Resolubilidade global de uma classe de campos vetoriais

Abstract

The topic under study is the global solvability of vector fields of the form L = ∂t+a(x)∂x on the 2−torus T(x,t), where a ∈ C∞(T1) is a real valued function. We consider the operator L acting on both spaces of functions and distributions. Using distribution theory we give necessary and sufficient conditions for the closedness of the range of L, ie, for L to be globally solvable. The most interesting case occurs when a vanishes somewhere but not everywhere; in this case, we show that a necessary and sufficient condition for L to be globally solvable is that each zero of a is of finite order. We also study the global solvability of operators of the form P = ∂t+∂x(a·), which are perturbations of L by a term of zero order. The operators P appear when we consider the transpose operator of L.