Global solvability for a class of vector field

Author(s)

Rafael Borro Gonzalez

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2011-02-25

Examining board members

Adalberto Panobianco Bergamasco; José Ruidival Soares dos Santos Filho; Evandro Raimundo da Silva

• Advisor: Adalberto Panobianco Bergamasco

Abstract

The topic under study is the global solvability of vector fields of the form L = \'\\PARTIAL IND. t\'+a(x)\'\\PARTIAL IND.x\' on the 2-torus \'T POT. 2 IND. (x;t)\' ; where a \'IT BELONGS\' \'C POT. INFINITY\' (\'T POT. 1\') 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 = \'\\ PARTIAL IND. t\'+\'\\ PARTIAL IND. x(\'a AST .\' which are perturbations of L by a term of zero order. The operators P appear when we consider the transpose operator of L (AU)