Global Existence of Weak Solutions to the Incompressible Axisymmetric Euler Equations Without Swirl

Jiu, Quansen; Liu, Jitao; Niu, Dongjuan

doi:10.1007/s00332-021-09687-4

Cited by 6 publications

(3 citation statements)

References 30 publications

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“…2 in [9] and [27] separately. However, for the incompressible axisymmetric Euler equations with swirl, the unique existence of global solution is still unknown, even for the exterior of a cylinder.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinder

Liu

2024

Nonlinearity

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In this paper, we study the initial boundary value problem and vanishing viscosity limit for incompressible axisymmetric Navier–Stokes equations with swirls in the exterior of a cylinder under Navier-slip boundary condition. In the first part, we prove the existence of a unique global solution with the axisymmetric initial data u 0 ν ∈ L σ 2 ( Ω ) and axisymmetric force f ∈ L 2 ( [ 0 , T ] ; L 2 ( Ω ) ) . This result improves the initial regularity condition on the global well-posedness result obtained by K Abe and G Seregin (2020 Proc. R. Soc. Edinburgh A 150 1671–98) and extends their boundary condition. In the second part, we make the first attempt to investigate the inviscid limit of unforced viscous axisymmetric flows with swirls and prove that the viscous axisymmetric flows with swirls converge to inviscid axisymmetric flows without swirls under the condition ‖ r u 0 θ ν ‖ L 2 ( Ω ) = O ( ν ) . Some new uniform estimates, independent of the viscosity, are obtained here. The second result can be thought as a follow-up work to the previous work by K Abe (2020 J. Math. Pures Appl. 137 1–32), where the inviscid limit for the same equations without swirls in an infinite cylinder was studied.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinder

Liu

2024

Nonlinearity

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“…We remark that the investigation is restricted to the orthogonal coordinate systems for specifying the solution. However, a more general curvilinear coordinates or even three-dimensional axisymmetric case [7] can be studied by a similar manner in [14].…”

Section: Concluding Remarkmentioning

confidence: 99%

Exact solutions of steady Euler equations in two dimension by functional separation

Kim

2024

J. Phys. A: Math. Theor.

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Exact solutions of two dimensional Euler equations for incompressible fluid flows arefound by the functional separation of variables. More specifically, two independent variablesare considered in the Cartesian coordinates, polar coordinates and also general orthogonalcurvilinear coordinates which include elliptic, parabolic and hyperbolic systems. Several newsolutions are obtained and other classical solutions are rediscovered.

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“…The motion of water can be described by the Euler equations, a mathematical model developed by Leonhard Euler to describe the flow of inviscid fluids. As nicely described in [5,17], in the twodimensional (2D) case the vorticity of the fluid is a scalar field satisfying a transport equation, while in the three-dimensional (3D) case the vorticity becomes a vector field satisfying the vorticity equation including the vortex stretching term. The vortex stretching term is the cause of the difference between 2D and 3D flows and causes difficulties for instance while proving global regularity.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

Erhardt¹,

Wahlén²,

Weber³

2022

Preprint

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We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinateinduced singularities and to cast the problem in the form "identity plus compact", which is amenable to Rabinowitz' global bifurcation theorem, while no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local and global solution curves, bifurcating from laminar flows with a flat surface, are constructed.

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Global Existence of Weak Solutions to the Incompressible Axisymmetric Euler Equations Without Swirl

Cited by 6 publications

References 30 publications

Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinder

Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinder

Exact solutions of steady Euler equations in two dimension by functional separation

Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

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