Uniformly Rotating Smooth Solutions for the Incompressible 2D Euler Equations

Castro, Ángel; Córdoba, Diego; Gómez-Serrano, Javier

doi:10.1007/s00205-018-1288-3

Cited by 46 publications

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“…in Rankine vortices) and non-normal algebraic instabilities. See also the recent nonlinear counter examples to inviscid damping around a vortex constructed in [21] without monotonicity.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations

2019

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Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the 2D Euler equations. We consider the incompressible 2D Euler equations linearized around a radially symmetric, strictly monotone decreasing vorticity distribution. For sufficiently regular data, we prove the inviscid damping of the θ-dependent radial and angular velocity fields with the optimal rates u r (t) t −1 and u θ (t) t −2 in the appropriate radially weighted L 2 spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as t → ∞, a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the θ-dependent angular Fourier modes in the vorticity are ejected from the origin as t → ∞, resulting in faster inviscid damping rates than those possible with passive scalar evolution. This non-local effect is called vorticity depletion. Our work appears to be the first to find vorticity depletion relevant for the dynamics of vortices.Remark 1.2. By density we can extend the results to cover any ω in k ∈ L 2 (satisfying (1.7)) for which the norms appearing on the right-hand sides above are finite.Remark 1.3. The L 2 norms we are using in Theorem 1.1, namely (1.10), are natural in light of (1.8) and are wellsuited for studying vorticity depletion. However, these norms are quite strong at the origin (and infinity). Note that≈ k −1 (as opposed to k −2 as one might expect), which explains why some of the powers of k in Theorem 1.1 are slightly higher than might be at first expected. Similarly, note that F k contains information about the second term in the expansion (1.8).Remark 1.4. The correct analogue of propagation of regularity for mixing problems is the regularity of: e iktu(r) ω k (t, r), the object which measures the difference between the passive scalar and full linearized (or nonlinear) dynamics. Regularity of this object is often studied in dispersive equations and it is sometimes called the 'profile'; see e.g.[14] for more discussions (note that regularity of this type was called 'gliding regularity' in [52]). Understanding regularity of the profile plays a major role in all of the works involving nonlinear inviscid/Landau damping [5-8, 10, 14, 15, 19, 52] including those which obtain results in Sobolev spaces [9,31]. Theorem 1.1 deduces higher regularity of the vorticity profile than is necessary to prove the (1.11), at least for k ≥ 3. However, as regularity plays a crucial role in the nonlinear theory, it seems appropriate to study it as carefully as possible in the linear problem. This goal has motivated many of the primary aspects of our approach.Remark 1.5. Because in this work we were only concerned with obtaining finite Sobolev regularity of the vorticity profile, we have not carefully quantified how the constants in (1.13) depend on n. This is...

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“…in Rankine vortices) and non-normal algebraic instabilities. See also the recent nonlinear counter examples to inviscid damping around a vortex constructed in [21] without monotonicity.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations

2019

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“…The work of e.g. [30] moreover construct a variety of interesting smooth, rigidly rotating solutions. Any solutions of these types clearly do not involve vorticity mixing as t → ∞, and so we can expect a large set of solutions which do not display any mixing.…”

Section: Open Problemsmentioning

confidence: 99%

Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions

Bedrossian¹,

Germain

Masmoudi

2018

Bull. Amer. Math. Soc.

