Abstract
We consider the family of active scalar equations on the plane and study the dynamics of two centrally symmetric patches. We focus on the two-dimensional Euler equation written in the vorticity form and consider its truncated version. For this model, a non-linear and non-local evolution equation is studied and a family of stationary solutions \({\{y(x,\lambda)\}, x\in [-1,1], \lambda\in (0,\lambda_0)}\) is found. For these functions, we have \({y(x,\lambda)\in C^\infty(-1,1)}\) and \({\|y(x,\lambda)-|x|\|_{C[-1,1]}\to 0, \,\,\lambda\to 0}\) . The relation to the V-states observed numerically in Wu et al. (J Comput Phys 53:1–42, 1984), Cerretelli and Williamson (J Fluid Mech 493:219–229, 2003) is discussed.
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Denisov, S.A. The Centrally Symmetric V-States for Active Scalar Equations. Two-Dimensional Euler with Cut-Off. Commun. Math. Phys. 337, 955–1009 (2015). https://doi.org/10.1007/s00220-015-2298-8
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DOI: https://doi.org/10.1007/s00220-015-2298-8