The Derivation of the Compressible Euler Equation from Quantum Many-Body Dynamics

We study the three dimensional many-particle quantum dynamics in meanfield setting. We forge together the hierarchy method and the modulated energy method. We prove rigorously that the compressible Euler equation is the limit as the particle number tends to infinity and the Planck’s constant tends to zero. We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation. We justify that the macroscopic pressure emerges from the space-time averages of microscopic interactions, which are in fact, Strichartz-type bounds. We have hence found a physical meaning for Strichartz type bounds which were first raised by Klainerman and Machedon in this context.

[1]  O. Lanford.  Time evolution of large classical systems , 1975 .

[2]  Xuwen Chen.  On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with a Quadratic Trap , 2012, 1204.0125.

[3]  Paul Smith,et al.  On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy , 2014, 1406.2649.

[4]  T. Paul,et al.  Mean‐Field and Classical Limit for the N‐Body Quantum Dynamics with Coulomb Interaction , 2019, Communications on Pure and Applied Mathematics.

[5]  Matthew Rosenzweig,et al.  From Quantum Many-Body Systems to Ideal Fluids , 2021, 2110.04195.

[6]  E. Madelung,et al.  Quantentheorie in hydrodynamischer Form , 1927 .

[7]  Zhifei Zhang,et al.  The unconditional uniqueness for the energy-supercritical NLS , 2021 .

[8]  Philip T. Gressman,et al.  On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy , 2012, 1212.2987.

[9]  Laure Saint-Raymond,et al.  Hydrodynamic Limits of the Boltzmann Equation , 2009 .

[10]  Justin Holmer,et al.  Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space , 2021 .

[11]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[12]  J. Holmer,et al.  The Derivation of the Energy-critical NLS from Quantum Many-body Dynamics , 2018 .

[13]  Thomas Chen,et al.  The quintic NLS as the mean field limit of a boson gas with three-body interactions , 2008, 0812.2740.

[14]  Shunlin Shen,et al.  The rigorous derivation of the T2 focusing cubic NLS from 3D , 2021 .

[15]  Horng-Tzer Yau,et al.  Derivation of the Gross‐Pitaevskii hierarchy for the dynamics of Bose‐Einstein condensate , 2004, math-ph/0410005.

[16]  Zhihui Xie,et al.  Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity , 2014, SIAM J. Math. Anal..

[17]  Sergiu Klainerman,et al.  On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy , 2007 .

[18]  Horng-Tzer Yau,et al.  Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate , 2004, math-ph/0606017.

[19]  Phan Thanh Nam,et al.  Derivation of Hartree's theory for generic mean-field Bose systems , 2013, 1303.0981.

[20]  Pair Excitations and the Mean Field Approximation of Interacting Bosons, I , 2012, 1208.3763.

[21]  J. Holmer,et al.  On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction , 2013, 1303.5385.

[22]  Sergiu Klainerman,et al.  Space-time estimates for null forms and the local existence theorem , 1993 .

[23]  Mean field limit for Coulomb-type flows , 2018, 1803.08345.

[24]  Z. Xie.  Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in $d=1,2$ , 2015, Differential and Integral Equations.

[25]  M. Grillakis,et al.  Pair Excitations and the Mean Field Approximation of Interacting Bosons, I , 2012, 1208.3763.

[26]  Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons , 2004, math-ph/0410038.

[27]  C. David Levermore,et al.  The Semiclassical Limit of the Defocusing NLS Hierarchy , 1999 .

[28]  J. Holmer,et al.  Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schr\"{o}dinger Equation from 3D , 2014, 1407.8457.

[29]  On the Quantum Boltzmann Equation , 2003, math-ph/0302034.

[30]  B. Schlein,et al.  Gross–Pitaevskii dynamics for Bose–Einstein condensates , 2017, Analysis & PDE.

[31]  J. Holmer,et al.  Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy , 2014, 1409.1425.

[32]  B. Schlein,et al.  Bogoliubov theory in the Gross–Pitaevskii limit , 2018, Acta Mathematica.

[33]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[34]  F. Lin,et al.  Semiclassical Limit of the Gross-Pitaevskii Equation in an Exterior Domain , 2006 .

[35]  Xuwen Chen.  Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps , 2011, 1102.0593.

[36]  V. Sohinger,et al.  Local existence of solutions to Randomized Gross-Pitaevskii hierarchies , 2014, 1401.0326.

[37]  J. Holmer,et al.  On the Rigorous Derivation of the 2D Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-Body Dynamics , 2012, 1212.0787.

[38]  H. Yau,et al.  Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential , 2008, 0802.3877.

[39]  G. Staffilani,et al.  Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics , 2008, 0808.0505.

[40]  Thomas Chen,et al.  Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms , 2011, 1111.6222.

[41]  N. Masmoudi.  Hydrodynamic Limits of the Boltzmann Equation , 2004 .

[42]  Horng-Tzer Yau,et al.  Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems , 2005, math-ph/0508010.

[43]  Rémi Carles,et al.  Semi-Classical Analysis For Nonlinear Schrodinger Equations , 2008 .

[44]  Isabelle Gallagher,et al.  From Newton to Boltzmann: Hard Spheres and Short-range Potentials , 2012, 1208.5753.

[45]  J. Holmer,et al.  Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation , 2013, 1308.3895.

[46]  S. Herr,et al.  The Gross–Pitaevskii Hierarchy on General Rectangular Tori , 2014, 1410.5338.

[47]  R. Seiringer,et al.  Unconditional Uniqueness for the Cubic Gross‐Pitaevskii Hierarchy via Quantum de Finetti , 2013, 1307.3168.

[48]  V. Sohinger,et al.  A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on T3 from the dynamics of many-body quantum systems , 2014, 1405.3003.

[49]  G. Staffilani,et al.  Randomization and the Gross–Pitaevskii Hierarchy , 2013, 1308.3714.

[50]  Robert S. Strichartz,et al.  Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations , 1977 .

[51]  B. Schlein,et al.  Quantitative Derivation of the Gross‐Pitaevskii Equation , 2012, 1208.0373.

[52]  H. Spohn.  Kinetic equations from Hamiltonian dynamics: Markovian limits , 1980 .

[53]  Ping Zhang,et al.  Wigner Measure and Semiclassical Limits of Nonlinear Schrodinger Equations , 2008 .

[54]  E. Grenier,et al.  Semiclassical limit of the nonlinear Schrödinger equation in small time , 1998 .

[55]  Yan Guo,et al.  On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation , 2014, 1412.1775.

[56]  S. Herr,et al.  Unconditional uniqueness results for the nonlinear Schrödinger equation , 2018, Communications in Contemporary Mathematics.

[57]  Derivation of the Euler equations from many-body quantum mechanics , 2002, math-ph/0210036.

[58]  Thomas Chen,et al.  On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate , 2021 .

[59]  B. Schlein,et al.  Quantum Many-Body Fluctuations Around Nonlinear Schrödinger Dynamics , 2015, 1509.03837.

[60]  F. Golse,et al.  Rigorous Derivation of the Cubic NLS in Dimension One , 2007 .

[61]  Younghun Hong,et al.  Uniqueness of solutions to the 3D quintic Gross-Pitaevskii Hierarchy , 2014, 1410.6961.