On self-similar solutions to the incompressible Euler equations

Bressan, Alberto; Murray, Ryan

doi:10.1016/j.jde.2020.04.005

Cited by 34 publications

(35 citation statements)

References 16 publications

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“…Uniqueness is known only for p = ∞ and was proved by Yudovich [43]. The uniqueness for unbounded vorticities is an old and outstanding open problem and only very recently some partial progress towards nonuniqueness has been achieved; see [7,8,34,40,41].…”

Section: Introductionmentioning

confidence: 99%

Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

Ciampa

Crippa

Spirito

2021

Arch Rational Mech Anal

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In this paper we prove the uniform-in-time $$L^p$$ L p convergence in the inviscid limit of a family $$\omega ^\nu $$ ω ν of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ ω of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $$\omega ^{\nu }$$ ω ν to $$\omega $$ ω in $$L^p$$ L p . Finally, we show that solutions of the Euler equations with $$L^p$$ L p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

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Section: Introductionmentioning

confidence: 99%

Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

Ciampa

Crippa

Spirito

2021

Arch Rational Mech Anal

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“…It is well known that, for general initial data u0∈L2, weak solutions are not unique, see [14] and references therein. Moreover, uniqueness has not been established even under additional restrictions on integrability of vorticity, with the exception of ω0=curl u0∈L∞, see [15], and slightly weaker spaces close to L∞, see [16], and there is some indication of non-uniqueness of weak solutions if, for any 1≤p<normal∞, ω0∈Lp, see [17]. It is natural, therefore, to focus on special weak solutions, such as those obtained as limits of solutions of the more physically realistic Navier–Stokes equations.…”

Section: Preliminariesmentioning

confidence: 99%

Energy balance for forced two-dimensional incompressible ideal fluid flow

Filho¹,

Lopes²

2022

Phil. Trans. R. Soc. A.

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Cheskidov et al. (2016 Commun. Math. Phys. 348 , 129–143. ( doi:10.1007/s00220-016-2730-8 )) proved that physically realizable weak solutions of the incompressible two-dimensional Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothesis was boundedness of the initial vorticity in L p , p > 1 . In this work, we extend their result, by adding forcing to the flow. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.

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“…These three components are then patched together by suitable matching conditions. A detailed analysis of the solution to (1.10) in a neighborhood of infinity and near the spirals' centers will appear in the companion paper [5], relying on the approach developed in [15,16,17]. In the present paper we focus on the derivation of a posteriori error estimates for a numerically computed solution to (1.10), on a bounded domain D ⊂ R 2 with smooth boundary ∂D.…”

Section: Remark 11mentioning

confidence: 99%

A posteriori error estimates for self-similar solutions to the Euler equations

Bressan¹,

Shen²

2021

Discrete &Amp; Continuous Dynamical Systems - A

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The main goal of this paper is to analyze a family of "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.

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On self-similar solutions to the incompressible Euler equations

Cited by 34 publications

References 16 publications

Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

Energy balance for forced two-dimensional incompressible ideal fluid flow

A posteriori error estimates for self-similar solutions to the Euler equations

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