Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow

Xu, Xiaoqian

doi:10.1016/j.jmaa.2016.02.066

Cited by 33 publications

(23 citation statements)

References 15 publications

(29 reference statements)

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“…After that Elgindi & Masmoudi (2014) and Bourgain & Li (2015b) produced similar results for the C 1 case (also the C m case). On the other hand, (also refer to Itoh, Miura & Yoneda (2014), Kiselev & Zlatos (2015) and Xu (2014) for related topics) showed a 2D Euler flow with a saddle point (with hyperbolic flow configuration) in a disk for which the gradient of vorticity exhibited a double-exponential growth in time for all times. Their estimate is known to be sharp, namely, the double-exponential growth is the fastest possible growth rate.…”

Section: Introductionmentioning

confidence: 99%

A local analysis of the axisymmetric Navier–Stokes flow near a saddle point and no-slip flat boundary

2016

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Tornadoes are one type of violent flow phenomenon and occur in many places in the world. There are many research methods that aim to reduce the loss of human lives and material damage caused by tornadoes. One effective method is numerical simulation such as that in Ishihara et al. (J. Wind Engng Ind. Aerodyn., vol. 99, 2011, pp. 239-248). The swirling structure of the Navier-Stokes flow is significant for both the mathematical analysis and numerical simulations of tornadoes. In this paper, we try to clarify the swirling structure. More precisely, we performed numerical computations on axisymmetric Navier-Stokes flows with a no-slip flat boundary. We compared a hyperbolic flow with swirl and one without swirl, and observed that the following phenomenon occurs only in the swirl case: the distance between the point with the maximum magnitude of velocity |v| and the z-axis changed drastically at a specific time (which we call the turning point). Besides, an 'increasing velocity phenomenon' occurred near the boundary, and the maximum value of |v| was obtained near the axis of symmetry and the boundary when the time was close to the turning point in the swirl case.

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Section: Introductionmentioning

confidence: 99%

A local analysis of the axisymmetric Navier–Stokes flow near a saddle point and no-slip flat boundary

2016

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“…Proposition 14 in [16] for more details. The second half it is not difficult to verify and has been proved in [18]. If y ∈ T (r), the reflection S(y) ≡ y = 2e(y) − y is well defined and C k−1 regular in all T (r).…”

Section: (23)mentioning

confidence: 94%

“…Since the argument is largely parallel to [11], we refer to it for the details. In [18], the following estimate is proved. Let D ∈ C 3 , and fix any z ∈ ∂D.…”

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confidence: 90%

“…The argument is fairly direct and uses estimates on y(y) and local representation of ∂D as a graph of a function f ∈ C 3 ; the idea is that when x ∈ ∂D ∩ B r/2 (z) and y ∈ D ∩ B r (z) then |x − y| and |x − y| are very close. Note that even though the statement of Proposition 1 in [18] makes an assumption that D has a symmetry axis and z belongs to it, this assumption is never used in the proof (it is needed for later applications in [18]). Now it is straightforward to extend the estimate (6.1) to the function (6.2) ϕ(x) := 1 2πˆT (r)∩D log |x − y| |x − y| ω(y) dy, yielding (6.3) ϕ C 2,α (∂D∩B r/2 (z)) ≤ C ω L ∞ .…”

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confidence: 99%

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Global regularity and fast small-scale formation for Euler patch equation in a smooth domain

Kiselev

2019

Communications in Partial Differential Equations

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It is well known that the Euler vortex patch in R 2 will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this paper, we study Euler vortex patches in a general smooth bounded domain. We prove global in time regularity by providing an upper bound on the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.

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“…We will consider here (1.1) on a disc, as in [5], although our results easily extend to general smooth two-dimensional domains with a symmetry axis via [15]. For convenience we will work with the unit disc D := B 1 (e 2 ) centered at e 2 = (0, 1), and we will denote its right/left halves by D ± := D ∩ (R ± × R).…”

Section: Introductionmentioning

confidence: 99%

On the Rate of Merging of Vorticity Level Sets for the 2D Euler Equations

Zlatoš

2017

J Nonlinear Sci

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We show that two distinct level sets of the vorticity of a solution to the 2D Euler equations on a disc can approach each other along a curve at an arbitrarily large exponential rate.on D × (0, ∞), with initial data ω(·, 0) = ω 0 .(1.2) We will consider the case when the vorticity ω = −∇ × u (which will be more convenient for us than the more standard ω = ∇ × u) is bounded, that is, ω 0 ∈ L ∞ (D). The customary no-flow boundary condition u · n = 0 on ∂D × (0, ∞), with n the unit outer normal vector, then yields the Biot-Savart lawIt has been known since the works of Hölder [4] and Wolibner [11] that solutions to the Euler equations on smooth two-dimensional domains remain globally regular, and that ∇ω(·, t) L ∞ cannot grow faster than double-exponentially as t → ∞ (although this bound seems to have first explicitly appeared in [12]). That is, for eachCt for each t ≥ 0. Whether the double-exponential rate of growth is attainable had been a long-standing open problem. The first examples of smooth solutions for which the vorticity gradient grows without bound as t → ∞ were constructed by Yudovich [13, 14]. Later Nadirashvili [10] and Denisov [1] provided examples with at least linear and superlinear growth, respectively. The period of relatively slow progress in this direction was ended by a striking recent result of 1

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Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow

Abstract: Abstract. We construct an initial data for the two-dimensional Euler equation in a bounded smooth symmetric domain such that the gradient of vorticity in L ∞ grows as a double exponential in time for all time. Our construction is based on the recent result by Kiselev andŠverák [9].

Cited by 33 publications

References 15 publications

A local analysis of the axisymmetric Navier–Stokes flow near a saddle point and no-slip flat boundary

A local analysis of the axisymmetric Navier–Stokes flow near a saddle point and no-slip flat boundary

Global regularity and fast small-scale formation for Euler patch equation in a smooth domain

On the Rate of Merging of Vorticity Level Sets for the 2D Euler Equations

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