Global regularity for vortex patches. (English) Zbl 0771.76014

Bertozzi, A. L.; Constantin, P.

Commun. Math. Phys. 152, No. 1, 19-28 (1993).

A vortex patch is a simply connected, open and bounded domain \(D\) of \(\mathbb{R}^ 2\) that moves with velocity \(v\) that is derived from the stream function \(\Psi(x)=(2\pi)^{-1}\omega_ 0\int_ D\log| x-y| dy\), where \(\omega_ 0\) is a constant. The paper seeks a scalar function \(\phi(x,t)\) governed by the equation \(\bigl({\partial\over \partial t}+v\cdot\text{grad}\bigr)\phi=0\), \(\phi(x,0)=\phi_ 0(x)\), where \(v\) is given the Biot-Savart formula \(2\pi\;v(x,t)=\omega_ 0\int_ D\nabla^ \perp_ x\log| x-y| dy\), with \(D=D(t)=\bigl\{x\in\mathbb{R}^ 2\mid\phi(x)>0\bigr\}\), and proves the main result that the boundary of a vortex patch remains smooth for all time if it is initially smooth.

Reviewer: S.K.Lakshmana Rao (Bangalore)

MSC:

76B47	Vortex flows for incompressible inviscid fluids
35Q35	PDEs in connection with fluid mechanics

Keywords:

stream function; Biot-Savart formula

References:

[1]	Alinhac, S.: Remarques sur l’instabilité du problème des poches de tourbillon. Prépublication de l’Université d’Orsay, à paraitre dans J. Funct. Anal. 1989.
[2]	Bertozzi, A.: Existence, Uniqueness, and a Characterization of Solutions to the Contour Dynamics Equation. PhD thesis, Princeton University 1991
[3]	Buttke, T.F.: The observation of singularities in the boundary of patches of constant vorticity. Phys. Fluids A1, 1283–1285 (1989)
[4]	Chemin, J.-Y.: Persistance de structures geometriques dans les fluides incompressibles bidimensionnels. Preprint 1991
[5]	Constantin, P., Lax, P.D., Majda, A.: A simple one dimensional model for the three dimensional vorticity equation. Commun. Pure Applied Math.38, 715–724 (1985) · Zbl 0615.76029
[6]	Constantin, P. and Titi, E.S.: On the evolution of nearly circular vortex patches. Commun. Math. Phys.119, 177–198 (1988) · Zbl 0673.76025
[7]	Dritschel, D.G., McIntyre, M.E.: Does contour dynamics go singular? Phys. Fluids A,2(5) 748–753 (1990)
[8]	Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Commun. Pure Appl. Math.39, 5187–5220 (1986) · Zbl 0595.76021
[9]	Torchinsky, A.: Real Variable Methods in Harmonic Analysis. New York: Academic Press 1986 · Zbl 0621.42001
[10]	Yudovich, V.I.: Non-stationary flow of an ideal incompressible liquid. Zh. Vych. Mat.3, 1032–1066 (1963) (in Russian)
[11]	Zabusky, N., Hughes, M.H., Roberts, K.V.: Contour dynamics for the Euler equations in two dimensions. J. Comp. Phys. 96–106 (1979) · Zbl 0405.76014