermion, and point out that pumping chiralp+ip p + i p topological states gives rise to a\mathbb{Z}_{8} ℤ 8 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization ofT^3 T 3 (3-torus) andT^2 \rtimes_{C_2} S^1 T 2 ⋊ C 2 S 1 (2-torus bundle over the circle) respectively, and pumpingp+ip p + i p states. Our considerations also imply the possibility of a logicalT T gate by placing the code on\mathbb{RP}^3 ℝ ℙ 3 and pumping ap+ip p + i p topological state. 
    more » « less
  2. ; ; (, Archive for Rational Mechanics and Analysis)
    Abstract Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in$$\mathbb R^3$$ R 3 . The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the$$x_1$$ x 1 -direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting$$H^3({\mathbb {R}}^3)$$ H 3 ( R 3 ) . In addition, explicit decay rates in$$H^2({\mathbb {R}}^3)$$ H 2 ( R 3 ) are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in$$\mathbb R^3$$ R 3 remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid. 
    more » « less
  3. ; ; (, Stochastics and Partial Differential Equations: Analysis and Computations)
    Abstract This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on$$\mathbb {R}^+\times \mathbb {T}^2$$ R + × T 2 within the framework of paracontrolled calculus (Gubinelli et al. in Forum Math, 2015). The model is given by the equation:$$\begin{aligned} (\partial _t-\Delta ) u=F(u)\eta \end{aligned}$$ ( ∂ t - Δ ) u = F ( u ) η where$$\eta \in C^{-1-\kappa }$$ η ∈ C - 1 - κ with$$1/6>\kappa >0$$ 1 / 6 > κ > 0 , and$$F\in C_b^2(\mathbb {R})$$ F ∈ C b 2 ( R ) . Assume that$$\eta \in C^{-1-\kappa }$$ η ∈ C - 1 - κ and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work by Chandra et al. (A priori bounds for 2-d generalised Parabolic Anderson Model,,2024), to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument). 
    more » « less
  4. ; ; ; (, Advanced Nonlinear Studies)
    Abstract In this paper, we establish the existence of a bounded, linear extension operator T : L 2 , p ( E ) → L 2 , p ( R 2 ) $$T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$$when 1 <p< 2 andEis a finite subset of R 2 $${\mathbb{R}}^{2}$$contained in a line. 
    more » « less
  5. (, Communications in Mathematical Physics)
    Abstract Given a suitable solutionV(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$ u ( 0 , x ) ∈ V ( 0 , x ) + H - 1 ( R ) . Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ V ( 0 , x ) ∈ H 5 ( R / Z ) satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4) where$$V\equiv 0$$ V ≡ 0 . In that setting, it is known that$$H^{-1}(\mathbb {R})$$ H - 1 ( R ) is sharp in the class of$$H^s(\mathbb {R})$$ H s ( R ) spaces. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email