February 2018, 38(2): 723-748. doi: 10.3934/dcds.2018032

On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs

1. 

IRMAR, UMR-CNRS 6625, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France

2. 

LAGA, UMR-CNRS 9345, Université de Paris 13, av. J. B. Clément, 93430 Villetaneuse, France

* Corresponding author: Ammari Zied

Received  October 2016 Revised  August 2017 Published  February 2018

In this paper, the Cauchy problem of classical Hamiltonian PDEs is recast into a Liouville's equation with measure-valued solutions. Then a uniqueness property for the latter equation is proved under some natural assumptions. Our result extends the method of characteristics to Hamiltonian systems with infinite degrees of freedom and it applies to a large variety of Hamiltonian PDEs (Hartree, Klein-Gordon, Schrödinger, Wave, Yukawa $\dots$). The main arguments in the proof are a projective point of view and a probabilistic representation of measure-valued solutions to continuity equations in finite dimension.

Citation: Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032
References:
[1]

R. AdamiC. BardosF. Golse and A. Teta, Towards a rigorous derivation of the cubic NLSE in dimension one, Asymptot. Anal., 40 (2004), 93-108.

[2]

L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1191-1244. doi: 10.1017/S0308210513000085.

[3]

L. Ambrosio and A. Figalli, On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions, J. Funct. Anal., 256 (2009), 179-214. doi: 10.1016/j.jfa.2008.05.007.

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford mathematical monographs Clarendon Press, 2000.

[5]

L. Ambrosio, N. Gigli and G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, second edition, 2008.

[6]

Z. Ammari and S. Breteaux, Propagation of chaos for many-boson systems in one dimension with a point pair-interaction, Asymptot. Anal., 76 (2012), 123-170.

[7]

Z. Ammari and M. Falconi, Wigner measures approach to the classical limit of the Nelson model: convergence of dynamics and ground state energy, J. Stat. Phys., 157 (2014), 330-362. doi: 10.1007/s10955-014-1079-7.

[8]

Z. Ammari and M. Falconi, Bohr's correspondence principle for the renormalized Nelson model, arXiv: 1602.03212.

[9]

Z. Ammari and F. Nier, Mean field limit for bosons and infinite dimensional phase-space analysis, Ann. Henri Poincaré, 9 (2008), 1503-1574. doi: 10.1007/s00023-008-0393-5.

[10]

Z. Ammari and F. Nier, Mean field limit for bosons and propagation of Wigner measures J. Math. Phys. 50 (2009), 042107, 16pp. doi: 10.1063/1.3115046.

[11]

Z. Ammari and F. Nier, Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states, J. Math. Pures Appl., 95 (2011), 585-626. doi: 10.1016/j.matpur.2010.12.004.

[12]

Z. Ammari and F. Nier, Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential, Ann. Sc. Norm. Super. Pisa Cl. Sci., 14 (2015), 155-220.

[13]

Z. Ammari and M. Zerzeri, On the classical limit of self-interacting quantum field Hamiltonians with cutoffs, Hokkaido Math. J., 43 (2014), 385-425. doi: 10.14492/hokmj/1416837571.

[14]

H. Bahouri and J.-Y. Chemin, Equations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Rational Mech. Anal., 127 (1994), 159-181. doi: 10.1007/BF00377659.

[15]

C. BardosF. Golse and N. J. Mauser, Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal., 7 (2000), 275-293. doi: 10.4310/MAA.2000.v7.n2.a2.

[16]

P. Bernard, Young measures, superposition and transport, Indiana Univ. Math. J., 57 (2008), 247-275. doi: 10.1512/iumj.2008.57.3163.

[17]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.

[18]

N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc., 16 (2014), 1-30. doi: 10.4171/JEMS/426.

[19]

E. CarlenJ. Fröhlich and J. Lebowitz, Exponential relaxation to equilibrium for a one-dimensional focusing non-linear Schrödinger equation with noise, Comm. Math. Phys., 342 (2016), 303-332. doi: 10.1007/s00220-015-2511-9.

