Singular integral system approach to regularity of 3D vortex patches

Abstract

The regularity of 3D vortex patches, which are weak solutions to the three dimensional incompressible Euler system with discontinuous initial data, is studied. The key idea is to introduce a singular integral system formulation to 3D Euler systems. Existence and regularity of solutions of the singular integral system are established in general settings. As a result, striated regularity of uniform Hölder type for weak solutions ω (x, t) to 3D Euler system is established for a short time. Furthermore, it is shown that this regularity persists as long as the quantity |ω(·, t)|L∞ is integrable. In particular, global regularity of solutions for axisymmetric flows is established.