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Lagrangian dynamics, Random dynamical systems, Invariant measure, Hyperbolicity
We consider a class of 1d Lagrangian systems with random forcing in the spaceperiodic setting: \begin{equation} \nonumber \phi_t+\phi_x^2/2=F^{\omega},\ x \in S^1=\R / \Z. \end{equation} These systems have been studied since the 1990s by Khanin, Sinai and their collaborators [7, 9, 11, 12, 15]. Here we give an overview of their results and then we expose our recent proof of the exponential convergence to the stationary measure [6]. This is the first such result in a classical setting, i.e. in the dual-Lipschitz metric with respect to the Lebesgue space $L_p$ for finite $p$, partially answering the conjecture formulated in [11]. In the multidimensional setting, a more technically involved proof has been recently given by Iturriaga, Khanin and Zhang [13].
[1] Bec, J., Frisch, U., HASH(0x24c1f68), Khanin, K.: Kicked Burgers turbulence. Journal of Fluid Mechanics, 416(8) (2000), pp. 239–267. DOI 10.1017/S0022112000001051 | MR 1777053
[2] Boritchev, A.: Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation. Proceedings of the Royal Society of Edinburgh A, 143(2) (2013), pp. 253–268. DOI 10.1017/S0308210511000989 | MR 3039811
[3] Boritchev, A.: Sharp estimates for turbulence in white-forced generalised Burgers equation. Geometric and Functional Analysis, 23(6) (2013), pp. 1730–1771. DOI 10.1007/s00039-013-0245-4 | MR 3132902
[4] Boritchev, A.: Erratum to: Multidimensional Potential Burgers Turbulence. Communicationsin Mathematical Physics, 344(1) (2016), pp. 369–370, see [5]. DOI 10.1007/s00220-016-2621-z | MR 3493146
[5] Boritchev, A.: Multidimensional Potential Burgers Turbulence. Communications in Mathematical Physics, 342 (2016), pp. 441–489, with erratum: see [4]. DOI 10.1007/s00220-015-2521-7 | MR 3459157
[6] Boritchev, A.: Exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing. accepted to Stochastic and Partial Differential Equations: Analysis and Computations. MR 3768996
[7] Boritchev, A., Khanin, K.: On the hyperbolicity of minimizers for 1D random Lagrangian systems. Nonlinearity, 26(1) (2013), pp. 65–80. DOI 10.1088/0951-7715/26/1/65 | MR 3001762
[8] Doering, C., Gibbon, J. D.: Applied analysis of the Navier-Stokes equations. Cambridge Texts in Applied Mathematics, Cambridge University Press, 1995. MR 1325465
[9] E, Weinan, Khanin, K., Mazel, A., HASH(0x24dff00), Sinai, Ya.: Invariant measures for Burgers equation with stochastic forcing. Annals of Mathematics, 151 (2000), pp. 877–960. DOI 10.2307/121126 | MR 1779561
[10] Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics. preliminary version, 2005.
[11] Gomes, D., Iturriaga, R., Khanin, K., HASH(0x24e24c0), Padilla, P.: Viscosity limit of stationary distributions for the random forced Burgers equation. Moscow Mathematical Journal, 5 (2005), pp. 613–631. DOI 10.17323/1609-4514-2005-5-3-613-631 | MR 2241814
[12] Iturriaga, R., Khanin, K.: Burgers turbulence and random Lagrangian systems. Communications in Mathematical Physics, 232:3 (2003), pp. 377–428. DOI 10.1007/s00220-002-0748-6 | MR 1952472
[13] Iturriaga, R., Khanin, K., HASH(0x24e6f80), Zhang, K.: Exponential convergence of solutions for random Hamilton-Jacobi equation. Preprint, arxiv: 1703.10218, 2017.
[14] Iturriaga, R., Sanchez-Morgado, H.: Hyperbolicity and exponential convergence of the Lax-Oleinik semigroup. Journal of Differential Equations, 246(5) (2009), pp. 1744–1753. DOI 10.1016/j.jde.2008.12.012 | MR 2494686
[15] Khanin, K., Zhang, K.: Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations. Communications in Mathematical Physics, 355 (2017), pp. 803. DOI 10.1007/s00220-017-2919-5 | MR 3681391
[16] Sinai, Y.: Two results concerning asymptotic behavior of solutions of the Burgers equation with force. Journal of Statistical Physics, 64, 1991, pp. 1–12. DOI 10.1007/BF01057866 | MR 1117645
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