# American Institute of Mathematical Sciences

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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