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

Viscous Aubry-Mather theory and the Vlasov equation

Ugo Bessi 1,

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.
Keywords: Vlasov equation., Viscous Aubry-Mather theory.
Mathematics Subject Classification: Primary: 70H20; Secondary: 35Q8.
Citation: Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
References:
[1]

L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52. 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), 207-276.  Google Scholar

[3]

G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 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, Edizioni della Normale, Pisa, 2007.  Google Scholar

[7]

R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58.  Google Scholar

[8]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003. Google Scholar

[9]

W. H. Fleming, The Cauchy problem for a nonlinear first order PDE, Journal of Differential Equations, 5 (1969), 515-530. doi: 10.1016/0022-0396(69)90091-6.  Google Scholar

[10]

A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992. Google Scholar

[11]

W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem, Adv. Math., 224 (2010), 260-292. 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), 155-183.  Google Scholar

[13]

D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.  Google Scholar

[14]

T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[15]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. 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, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

show all references

References:
[1]

L. Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52. 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), 207-276.  Google Scholar

[3]

G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 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, Edizioni della Normale, Pisa, 2007.  Google Scholar

[7]

R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58.  Google Scholar

[8]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003. Google Scholar

[9]

W. H. Fleming, The Cauchy problem for a nonlinear first order PDE, Journal of Differential Equations, 5 (1969), 515-530. doi: 10.1016/0022-0396(69)90091-6.  Google Scholar

[10]

A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992. Google Scholar

[11]

W. Gangbo and A. Tudorascu, Lagrangian dynamics on an infinite-dimensional torus; a weak KAM theorem, Adv. Math., 224 (2010), 260-292. 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), 155-183.  Google Scholar

[13]

D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.  Google Scholar

[14]

T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[15]

J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. 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, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.  Google Scholar

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