Non-differentiable embedding of Lagrangian systems and partial differential equations

Jacky Cresson and Isabelle Greff

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Submission date: 24. Feb. 2010
Pages: 29
published in: Journal of mathematical analysis and applications, 384 (2011) 2, p. 626-646 
DOI number (of the published article): 10.1016/j.jmaa.2011.06.008
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Abstract:
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 PDE's. 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.