×

Non-differentiable embedding of Lagrangian systems and partial differential equations. (English) Zbl 1251.49020

Summary: 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.

MSC:

49J52 Nonsmooth analysis
35Q93 PDEs in connection with control and optimization
35Q30 Navier-Stokes equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnold, V., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à lʼhydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16, 1, 319-361 (1966) · Zbl 0148.45301
[2] Arnold, V. I., Mathematical Methods of Classical Mechanics, Grad. Texts in Math., vol. 60 (1989), Springer-Verlag: Springer-Verlag New York, translated from Russian by K. Vogtmann and A. Weinstein
[3] Brenier, Yann, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc., 2, 2, 225-255 (1989) · Zbl 0697.76030
[4] Cresson, Jacky, Scale calculus and the Schrödinger equation, J. Math. Phys., 44, 11, 4907-4938 (2003) · Zbl 1062.39022
[5] Cresson, Jacky, Non-differentiable variational principles, J. Math. Anal. Appl., 307, 1, 48-64 (2005) · Zbl 1077.49033
[6] Jacky Cresson, Théories de plongement des systèmes dynamiques - un programme, 2005, 21 pp.; Jacky Cresson, Théories de plongement des systèmes dynamiques - un programme, 2005, 21 pp.
[7] Cresson, Jacky, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48, 3, 033504 (2007), 34 pp · Zbl 1137.37322
[8] Cresson, Jacky; Darses, Sébastien, Plongement stochastique des systèmes lagrangiens, C. R. Math. Acad. Sci. Paris, 342, 5, 333-336 (2006) · Zbl 1086.60042
[9] Cresson, Jacky; Darses, Sébastien, Stochastic embedding of dynamical systems, J. Math. Phys., 48, 7, 072703 (2007), 54 pp · Zbl 1144.81333
[10] Ebin, D.; Marsden, J., Group of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 40, 102-163 (1970) · Zbl 0211.57401
[11] Gérard, Patrick, Résultats récents sur les fluides parfaits incompressibles bidimensionnels (dʼaprès J.-Y. Chemin et J.-M. Delort), Séminaire Bourbaki, vol. 1991/1992. Séminaire Bourbaki, vol. 1991/1992, Astérisque, 206, 411-444 (1992), Exp. No. 757, 5 · Zbl 1154.76321
[12] Karatzas, I.; Schreve, S. E., Brownian Motion and Stochastic Calculus, Grad. Texts in Math., vol. 113 (1991), Springer
[13] Kac, Victor; Pokman, Cheung, Quantum Calculus, Universitext (2002), Springer-Verlag: Springer-Verlag New York · Zbl 0986.05001
[14] Nottale, Laurent, The theory of scale relativity, Internat. J. Modern Phys. A, 7, 20, 4899-4936 (1992) · Zbl 0954.81501
[15] Nottale, Laurent, The scale-relativity program, Superstrings, M, F, Słots Theory. Superstrings, M, F, Słots Theory, Chaos Solitons Fractals, 10, 2-3, 459-468 (1999) · Zbl 0997.81526
[16] Erwin Schrödinger, Physique quantique et représentation du monde, in: Collection Points-Sciences, Editions du Seuil, Paris, 1992.; Erwin Schrödinger, Physique quantique et représentation du monde, in: Collection Points-Sciences, Editions du Seuil, Paris, 1992.
[17] Tricot, Claude, Courbes et dimension fractale (1999), Springer-Verlag: Springer-Verlag Berlin, with a preface by Michel Mendès France · Zbl 0927.28004
[18] Villani, Cédric, Limites hydrodynamiques de lʼéquation de Boltzmann (dʼaprès C. Bardos, F. Golse, C.D. Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond), Séminaire Bourbaki, vol. 2000/2001. Séminaire Bourbaki, vol. 2000/2001, Astérisque, 282, 365-405 (2002), Exp. No. 893, ix · Zbl 1119.82037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.