A Generalized Dynamical Approach to the Large Time Behavior of Solutions of Hamilton-Jacobi Equations
A Generalized Dynamical Approach to the Large Time Behavior of Solutions of Hamilton-Jacobi Equations
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DOI：
10.1137/050621955
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
A. Davini;A. Siconolfi
中科院分区：
文献类型：
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作者：
A. Davini;A. Siconolfi
考虑汉密尔顿-Jacobi方程\[ \partial_t u+H（x，Du）=0\qquad \hbox{in $（0，+\infty）\times\T^{N}$}，\]其中$\T^{N}$是平坦的N维环面，Hamilton算子$H（x，p）$在x上连续，在p上严格凸和强制。对于H唯一凸的情形，也给出了一些收敛性结果.我们的定性方法是基于对Aubry集的动力学性质的分析，在[A. Fathi和A. Siconolfi，Calc.变种偏微分方程，22（2005），pp。185-228]。这可以被看作是[A]中使用的技术的推广。法特希角R. Acad. Sci.巴黎第一系数学，327（1998），pp. 267-270]和[J. M. Roquejoffre，J. Math. Pures Appl.（9），80（2001），pp. 85-104]。文[G.巴尔斯和体育Souganidis，SIAM J. Math. Anal.，31（2000），pp. 925-939]使用PDE方法。
We consider the Hamilton--Jacobi equation \[ \partial_t u+H(x,Du)=0\qquad \hbox{in $(0,+\infty)\times\T^{N}$}, \] where $\T^{N}$ is the flat N-dimensional torus, and the Hamiltonian $H(x,p)$ is assumed continuous in x and strictly convex and coercive in p. We study the large time behavior of solutions, and we identify the limit through a Lax-type formula. Some convergence results are also given for H solely convex. Our qualitative method is based on the analysis of the dynamical properties of the Aubry set, performed in the spirit of [A. Fathi and A. Siconolfi, Calc. Var. Partial Differential Equations, 22 (2005), pp. 185-228]. This can be viewed as a generalization of the techniques used in [A. Fathi, C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 267-270] and [J. M. Roquejoffre, J. Math. Pures Appl. (9), 80 (2001), pp. 85-104]. Analogous results have been obtained in [G. Barles and P. E. Souganidis, SIAM J. Math. Anal., 31 (2000), pp. 925-939] using PDE methods.