Singularity formation in the incompressible Euler equation in finite and infinite time

Drivas, Theodore D.; Elgindi, Tarek M.

doi:10.48550/arxiv.2203.17221

Cited by 7 publications

(11 citation statements)

References 164 publications

(336 reference statements)

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“…Classical solutions to the 3D Euler equations could develop finite-time singularities ( [14,28]). Many more results in this direction can be found in a review paper by Drivas and Elgindi [25]. As a special consequence, perturbations governed by the Euler equations near the trivial solution are generally not stable.…”

Section: Introductionmentioning

confidence: 99%

Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

Lin¹,

Wu²,

Zhu³

2022

Preprint

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Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and damps electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in R 3 . The velocity equation in this system is the 3D Navier-Stokes equation with dissipation only in the x 1 -direction while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting H 3 (R 3 ). In addition, explicit decay rates in H 2 (R 3 ) are also obtained. When there is no presence of the magnetic field, the 3D anisotropic Navier-Stokes equation in R 3 is not well understood and the small data global well-posedness remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.

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Section: Introductionmentioning

confidence: 99%

Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

Lin¹,

Wu²,

Zhu³

2022

Preprint

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“…where u only depends on x, y, not z) can lead to infinite-in-time linear growth of ω; see Bardos-Titi [2, Remark 3.1] for example. See the excellent survey by Drivas-Elgindi [18] for more results on growth and singularity formation for 2D and 3D Euler equations.…”

Section: 3mentioning

confidence: 99%

Small scale formation for the 2D Boussinesq equation

Kiselev¹,

Park²,

Yao³

2022

Preprint

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We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying sup τ ∈[0,t] ∇ρ(τ ) L 2 t α for some α > 0. For the inviscid equation in the strip, we construct examples satisfying ω(t) L ∞ t 3 and sup τ ∈[0,t] ∇ρ(τ ) L ∞ t 2 during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth.

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“…See Sect. 5 of [14] for additional discussion. The second result, Theorem 6, considers initial data with a confined singular set and establishes the propagation of smoothness outside the domain of influence of the singularities (a set which can be confined to an expanding ball) for weak solutions arising by smooth approximations of the data.…”

Section: Singularities Determine the Solution In The Diperna-majda Classmentioning

confidence: 99%

Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids

Drivas

Elgindi

2022

Math. Ann.

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We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of $$\log \log _+(1/|x|)$$ log log + ( 1 / | x | ) vortices (and those that are slightly less singular) travel with the fluid in a nonlinear fashion, up to bounded perturbations. We also give stability results for weak Euler solutions away from their singular set.

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Singularity formation in the incompressible Euler equation in finite and infinite time

Cited by 7 publications

References 164 publications

Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field

Small scale formation for the 2D Boussinesq equation

Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids

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