Non-differentiable embedding of Lagrangian systems and partial differential equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:384

Article

Cresson, Jacky^1,2  Greff, Isabelle^1,3 

[1] Univ Pau & Pays Adour, Lab Math Appl Pau, F-64013 Pau, France

[2] Observ Paris, Inst Mecan Celeste & Calcul Ephemerides, F-75014 Paris, France

[3] Max Planck Inst Math Nat Wissensch Leipzig, D-04103 Leipzig, Germany

关键词: Non-differentiable calculus of variations;    Lagrangian systems;    Navier-Stokes equation;    Schrodinger equation;   

DOI  :  10.1016/j.jmaa.2011.06.008

来源: Elsevier

【 摘 要 】

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 Schrodinger equation are non-differentiable extremals of the Newton's Lagrangian. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free