An official website of the United States government
Abstract While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in with afixedspeed of rotation. We show that for any , axisymmetric initial data of sufficiently small size ε lead to solutions that exist for a long time at least and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.
more »
« less
Global axisymmetric Euler flows with rotation
Abstract We construct a class of global, dynamical solutions to the 3 d Euler equations near the stationary state given by uniform “rigid body” rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is crucial for the construction of global Euler flows.
more »
« less
- PAR ID:
- 10410324
- Date Published:
- Journal Name:
- Inventiones mathematicae
- Volume:
- 231
- Issue:
- 1
- ISSN:
- 0020-9910
- Page Range / eLocation ID:
- 169 to 262
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.more » « less
-
This article is concerned with one dimensional dispersive flows with cubic non- linearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small initial data yields global, scattering solutions. Then this conjecture was proved in the case of a Schr¨odinger dispersion relation. In terms of scattering, our global solutions were proved to satisfy both global L6 Strichartz estimates and bilinear L2 bounds. Notably, no localization assumption is made on the initial data. In this article we consider the focusing scenario. There potentially one may have small solitons, so one cannot hope to have global scattering solutions in general. Instead, we look for long time solutions, and ask what is the time-scale on which the solutions exist and satisfy good dispersive estimates. Our main result, which also applies in the case of the Schr¨odinger dispersion relation, asserts that for initial data of size ϵ, the solutions exist on the time-scale ϵ^{-8}, and satisfy the desired L^6 Strichartz estimates and bilinear L^2 bounds on the time-scale ϵ^{−6}. To the best of our knowledge, this is the first result to reach such a threshold.more » « less
-
null (Ed.)Abstract Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.more » « less
-
Shatah, Jalal (Ed.)We prove asymptotic stability of point vortex solutions to the full Euler equation in two dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as $$t\to\infty$$ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping. © 2021 Wiley Periodicals LLC.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00222-022-01145-6
