Improved quantitative unique continuation for complex-valued drift equations in the plane

Abstract

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form Δu+W⋅∇u=0 in R2 , where W=W1+iW2 with each Wj being real-valued. Under the assumptions that Wj∈Lqj for some q1∈[2,∞] , q2∈(2,∞] and that W2 exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.