| Title:
|
Exponential convergence to the stationary measure and hyperbolicity of the minimisers for random Lagrangian Systems (English) |
| Author:
|
Boritchev, Alexandre |
| Language:
|
English |
| Journal:
|
Proceedings of Equadiff 14 |
| Volume:
|
Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |
| Issue:
|
2017 |
| Year:
|
|
| Pages:
|
117-126 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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]. (English) |
| Keyword:
|
Lagrangian dynamics, Random dynamical systems, Invariant measure, Hyperbolicity |
| MSC:
|
35Q35 |
| MSC:
|
35Q53 |
| MSC:
|
35R60 |
| MSC:
|
37H10 |
| MSC:
|
76M35 |
| . |
| Date available:
|
2019-09-27T07:46:11Z |
| Last updated:
|
2019-09-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/703036 |
| . |
| Reference:
|
[1] Bec, J., Frisch, U., HASH(0x24c1f68), Khanin, K.: Kicked Burgers turbulence., Journal of Fluid Mechanics, 416(8) (2000), pp. 239–267. MR 1777053, 10.1017/S0022112000001051 |
| Reference:
|
[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. MR 3039811, 10.1017/S0308210511000989 |
| Reference:
|
[3] Boritchev, A.: Sharp estimates for turbulence in white-forced generalised Burgers equation., Geometric and Functional Analysis, 23(6) (2013), pp. 1730–1771. MR 3132902, 10.1007/s00039-013-0245-4 |
| Reference:
|
[4] Boritchev, A.: Erratum to: Multidimensional Potential Burgers Turbulence., Communicationsin Mathematical Physics, 344(1) (2016), pp. 369–370, see [5]. MR 3493146, 10.1007/s00220-016-2621-z |
| Reference:
|
[5] Boritchev, A.: Multidimensional Potential Burgers Turbulence., Communications in Mathematical Physics, 342 (2016), pp. 441–489, with erratum: see [4]. MR 3459157, 10.1007/s00220-015-2521-7 |
| Reference:
|
[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 |
| Reference:
|
[7] Boritchev, A., Khanin, K.: On the hyperbolicity of minimizers for 1D random Lagrangian systems., Nonlinearity, 26(1) (2013), pp. 65–80. MR 3001762, 10.1088/0951-7715/26/1/65 |
| Reference:
|
[8] Doering, C., Gibbon, J. D.: Applied analysis of the Navier-Stokes equations., Cambridge Texts in Applied Mathematics, Cambridge University Press, 1995. MR 1325465 |
| Reference:
|
[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. MR 1779561, 10.2307/121126 |
| Reference:
|
[10] Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics., preliminary version, 2005. |
| Reference:
|
[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. MR 2241814, 10.17323/1609-4514-2005-5-3-613-631 |
| Reference:
|
[12] Iturriaga, R., Khanin, K.: Burgers turbulence and random Lagrangian systems., Communications in Mathematical Physics, 232:3 (2003), pp. 377–428. MR 1952472, 10.1007/s00220-002-0748-6 |
| Reference:
|
[13] Iturriaga, R., Khanin, K., HASH(0x24e6f80), Zhang, K.: Exponential convergence of solutions for random Hamilton-Jacobi equation., Preprint, arxiv: 1703.10218, 2017. |
| Reference:
|
[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. MR 2494686, 10.1016/j.jde.2008.12.012 |
| Reference:
|
[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. MR 3681391, 10.1007/s00220-017-2919-5 |
| Reference:
|
[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. MR 1117645, 10.1007/BF01057866 |
| . |
