February  2014, 34(2): 379-420. doi: 10.3934/dcds.2014.34.379

Viscous Aubry-Mather theory and the Vlasov equation

1. 

Dip. di Matematica, Università di Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Roma, Italy

Received  November 2012 Revised  May 2013 Published  August 2013

The Vlasov equation models a group of particles moving under a potential $V$; moreover, each particle exerts a force, of potential $W$, on the other ones. We shall suppose that these particles move on the $p$-dimensional torus $T^p$ and that the interaction potential $W$ is smooth. We are going to perturb this equation by a Brownian motion on $T^p$; adapting to the viscous case methods of Gangbo, Nguyen, Tudorascu and Gomes, we study the existence of periodic solutions and the asymptotics of the Hopf-Lax semigroup.
Citation: Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
References:
[1]

L. Ambrosio, Lecture notes on optimal transport problems,, in, 1812 (2003), 1.  doi: 10.1007/978-3-540-39189-0_1.  Google Scholar

[2]

N. Anantharaman, On the zero-temperature vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics,, J. Eur. Math. Soc. (JEMS), 6 (2004), 207.   Google Scholar

[3]

G. Birkhoff, "Lattice Theory,", Third edition, (1967).   Google Scholar

[4]

P. Cardialiaguet, "Notes on Mean Field Games,", from P.-L. Lions' lectures at the Collège de France, ().   Google Scholar

[5]

E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications,", mimeographed notes, (2009).   Google Scholar

[6]

G. Da Prato, "Introduction to Stochastic Analysis and Malliavin Calculus,", Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 6 (2007).   Google Scholar

[7]

R. L. Dobrušin, Vlasov equations,, Functional Analysis and its Applications, 13 (1979), 45.   Google Scholar

[8]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Fourth preliminary version, (2003).   Google Scholar

[9]

W. H. Fleming, The Cauchy problem for a nonlinear first order PDE,, Journal of Differential Equations, 5 (1969), 515.  doi: 10.1016/0022-0396(69)90091-6.  Google Scholar

[10]

A. Friedman, "Partial Differential Equations of Parabolic Type,", Dover, (1992).   Google Scholar

[11]

W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem,, Adv. Math., 224 (2010), 260.  doi: 10.1016/j.aim.2009.11.005.  Google Scholar

[12]

W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space,, Methods Appl. Anal., 15 (2008), 155.   Google Scholar

[13]

D. Gomes, A stochastic analog of Aubry-Mather theory,, Nonlinearity, 15 (2002), 581.  doi: 10.1088/0951-7715/15/3/304.  Google Scholar

[14]

T. Hida, "Brownian Motion,", Applications of Mathematics, 11 (1980).   Google Scholar

[15]

J. N. Mather, Variational construction of connecting orbits,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349.  doi: 10.5802/aif.1377.  Google Scholar

[16]

M. Viana, "Stochastic Dynamics of Deterministic Systems,", mimeographed notes, (2000).   Google Scholar

[17]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

show all references

References:
[1]

L. Ambrosio, Lecture notes on optimal transport problems,, in, 1812 (2003), 1.  doi: 10.1007/978-3-540-39189-0_1.  Google Scholar

[2]

N. Anantharaman, On the zero-temperature vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics,, J. Eur. Math. Soc. (JEMS), 6 (2004), 207.   Google Scholar

[3]

G. Birkhoff, "Lattice Theory,", Third edition, (1967).   Google Scholar

[4]

P. Cardialiaguet, "Notes on Mean Field Games,", from P.-L. Lions' lectures at the Collège de France, ().   Google Scholar

[5]

E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications,", mimeographed notes, (2009).   Google Scholar

[6]

G. Da Prato, "Introduction to Stochastic Analysis and Malliavin Calculus,", Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 6 (2007).   Google Scholar

[7]

R. L. Dobrušin, Vlasov equations,, Functional Analysis and its Applications, 13 (1979), 45.   Google Scholar

[8]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Fourth preliminary version, (2003).   Google Scholar

[9]

W. H. Fleming, The Cauchy problem for a nonlinear first order PDE,, Journal of Differential Equations, 5 (1969), 515.  doi: 10.1016/0022-0396(69)90091-6.  Google Scholar

[10]

A. Friedman, "Partial Differential Equations of Parabolic Type,", Dover, (1992).   Google Scholar

[11]

W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem,, Adv. Math., 224 (2010), 260.  doi: 10.1016/j.aim.2009.11.005.  Google Scholar

[12]

W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space,, Methods Appl. Anal., 15 (2008), 155.   Google Scholar

[13]

D. Gomes, A stochastic analog of Aubry-Mather theory,, Nonlinearity, 15 (2002), 581.  doi: 10.1088/0951-7715/15/3/304.  Google Scholar

[14]

T. Hida, "Brownian Motion,", Applications of Mathematics, 11 (1980).   Google Scholar

[15]

J. N. Mather, Variational construction of connecting orbits,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349.  doi: 10.5802/aif.1377.  Google Scholar

[16]

M. Viana, "Stochastic Dynamics of Deterministic Systems,", mimeographed notes, (2000).   Google Scholar

[17]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Mathematics, 58 (2003).  doi: 10.1007/b12016.  Google Scholar

[1]

Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135

[2]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807

[3]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683

[4]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155

[5]

Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823

[6]

Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103

[7]

Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983

[8]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018

[9]

Miroslav Grmela, Michal Pavelka. Landau damping in the multiscale Vlasov theory. Kinetic & Related Models, 2018, 11 (3) : 521-545. doi: 10.3934/krm.2018023

[10]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[11]

Eduard Feireisl. Mathematical theory of viscous fluids: Retrospective and future perspectives. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 533-555. doi: 10.3934/dcds.2010.27.533

[12]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[13]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[14]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[15]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[16]

Frédérique Charles, Bruno Després, Benoît Perthame, Rémis Sentis. Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinetic & Related Models, 2013, 6 (2) : 269-290. doi: 10.3934/krm.2013.6.269

[17]

Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169

[18]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207

[19]

Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159

[20]

Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic & Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]