Differential Equations with Constraints

Abstract: We study various differential equations subject to constraints. In the first part we study a partial differential equation, Burgers equation, subject to time-periodicity constraint. The forcing term is time-periodic and may be highly irregular. We prove an existence and uniqueness result which in a sense is optimal since we show that the operator corresponding to the Burgers equation is a diffeomorphism from a functional space to its dual. In the second part we study general ordinary differential equations subject to general constraints. We first describe precisely what the index is. Subsequently we investigate the particular case of linear ordinary differential equation and derive a new normal form. We show that it is characterized by defect indices and we show the relation with the Kronecker normal form.