Large time behavior, bi-Hamiltonian structure, and kinetic formulation for a complex Burgers equation
Authors:
Yu Gao, Yuan Gao and Jian-Guo Liu
Journal:
Quart. Appl. Math. 79 (2021), 55-102
MSC (2010):
Primary 35L65, 35R60, 37K05, 82B40, 15B52
DOI:
https://doi.org/10.1090/qam/1573
Published electronically:
May 21, 2020
MathSciNet review:
4188624
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Abstract: We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner’s semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained, and some explicit solutions are given by Wigner’s semicircle laws. We also construct a bi-Hamiltonian structure for the system of real and imaginary components of the complex Burgers equation (coupled Burgers system). We establish a kinetic formulation for the coupled Burgers system and prove the existence and uniqueness of entropy solutions. The coupled Burgers system in Lagrangian variable naturally leads to two interacting particle systems, the Fermi–Pasta–Ulam–Tsingou model with nearest-neighbor interactions, and the Calogero–Moser model. These two particle systems yield the same Lagrangian dynamics in the continuum limit.
References
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- Sandro Salsa, Partial differential equations in action, 3rd ed., Unitext, vol. 99, Springer, [Cham], 2016. From modelling to theory; La Matematica per il 3+2. MR 3497072
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- Luis Silvestre and Vlad Vicol, On a transport equation with nonlocal drift, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6159–6188. MR 3461030, DOI https://doi.org/10.1090/S0002-9947-2015-06651-3
- Terence Tao, Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012. MR 2906465
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References
- Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. MR 2401600
- Robert J. Berman and Magnus Önnheim, Propagation of chaos for a class of first order models with singular mean field interactions, SIAM J. Math. Anal. 51 (2019), no. 1, 159–196. MR 3904425, DOI https://doi.org/10.1137/18M1196662
- José A. Carrillo, Lucas C. F. Ferreira, and Juliana C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math. 231 (2012), no. 1, 306–327. MR 2935390, DOI https://doi.org/10.1016/j.aim.2012.03.036
- A. Castro and D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math. 219 (2008), no. 6, 1916–1936. MR 2456270, DOI https://doi.org/10.1016/j.aim.2008.07.015
- Emmanuel Cépa and Dominique Lépingle, Diffusing particles with electrostatic repulsion, Probab. Theory Related Fields 107 (1997), no. 4, 429–449. MR 1440140, DOI https://doi.org/10.1007/s004400050092
- Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (2005), no. 3, 1377–1389. MR 2179734, DOI https://doi.org/10.4007/annals.2005.162.1377
- Freeman J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys. 3 (1962), 1191–1198. MR 148397, DOI https://doi.org/10.1063/1.1703862
- László Erdős and Horng-Tzer Yau, A dynamical approach to random matrix theory, Courant Lecture Notes in Mathematics, vol. 28, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2017. MR 3699468
- David J. Gross and Andrei Matytsin, Some properties of large-$N$ two-dimensional Yang-Mills theory, Nuclear Phys. B 437 (1995), no. 3, 541–584. MR 1321333, DOI https://doi.org/10.1016/0550-3213%2894%2900570-5
- Richard Jordan, David Kinderlehrer, and Felix Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), no. 1, 1–17. MR 1617171, DOI https://doi.org/10.1137/S0036141096303359
- Richard Kenyon and Andrei Okounkov, Limit shapes and the complex Burgers equation, Acta Math. 199 (2007), no. 2, 263–302. MR 2358053, DOI https://doi.org/10.1007/s11511-007-0021-0
- Steven G. Krantz and Harold R. Parks, The implicit function theorem: History, theory, and applications; Reprint of the 2003 edition, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2013. MR 2977424
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI https://doi.org/10.1002/cpa.3160100406
- Lei Li, Jian-Guo Liu, and Pu Yu, On the mean field limit for Brownian particles with Coulomb interaction in 3D, J. Math. Phys. 60 (2019), no. 11, 111501, 34. MR 4026330, DOI https://doi.org/10.1063/1.5114854
- P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790
- Jian-Guo Liu and Rong Yang, Propagation of chaos for large Brownian particle system with Coulomb interaction, Res. Math. Sci. 3 (2016), Paper No. 40, 33. MR 3572548, DOI https://doi.org/10.1186/s40687-016-0086-5
- Jian-Guo Liu and Robert L. Pego, On generating functions of Hausdorff moment sequences, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8499–8518. MR 3551579, DOI https://doi.org/10.1090/tran/6618
- Franco Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), no. 5, 1156–1162. MR 488516, DOI https://doi.org/10.1063/1.523777
- G. Menon, The complex Burgers equation, the HCIZ integral and the Calogero-Moser system, preprint.
- Govind Menon, Lesser known miracles of Burgers equation, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 1, 281–294. MR 2921877, DOI https://doi.org/10.1016/S0252-9602%2812%2960017-4
- J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Surveys in applied mathematics (Proc. First Los Alamos Sympos. Math. in Natural Sci., Los Alamos, N.M., 1974) Academic Press, New York, 1976, pp. 235–258. MR 0665768
- Peter J. Olver and Yavuz Nutku, Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys. 29 (1988), no. 7, 1610–1619. MR 946335, DOI https://doi.org/10.1063/1.527909
- J. N. Pandey, The Hilbert transform of Schwartz distributions and applications, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. MR 1363489
- Benoit Perthame and Athanasios E. Tzavaras, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch. Ration. Mech. Anal. 155 (2000), no. 1, 1–48. MR 1799273, DOI https://doi.org/10.1007/s002050000109
- L. C. G. Rogers and Z. Shi, Interacting Brownian particles and the Wigner law, Probab. Theory Related Fields 95 (1993), no. 4, 555–570. MR 1217451, DOI https://doi.org/10.1007/BF01196734
- Sandro Salsa, Partial differential equations in action, 3rd ed., with From modelling to theory; La Matematica per il 3+2, Unitext, vol. 99, Springer, [Cham], 2016. MR 3497072
- D. Serre, Systems of conservation laws 1, Systems of Conservation Laws 255 (1999), no. 7, 286.
- Luis Silvestre and Vlad Vicol, On a transport equation with nonlocal drift, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6159–6188. MR 3461030, DOI https://doi.org/10.1090/tran6651
- Terence Tao, Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012. MR 2906465
- Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2) 62 (1955), 548–564. MR 77805, DOI https://doi.org/10.2307/1970079
- N. M. Zubarev and EA Karabut, Exact local solutions for the formation of singularities on the free surface of an ideal fluid, JETP Letters 107 (2018), no. 7, 412–417.
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Additional Information
Yu Gao
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong
ORCID:
0000-0002-8535-3889
Email:
gaoyu90@hku.hk
Yuan Gao
Affiliation:
Department of Mathematics, Duke University, Durham, North Carolina 27708
ORCID:
0000-0002-7231-5672
Email:
yuangao@math.duke.edu
Jian-Guo Liu
Affiliation:
Department of Mathematics and Department of Physics, Duke University, Durham, North Carolina 27708
MR Author ID:
233036
ORCID:
0000-0002-9911-4045
Email:
jliu@phy.duke.edu
Received by editor(s):
December 24, 2019
Received by editor(s) in revised form:
March 22, 2020
Published electronically:
May 21, 2020
Additional Notes:
The authors would like to thank the support by the National Science Foundation under grants DMS 1514826 and 1812573 (JGL)
Article copyright:
© Copyright 2020
Brown University