A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 183-200.

We investigate large-time asymptotics for viscous Hamilton–Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems.

DOI : 10.1016/j.anihpc.2013.10.005
Classification : 35B40, 35F55, 49L25
Mots-clés : Large-time behavior, Hamilton–Jacobi equations, Degenerate parabolic equations, Nonlinear adjoint methods, Viscosity solutions 
@article{AIHPC_2015__32_1_183_0,
     author = {Cagnetti, Filippo and Gomes, Diogo and Mitake, Hiroyoshi and Tran, Hung V.},
     title = {A new method for large time behavior of degenerate viscous {Hamilton{\textendash}Jacobi} equations with convex {Hamiltonians}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {183--200},
     publisher = {Elsevier},
     volume = {32},
     number = {1},
     year = {2015},
     doi = {10.1016/j.anihpc.2013.10.005},
     mrnumber = {3303946},
     zbl = {1312.35020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.10.005/}
}
TY  - JOUR
AU  - Cagnetti, Filippo
AU  - Gomes, Diogo
AU  - Mitake, Hiroyoshi
AU  - Tran, Hung V.
TI  - A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 183
EP  - 200
VL  - 32
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2013.10.005/
DO  - 10.1016/j.anihpc.2013.10.005
LA  - en
ID  - AIHPC_2015__32_1_183_0
ER  - 
%0 Journal Article
%A Cagnetti, Filippo
%A Gomes, Diogo
%A Mitake, Hiroyoshi
%A Tran, Hung V.
%T A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 183-200
%V 32
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2013.10.005/
%R 10.1016/j.anihpc.2013.10.005
%G en
%F AIHPC_2015__32_1_183_0
Cagnetti, Filippo; Gomes, Diogo; Mitake, Hiroyoshi; Tran, Hung V. A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 183-200. doi : 10.1016/j.anihpc.2013.10.005. http://www.numdam.org/articles/10.1016/j.anihpc.2013.10.005/

[1] G. Barles, H. Ishii, H. Mitake, A new PDE approach to the large time asymptotics of solutions of Hamilton–Jacobi equation, Bull. Sci. Math. (2013), http://dx.doi.org/10.1007/s13373-013-0036-0 | MR | Zbl

[2] G. Barles, P.E. Souganidis, On the large time behavior of solutions of Hamilton–Jacobi equations, SIAM J. Math. Anal. 31 no. 4 (2000), 925 -939 | MR | Zbl

[3] G. Barles, P.E. Souganidis, Space–time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal. 32 no. 6 (2001), 1311 -1323 | MR | Zbl

[4] F. Cagnetti, D. Gomes, H.V. Tran, Aubry–Mather measures in the nonconvex setting, SIAM J. Math. Anal. 43 no. 6 (2011), 2601 -2629 | MR | Zbl

[5] F. Cagnetti, D. Gomes, H.V. Tran, Adjoint methods for obstacle problems and weakly coupled systems of PDE, ESAIM Control Optim. Calc. Var. 19 no. 3 (2013), 754 -779 | EuDML | Numdam | MR | Zbl

[6] F. Camilli, O. Ley, P. Loreti, V. Nguyen, Large time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations, Nonlinear Differ. Equ. Appl. 19 (2012), 719 -749 | MR | Zbl

[7] A. Davini, A. Siconolfi, A generalized dynamical approach to the large-time behavior of solutions of Hamilton–Jacobi equations, SIAM J. Math. Anal. 38 no. 2 (2006), 478 -502 | MR | Zbl

[8] L.C. Evans, Adjoint and compensated compactness methods for Hamilton–Jacobi PDE, Arch. Ration. Mech. Anal. 197 (2010), 1053 -1088 | MR | Zbl

[9] L.C. Evans, Envelopes and nonconvex Hamilton–Jacobi equations, Calc. Var. Partial Differ. Equ. (2013), http://dx.doi.org/10.1007/s00526-013-0635-3 | MR

[10] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math. 327 no. 3 (1998), 267 -270 | MR

[11] A. Fathi, A. Siconolfi, Existence of C 1 critical subsolutions of the Hamilton–Jacobi equation, Invent. Math. 155 no. 2 (2004), 363 -388 | MR | Zbl

[12] H. Ishii, Asymptotic solutions for large-time of Hamilton–Jacobi equations in Euclidean n space, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25 no. 2 (2008), 231 -266 | EuDML | Numdam | MR | Zbl

[13] O. Ley, V.D. Nguyen, Large time behavior for some nonlinear degenerate parabolic equations, arXiv:1306.0748 | MR | Zbl

[14] H. Mitake, H.V. Tran, Remarks on the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations, Asymptot. Anal. 77 (2012), 43 -70 | MR | Zbl

[15] H. Mitake, H.V. Tran, Homogenization of weakly coupled systems of Hamilton–Jacobi equations with fast switching rates, Arch. Ration. Mech. Anal. (2013), arXiv:1204.2748 [math.AP] | MR | Zbl

[16] H. Mitake, H.V. Tran, A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton–Jacobi equations, J. Math. Pures Appl. (2013), http://dx.doi.org/10.1016/j.matpur.2013.05.004 | MR | Zbl

[17] H. Mitake, H.V. Tran, Large-time behavior for obstacle problems for degenerate viscous Hamilton–Jacobi equations, arXiv:1309.4831 | MR | Zbl

[18] G. Namah, J.-M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton–Jacobi equations, Commun. Partial Differ. Equ. 24 no. 5–6 (1999), 883 -893 | MR | Zbl

[19] V.D. Nguyen, Some results on the large time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations, arXiv:1209.5929 | MR

[20] D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Classics Math. , Springer-Verlag, Berlin (2006) | MR

[21] H.V. Tran, Adjoint methods for static Hamilton–Jacobi equations, Calc. Var. Partial Differ. Equ. 41 (2011), 301 -319 | MR | Zbl

Cité par Sources :