Global Existence of Weak Solutions to the Incompressible Axisymmetric Euler Equations Without Swirl

Abstract

In this paper, we consider solutions to the incompressible axisymmetric Euler equations without swirl. The main result is to prove the global existence of weak solutions if the initial vorticity $$w_0^\theta $$ satisfies that $$\frac{w_0^\theta }{r}\in L^1\cap L^p({\mathbb {R}}^3)$$ for some $$p>1$$ . It is not required that the initial energy is finite, that is, the initial velocity $$u_0$$ belongs to $$L^2({\mathbb {R}}^3)$$ here. We construct the approximate solutions by regularizing the initial data and show that the concentrations of energy do not occur in this case. The key ingredient in the proof lies in establishing the $$L_{\mathrm{loc}}^{2+\alpha }({\mathbb {R}}^3)$$ estimates of velocity fields for some $$\alpha >0$$ , which is new to the best of our knowledge.