Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in 2 / 2
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 3, p. 837-856
The paper investigates the long time average of the solutions of Hamilton–Jacobi equations with a noncoercive, nonconvex Hamiltonian in the torus 2 / 2 . We give nonresonance conditions under which the long-time average converges to a constant. In the resonant case, we show that the limit still exists, although it is nonconstant in general. We compute the limit at points where it is not locally constant.
Nous considérons le comportement en temps grand de la moyenne temporelle de solutions d'équations de Hamilton–Jacobi pour un hamiltonien non convexe et non coercif dans le tore 2 / 2 . Nous mettons en évidence des conditions de non-résonnance sous lesquelles cette moyenne converge vers une constante. Dans le cas où il y a résonnance, nous montrons que la limite existe, bien qu'étant non constante en général. Nous calculons la limite aux points où celle-ci est non localement constante.
@article{AIHPC_2010__27_3_837_0,
     author = {Cardaliaguet, Pierre},
     title = {Ergodicity of Hamilton--Jacobi equations with a noncoercive nonconvex Hamiltonian in $ {\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {3},
     year = {2010},
     pages = {837-856},
     doi = {10.1016/j.anihpc.2009.11.015},
     zbl = {1201.35089},
     mrnumber = {2629882},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_3_837_0}
}
Cardaliaguet, Pierre. Ergodicity of Hamilton–Jacobi equations with a noncoercive nonconvex Hamiltonian in $ {\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 3, pp. 837-856. doi : 10.1016/j.anihpc.2009.11.015. http://www.numdam.org/item/AIHPC_2010__27_3_837_0/

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