[20]

T. Cazenave, Semilinear Schrödinger Equations volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[21]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations Oxford Lecture Series in Mathematics and its Applications, 1998.

[22]

T. ChenC. HainzlN. Pavlovic and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, Commun. Pure Appl. Math., 68 (2015), 1845-1884. doi: 10.1002/cpa.21552.

[23]

J. CollianderJ. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[24]

F. Colombini and N. Lerner, Uniqueness of continuous solutions for BV vector fields, Duke Math. J., 111 (2002), 357-384. doi: 10.1215/S0012-7094-01-11126-5.

[25]

G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields volume 12 of Theses of Scuola Normale Superiore di Pisa. Edizioni della Normale, Pisa, 2009.

[26]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.

[27]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[28]

M. Donald, The classical field limit of $P{(\varphi )_2}$ quantum field theory, Comm. Math. Phys., 79 (1981), 153-165.

[29]

L. ErdösB. Schlein and H. T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-604. doi: 10.1007/s00222-006-0022-1.

[30]

J. FröhlichS. Graffi and S. Schwarz, Mean-field-and classical limit of many-body Schrödinger dynamics for bosons, Comm. Math. Phys., 271 (2007), 681-697. doi: 10.1007/s00220-007-0207-5.

[31]

J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems, I, Comm. Math. Phys., 66 (1979), 37-76.

[32]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.

[33]

K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277. doi: 10.1007/BF01646348.

[34]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Commun. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.

[35]

A. Knowles and P. Pickl, Mean-field dynamics: Singular potentials and rate of convergence, Comm. Math. Phys., 298 (2010), 101-138. doi: 10.1007/s00220-010-1010-2.

[36]

A. V. Kolesnikov and M. Röckner, On continuity equations in infinite dimensions with non-Gaussian reference measure, J. Funct. Anal., 266 (2014), 4490-4537. doi: 10.1016/j.jfa.2014.01.010.

[37]

J. LebowitzH. A. Rose and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., 50 (1988), 657-687. doi: 10.1007/BF01026495.

[38]

Q. Liard, Dérivation Des Équations de Schrödinger non Linéaires Par Une Méthode Des Caractéristiques en Dimension Infinie, PHD Thesis (Rennes) 2015.

[39]

Q. Liard, On the mean-field approximation of many-boson dynamics, J. Funct. Anal., 273 (2017), 1397-1442. doi: 10.1016/j.jfa.2017.04.016.

[40]

Q. Liard and B. Pawilowski, Mean field limit for bosons with compact kernels interactions by Wigner measures transportation Journal of Mathematical Physics 55 (2014), 092304, 23pp. doi: 10.1063/1. 4895467.

[41]

M. Mandelkern, On the uniform continuity of Tietze extensions, Arch. Math., 55 (1990), 387-388. doi: 10.1007/BF01198478.

[42]

S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, J. Math. Pures Appl., 87 (2007), 601-626. doi: 10.1016/j.matpur.2007.04.001.

[43]

H. P. McKean and K. L. Vaninsky, Statistical mechanics of nonlinear wave equations, in Trends and perspectives in applied mathematics, (eds. L. Sirovich), Appl. Math. Sci. AMS, 100 (1994), 239-264. doi: 10.1007/978-1-4612-0859-4_8.

[44]

H. Pecher, Some new well-posedness results for the Klein-Gordon-Schrödinger system, Differential Integral Equations, 25 (2012), 117-142.

[45]

F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358. doi: 10.1080/03605309708821265.

[46]

L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures Oxford University Press, London, 1973.

[47]

I. Segal, Construction of non-linear local quantum processes, I, Ann. of Math., 92 (1970), 462-481. doi: 10.2307/1970628.

[48]

B. Simon, The P(φ)2 Euclidean (Quantum) Field Theory Princeton University Press, Princeton, N. J., 1974.

[49]

H. Spohn, Kinetic equations from Hamiltonian dynamics: The Markovian approximations, in Kinetic theory and gas dynamics, volume 293 of CISM Courses and Lectures, Springer, (1988), 183-211. doi: 10.1007/978-3-7091-2762-9_6.

