The Centrally Symmetric V -states for Active Scalar Equations. Two-dimensional Euler with Cut-off

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, λ)}, x ∈ [−1, 1], λ ∈ (0, λ0) is found. For these functions, we have y(x, λ) ∈ C∞(−1, 1) and ∥y(x, λ) − |x|∥C[−1,1] → 0, λ → 0. The relation to the V -states observed numerically in [15, 3] is discussed.