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We review works on the asymptotic stability of the Couette flow. The majority of the paper is aimed towards a wide range of applied mathematicians with an additional section aimed towards experts in the mathematical analysis of PDEs. Contents Hydrodynamic stability at high Reynolds number1 2. Linear dynamics in dimension d = 2 8 3. Nonlinear dynamics in dimension d = 2 11 4. Linear dynamics in dimension d = 3 1 As remarked above, in order to get non-trivial equilibria, we usually need to impose boundary conditions or an external force field. However, let us ignore this for the moment, as it is not directly relevant to our discussions. STABILITY OF THE COUETTE FLOW AT HIGH REYNOLDS NUMBERS IN 2D AND 3D3 simply by dropping the quadratic nonlinearity in (1.1):( 1.3)The simplest notion of linear stability is spectral stability:Many classical theories of hydrodynamic stability are focused on studies of spectral stability (see [122] and the references therein). In the context of shear flows at high Reynolds number, the most famous results are Rayleigh's inflection point theorem [96,40] and its refinement due to Fjørtoft [48], regarding the stability of shear flows in the 2D Euler equations (ν = 0). These results (unfortunately) extend to planar 3D shear flows via Squire's theorem [107,39,40]. Squire's theorem for inviscid flows states: If the 3D Euler equations linearized around the shear flow u E = (f (y), 0, 0) have unstable eigenvalues, then so do the 2D Euler equations linearized around u E = (f (y), 0) (the converse implication being obvious). Therefore, for any shear flow of this type, spectral stability in 2D implies spectral stability in 3D. This is 'unfortunate' because it gives the false impression that most of the interesting aspects of hydrodynamic stability can be found in the 2D equations. As we will see, hydrodynamic stability in 2D and 3D, at high Reynolds numbers, are quite distinct.Spectral stability is not the only relevant definition of linear stability. In particular, we have the following distinct definition, which was suggested as more natural in fluid mechanics by Kelvin [68], Orr [92], and later Case [29]and Dikii [38].Definition 1.2 (Lyapunov linear stability). Given two norms X and Y (often assumed the same), the equilibrium u E is called linearly stable (from X to Y ) if

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“…Such solutions have only been constructed very recently by Castro-Córdoba-Gómez-Serrano [6] who found a smooth 3-fold solution that rotates uniformly (both in time and space) by perturbing from a smooth annular profile. See also [8]. We remark that Dritschel [21] had constructed nontrivial global rotating solutions with C 1/2 regularity.…”

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confidence: 92%

Global Solutions for the Generalized SQG Patch Equation

Córdoba

Gómez-Serrano

Ionescu

2019

Arch Rational Mech Anal

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We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter α ∈ (1, 2). The cases α = 0 and α = 1 correspond to 2d Euler and SQG respectively, and our choice of the parameter α results in a velocity more singular than in the SQG case.Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the patch setting are the so-called V-states, which are uniformly rotating and periodic in time solutions.that started arbitrarily small but grew arbitrarily large in finite time. More recently, Kiselev-Ryzhik-Yao-Zlatos [36] introduced a new gSQG-patch model, with a fixed boundary, and proved the formation of finite time singularities in this model for certain patches that touch the boundary at all times. At this point it is unclear whether such a scenario can lead to singularities in the classical gSQG models considered here.1.1.3. Global regularity and rotating solutions. The construction of nontrivial global solutions for the gSQG equations is also a challenging problem, both in the smooth and in the patch case. In fact, the only non-stationary global solutions that are known in the patch setting are very special rotating solutions. These solutions, which are periodic in time and evolve by rotating with constant angular velocity around their center of mass, are known as V-states.Deem-Zabusky [16] were the first to discover the V-states numerically in the patch case. Other authors have later improved the methods and numerically computed larger classes (see for example [49,22,39,44]).Hassainia-Hmidi [29] have rigorously proved the existence of V-states in the case 0 < α < 1. They were able to show the existence of convex V-states with C k boundary regularity. In [5], Castro-Córdoba-Gómez-Serrano were able to prove existence and C ∞ regularity of convex global rotating solutions for the remaining open cases: α ∈ [1, 2) for the existence, α ∈ (0, 2) for the regularity. This boundary regularity was subsequently improved to analytic in [7].The problem of constructing rotating periodic in time solutions is more challenging in the smooth case (1.1). Such solutions have only been constructed very recently by Castro-Córdoba-Gómez-Serrano [6] who found a smooth 3-fold solution that rotates uniformly (both in time and space) by perturbing from a smooth annular profile. See also [8]. We remark that Dritschel [21] had constructed nontrivial global rotating solutions with C 1/2 regularity.

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Uniformly Rotating Smooth Solutions for the Incompressible 2D Euler Equations

Abstract: In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equation and rotate uniformly in time and space.

Cited by 46 publications

References 28 publications

Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations

Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations

Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions

Global Solutions for the Generalized SQG Patch Equation

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