The Euler Equations in Planar Domains with Corners

Lacave, Christophe; Zlatoš, Andrej

doi:10.1007/s00205-019-01384-7

Cited by 14 publications

(28 citation statements)

References 29 publications

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“…Lacave first proved in [14] that if ∂Ω is C 1,1 except at finitely many corners that are all exact sectors with angles greater than π 2 , and ω 0 has a constant sign and is constant near ∂Ω, then ω will indeed remain constant near ∂Ω forever and weak solutions are unique. Then, together with Zlatoš, they showed the same result when ∂Ω is C 1,1 except at finitely many corners of arbitrary angles from (0, π) that do not need to be exact sectors, and without the sign restriction on ω 0 [17]. In both works, Euler particle trajectories for bounded solutions (general ones in [17] and with a constant sign in [14]) were shown to remain in Ω for all time (again approaching ∂Ω no faster than double-exponentially), and in [17] this was even proved to hold when ∂Ω is only C 1,γ (for some γ > 0) except at finitely many corners with angles from (0, π).…”

Section: Prior Existence and Uniqueness Resultsmentioning

confidence: 68%

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Euler Equations on General Planar Domains

Han¹,

Zlatoš²

2020

Preprint

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We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than π and, in particular, is satisfied by all convex domains. The main ingredient in our approach is showing that constancy of the vorticity near the boundary is preserved for all time because Euler particle trajectories on these domains, even for general bounded solutions, cannot reach the boundary in finite time. We then use this to show that no vorticity can be created by the boundary of such possibly singular domains for general bounded solutions. We also show that our condition is essentially sharp in this sense by constructing domains that come arbitrarily close to satisfying it, and on which particle trajectories can reach the boundary in finite time. In addition, when the condition is satisfied, we find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary.

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Section: Prior Existence and Uniqueness Resultsmentioning

confidence: 68%

“…Moreover, [17] also constructed examples of domains smooth everywhere except at a single corner with an arbitrary angle from (π, 2π) where Euler particle trajectories can reach ∂Ω in finite time, using an idea of Kiselev and Zlatoš [13].…”

Section: Prior Existence and Uniqueness Resultsmentioning

confidence: 99%

Euler Equations on General Planar Domains

Han¹,

Zlatoš²

2020

Preprint

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“…In addition, it is well-known that Yudovich [57] obtained the existence and uniqueness of global weak solutions if the initial vorticity ω 0 lies in L 1 ∩ L ∞ for the domain D = R 2 (see [47] for the bounded domain case). One can refer to [3,4,6,11,18,19,20,26,39,40,42,56,58] and the references therein for other related interesting and important aspects concerning with the two-dimensional incompressible Euler equations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global well-posedness to the two-dimensional incompressible vorticity equation in the half plane

Jiu¹,

Li²,

Zhang³

2021

Preprint

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This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity ω0 ∈ W k,p (R 2 + ) with k ≥ 3 and 1 < p < 2, it is shown that the two-dimensional incompressible vorticity equation admits a unique solution ω ∈ C([0, T ]; W k,p (R 2 + )) for any T > 0. An elementary and self-contained proof is presented and delicate estimates of the velocity and its derivatives are obtained in this paper. It should be emphasized that the uniform estimate ondτ is required to complete the global regularity of the solution. To do that, the double exponential growth in time of the gradient of the vorticity in the half plane is established and applied. This is different from the proof of global well-posedness of the Euler velocity equations in the Sobolev spaces, in which a Kato-type or logarithmic-type estimate of the gradient of the velocity is enough to close the energy estimates.

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“…For more details about renormalized solutions, we refer, for instance, to [16,Prop. 4.1] or [17,Sect. 3.2].…”

Section: Estimates With Precise Rate For Stronger Solutionsmentioning

confidence: 99%

A note on the regularity of the holes for permeability property through a perforated domain for the 2D Euler equations

Lacave

Wang

2019

Sci. China Math.

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For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes dε is equal or much larger than the size of the holes ε. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when (ε, dε) → (0, 0). Smaller distance was avoided for mathematical reasons and for theses large distances, the geometry of the holes does not affect -or few-the asymptotic result. Very recently, it was shown for the 2D-Euler equations that a porous medium is non-negligible only for inter-holes distance much smaller than the size of the holes. For this result, the regularity of holes boundary plays a crucial role, and the permeability criterium depends on the geometry of the lateral boundary. In this paper, we relax slightly the regularity condition, allowing a corner, and we note that a line of irregular obstacles cannot slow down a perfect fluid in any regime such that ε ln dε → 0.Date: April 12, 2019.

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The Euler Equations in Planar Domains with Corners

Cited by 14 publications

References 29 publications

Euler Equations on General Planar Domains

Euler Equations on General Planar Domains

Global well-posedness to the two-dimensional incompressible vorticity equation in the half plane

A note on the regularity of the holes for permeability property through a perforated domain for the 2D Euler equations

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