Global Well‐Posedness of the Spatially Homogeneous Hubbard‐Boltzmann Equation. Issue 5 (27th May 2014)

• Record Type: Journal Article
• Authors: Lukkarinen, Jani; Mei, Peng; Spohn, Herbert
• Is Part Of: Communications on pure and applied mathematics. Volume 68:Issue 5(2015:May)
• Page Start: 758
• Page End: 807
• Publication Date: 2014-05-27
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21524
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract : The Hubbard model is a simplified description for the evolution of interacting spin fermions on a d ‐dimensional lattice. In a kinetic scaling limit, the Hubbard model can be associated with a matrix‐valued Boltzmann equation, the Hubbard‐Boltzmann equation. Its collision operator is a sum of two qualitatively different terms: The first term is similar to the collision operator of the fermionic Boltzmann‐Nordheim equation. The second term leads to a momentum‐dependent rotation of the spin basis. The rotation is determined by a principal value integral that depends quadratically on the state of the system and might become singular for nonsmooth states. In this paper, we prove that the spatially homogeneous equation nevertheless has global solutions in for any initial data W 0 that satisfies the "Fermi constraint" in the sense that 0 ≤ W 0 ≤ 1 almost everywhere. We also prove that there is a unique "physical" solution for which the Fermi constraint holds at all times. For the proof, we need to make a number of assumptions about the lattice dispersion relation which, however, are satisfied by the nearest‐neighbor Hubbard model provided that d ≥ 3. These assumptions suffice to guarantee that, although possibly singular, the local rotation term is generated by a function in .© 2015 Wiley Periodicals, Inc.