[50]

C. Swartz, Measure, Integration and Function Spaces World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/2223.

[51]

J. Szczepański, On the basis of statistical mechanics. The Liouville equation for systems with an infinite countable number of degrees of freedom, Phys. A, 157 (1989), 955-982. doi: 10.1016/0378-4371(89)90075-7.

show all references

References:
[1]

R. AdamiC. BardosF. Golse and A. Teta, Towards a rigorous derivation of the cubic NLSE in dimension one, Asymptot. Anal., 40 (2004), 93-108.

[2]

L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1191-1244. doi: 10.1017/S0308210513000085.

[3]

L. Ambrosio and A. Figalli, On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions, J. Funct. Anal., 256 (2009), 179-214. doi: 10.1016/j.jfa.2008.05.007.

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford mathematical monographs Clarendon Press, 2000.

[5]

L. Ambrosio, N. Gigli and G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, second edition, 2008.

[6]

Z. Ammari and S. Breteaux, Propagation of chaos for many-boson systems in one dimension with a point pair-interaction, Asymptot. Anal., 76 (2012), 123-170.

[7]

Z. Ammari and M. Falconi, Wigner measures approach to the classical limit of the Nelson model: convergence of dynamics and ground state energy, J. Stat. Phys., 157 (2014), 330-362. doi: 10.1007/s10955-014-1079-7.

[8]

Z. Ammari and M. Falconi, Bohr's correspondence principle for the renormalized Nelson model, arXiv: 1602.03212.

[9]

Z. Ammari and F. Nier, Mean field limit for bosons and infinite dimensional phase-space analysis, Ann. Henri Poincaré, 9 (2008), 1503-1574. doi: 10.1007/s00023-008-0393-5.

[10]

Z. Ammari and F. Nier, Mean field limit for bosons and propagation of Wigner measures J. Math. Phys. 50 (2009), 042107, 16pp. doi: 10.1063/1.3115046.

[11]

Z. Ammari and F. Nier, Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states, J. Math. Pures Appl., 95 (2011), 585-626. doi: 10.1016/j.matpur.2010.12.004.

[12]

Z. Ammari and F. Nier, Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential, Ann. Sc. Norm. Super. Pisa Cl. Sci., 14 (2015), 155-220.

[13]

Z. Ammari and M. Zerzeri, On the classical limit of self-interacting quantum field Hamiltonians with cutoffs, Hokkaido Math. J., 43 (2014), 385-425. doi: 10.14492/hokmj/1416837571.

[14]

H. Bahouri and J.-Y. Chemin, Equations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Rational Mech. Anal., 127 (1994), 159-181. doi: 10.1007/BF00377659.

[15]

C. BardosF. Golse and N. J. Mauser, Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal., 7 (2000), 275-293. doi: 10.4310/MAA.2000.v7.n2.a2.

[16]

P. Bernard, Young measures, superposition and transport, Indiana Univ. Math. J., 57 (2008), 247-275. doi: 10.1512/iumj.2008.57.3163.

[17]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.

[18]

N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc., 16 (2014), 1-30. doi: 10.4171/JEMS/426.

[19]

E. CarlenJ. Fröhlich and J. Lebowitz, Exponential relaxation to equilibrium for a one-dimensional focusing non-linear Schrödinger equation with noise, Comm. Math. Phys., 342 (2016), 303-332. doi: 10.1007/s00220-015-2511-9.

[20]

T. Cazenave, Semilinear Schrödinger Equations volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[21]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations Oxford Lecture Series in Mathematics and its Applications, 1998.

[22]

T. ChenC. HainzlN. Pavlovic and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, Commun. Pure Appl. Math., 68 (2015), 1845-1884. doi: 10.1002/cpa.21552.

[23]

J. CollianderJ. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[24]

F. Colombini and N. Lerner, Uniqueness of continuous solutions for BV vector fields, Duke Math. J., 111 (2002), 357-384. doi: 10.1215/S0012-7094-01-11126-5.

