Viscous Aubry-Mather theory and the Vlasov equation

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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

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

Abstract
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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 and 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.
[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.
[3]	G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 1967.
[4]	P. Cardialiaguet, "Notes on Mean Field Games,", from P.-L. Lions' lectures at the Collège de France, ().
[5]	E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications," mimeographed notes, 2009.
[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.
[7]	R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58.
[8]	A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003.
[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.
[10]	A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992.
[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.
[12]	W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods Appl. Anal., 15 (2008), 155-183.
[13]	D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.
[14]	T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980.
[15]	J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.
[16]	M. Viana, "Stochastic Dynamics of Deterministic Systems," mimeographed notes, 2000.
[17]	C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

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.
[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.
[3]	G. Birkhoff, "Lattice Theory," Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I., 1967.
[4]	P. Cardialiaguet, "Notes on Mean Field Games,", from P.-L. Lions' lectures at the Collège de France, ().
[5]	E. Carlen, "Lectures on Optimal Mass Transportation and Certain of its Applications," mimeographed notes, 2009.
[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.
[7]	R. L. Dobrušin, Vlasov equations, Functional Analysis and its Applications, 13 (1979), 45-58.
[8]	A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Fourth preliminary version, mimeographed notes, Lyon, 2003.
[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.
[10]	A. Friedman, "Partial Differential Equations of Parabolic Type," Dover, New York, 1992.
[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.
[12]	W. Gangbo, T. Nguyen and A. Tudorascu, Hamilton-Jacobi equations in the Wasserstein space, Methods Appl. Anal., 15 (2008), 155-183.
[13]	D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity, 15 (2002), 581-603. doi: 10.1088/0951-7715/15/3/304.
[14]	T. Hida, "Brownian Motion," Applications of Mathematics, 11, Springer-Verlag, New York-Berlin, 1980.
[15]	J. N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386. doi: 10.5802/aif.1377.
[16]	M. Viana, "Stochastic Dynamics of Deterministic Systems," mimeographed notes, 2000.
[17]	C. Villani, "Topics in Optimal Transportation," Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016.

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