August  2005, 5(3): 513-528. doi: 10.3934/dcdsb.2005.5.513

Physical solutions of the Hamilton-Jacobi equation

1. 

École Normale Supérieure, U.M.P.A., 46, allée d'Italie, 69364 Lyon Cedex 07, France

2. 

CIMAT, A.P. 402, 3600, Guanajuato. Gto, Mexico

3. 

IIMAS, UNAM, Cd. Universitaria, México, D.F. 04510, Mexico

4. 

I. de Matemáticas, UNAM, Cd. Universitaria, México, D.F. 04510, Mexico

Received  January 2004 Revised  August 2004 Published  May 2005

We consider a Lagrangian system on the d-dimensional torus, and the associated Hamilton-Jacobi equation. Assuming that the Aubry set of the system consists in a finite number of hyperbolic periodic orbits of the Euler-Lagrange flow, we study the vanishing-viscosity limit, from the viscous equation to the inviscid problem. Under suitable assumptions, we show that solutions of the viscous Hamilton-Jacobi equation converge to a unique solution of the inviscid problem.
Citation: Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
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