Non-differentiable embedding of Lagrangian systems and partial differential equations

Jacky Cresson (1) , Isabelle Greff (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We develop the non-differentiable embedding theory of differential operators and Lagrangian systems using a new operator on non-differentiable functions. We then construct the corresponding calculus of variations and we derive the associated non-differentiable Euler-Lagrange equation, and apply this formalism to the study of PDEs. First, we extend the characteristics method to the non-differentiable case. We prove that non-differentiable characteristics for the Navier-Stokes equation correspond to extremals of an explicit non-differentiable Lagrangian system. Second, we prove that the solutions of the Schrödinger equation are non-differentiable extremals of the Newton's Lagrangian. © 2011 Elsevier Inc.

Domaines: Equations aux dérivées partielles [math.AP]

Jacky Cresson, Isabelle Greff. Non-differentiable embedding of Lagrangian systems and partial differential equations. Journal of Mathematical Analysis and Applications, 2011, 384 (2), pp.626-646. ⟨10.1016/j.jmaa.2011.06.008⟩. ⟨hal-00865004⟩