Steady Vortex Patches on Flat Torus with a Constant Background Vorticity

We construct a series of vortex patch solutions in a doubly-periodic rectangular domain (flat torus), which is accomplished by studying the contour dynamic equation for patch boundaries. We will illustrate our key idea by discussing single-layered patches as the most fundamental configuration and then investigate the general construction for N patches near a point vortex equilibrium. Unlike the case of bounded domains in $$\mathbb {R}^2$$ , a constant background vorticity will arise from the compact nature of a flat torus, and the two-dimensional translational invariance will cause trouble in determining the patch locations. To overcome these two difficulties, we will add additional terms for background vorticity and introduce a centralized condition for the location vector. By utilizing the regularity difference of terms in contour dynamic equations, we also obtain the $$C^\infty $$ regularity and convexity of the boundary curves.