[25]

G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields volume 12 of Theses of Scuola Normale Superiore di Pisa. Edizioni della Normale, Pisa, 2009.

[26]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.

[27]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.

[28]

M. Donald, The classical field limit of $P{(\varphi )_2}$ quantum field theory, Comm. Math. Phys., 79 (1981), 153-165.

[29]

L. ErdösB. Schlein and H. T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-604. doi: 10.1007/s00222-006-0022-1.

[30]

J. FröhlichS. Graffi and S. Schwarz, Mean-field-and classical limit of many-body Schrödinger dynamics for bosons, Comm. Math. Phys., 271 (2007), 681-697. doi: 10.1007/s00220-007-0207-5.

[31]

J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems, I, Comm. Math. Phys., 66 (1979), 37-76.

[32]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155.

[33]

K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), 265-277. doi: 10.1007/BF01646348.

[34]

S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Commun. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.

[35]

A. Knowles and P. Pickl, Mean-field dynamics: Singular potentials and rate of convergence, Comm. Math. Phys., 298 (2010), 101-138. doi: 10.1007/s00220-010-1010-2.

[36]

A. V. Kolesnikov and M. Röckner, On continuity equations in infinite dimensions with non-Gaussian reference measure, J. Funct. Anal., 266 (2014), 4490-4537. doi: 10.1016/j.jfa.2014.01.010.

[37]

J. LebowitzH. A. Rose and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., 50 (1988), 657-687. doi: 10.1007/BF01026495.

[38]

Q. Liard, Dérivation Des Équations de Schrödinger non Linéaires Par Une Méthode Des Caractéristiques en Dimension Infinie, PHD Thesis (Rennes) 2015.

[39]

Q. Liard, On the mean-field approximation of many-boson dynamics, J. Funct. Anal., 273 (2017), 1397-1442. doi: 10.1016/j.jfa.2017.04.016.

[40]

Q. Liard and B. Pawilowski, Mean field limit for bosons with compact kernels interactions by Wigner measures transportation Journal of Mathematical Physics 55 (2014), 092304, 23pp. doi: 10.1063/1. 4895467.

[41]

M. Mandelkern, On the uniform continuity of Tietze extensions, Arch. Math., 55 (1990), 387-388. doi: 10.1007/BF01198478.

[42]

S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, J. Math. Pures Appl., 87 (2007), 601-626. doi: 10.1016/j.matpur.2007.04.001.

[43]

H. P. McKean and K. L. Vaninsky, Statistical mechanics of nonlinear wave equations, in Trends and perspectives in applied mathematics, (eds. L. Sirovich), Appl. Math. Sci. AMS, 100 (1994), 239-264. doi: 10.1007/978-1-4612-0859-4_8.

[44]

H. Pecher, Some new well-posedness results for the Klein-Gordon-Schrödinger system, Differential Integral Equations, 25 (2012), 117-142.

[45]

F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358. doi: 10.1080/03605309708821265.

[46]

L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures Oxford University Press, London, 1973.

[47]

I. Segal, Construction of non-linear local quantum processes, I, Ann. of Math., 92 (1970), 462-481. doi: 10.2307/1970628.

[48]

B. Simon, The P(φ)2 Euclidean (Quantum) Field Theory Princeton University Press, Princeton, N. J., 1974.

[49]

H. Spohn, Kinetic equations from Hamiltonian dynamics: The Markovian approximations, in Kinetic theory and gas dynamics, volume 293 of CISM Courses and Lectures, Springer, (1988), 183-211. doi: 10.1007/978-3-7091-2762-9_6.

[50]

C. Swartz, Measure, Integration and Function Spaces World Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/2223.

[51]

J. Szczepański, On the basis of statistical mechanics. The Liouville equation for systems with an infinite countable number of degrees of freedom, Phys. A, 157 (1989), 955-982. doi: 10.1016/0378-4371(89)90075-7.

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