Flexibility and rigidity in steady fluid motion

Abstract

Flexibility and rigidity properties of steady (time-independent) solutions of the Euler, Boussinesq and Magnetohydrostatic equations are investigated. Specifically, certain Liouville-type theorems are established which show that suitable steady solutions with no stagnation points occupying a two-dimensional periodic channel, or axisymmetric solutions in (hollowed out) cylinder, must have certain structural symmetries. It is additionally shown that such solutions can be deformed to occupy domains which are themselves small perturbations of the base domain. As application of the general scheme, Arnol'd stable solutions are shown to be structurally stable.

Boundary value problems for two dimensional steady incompressible fluids

“…Employing the so called Clebsch variables the velocity is written as v = ∇f × ∇g to derive an elliptic non-linear system and perform a Nash-Moser scheme to solve it. Furthermore, ideas closely related to the method of Grad-Shafranov have been recently applied to study rigidity and flexibility properties solutions of the steady Euler equation in [16,17,7].…”

On quasisymmetric plasma equilibria sustained by small force

“…We start by giving an outline of the arguments used to establish the main theorem. All details can be found in [6].…”

“…However, if we can arrange for the metric g to be sufficiently close to the Euclidean metric δ, then B g will satisfy the usual MHS equations curl B g ×B g −∇P = 0 up to a small error. Our approach will be to solve the generalized Grad-Shafranov equation (1.20) by deforming an appropriate solution ψ 0 of the axisymmetric Grad-Shafranov equation (1.5), using the methods from [6]. In particular, we seek a diffeomorphism γ : D 0 → D and requiring that ψ = ψ 0 • γ −1 .…”

“…Specifically, in our theorem, we treat ξ as a fixed vector field sufficiently close to ξ 0 and we made the somewhat arbitrary choice that the map γ should be volume preserving. The results in [6] actually allow one to construct the map γ so that det ∇γ := ρ is any given function, sufficiently close to one; in fact by iterating that result, one can additionally achieve that det ∇γ = X(φ, η, ∂φ, ∂η, ∂∂ s φ, ∂∂ s η) for a suitable nonlinearity X sufficiently close to one when φ, η = 0. Using this freedom, it is possible to show that, under some (possibly restrictive and undesirable) assumptions on the field ξ, the Jacobian ρ can be used to achieve exact quasisymmetry on a slice of the torus (namely on the cross-section D).…”

Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

“…This type of rigidity question has been very lately understood for different equations and different settings such as in the papers by Koch-Nadirashvili-Sverak [20] for Navier-Stokes, Hamel-Nadirashvili [17,16,18] for the 2D Euler equation on a strip, punctured disk or the full plane, Gómez-Serrano-Park-Shi-Yao [14] for the 2D Euler and modified SQG in the full plane and Constantin-Drivas-Ginsberg [8] for the 2D and 3D Euler, as well as the 2D Boussinesq and the 3D Magnetohydrostatic (MHS) equations.…”