Abstract The seminal work of DiPerna and Lions (Invent Math 98(3):511–547, 1989) guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A longstanding open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio’s superposition principle, we relate the latter to the uniqueness of positive solutions of the continuity equation and we then provide a negative answer using tools introduced by Modena and Székelyhidi in the recent groundbreaking work (Modena and Székelyhidi in Ann PDE 4(2):38, 2018). On the opposite side, we introduce a new class of asymmetric Lusin–Lipschitz inequalities and use them to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna–Lions theory.
Large time behavior, biHamiltonian structure, and kinetic formulation for a complex Burgers equation
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 biHamiltonian 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 nearestneighbor interactions, and the Calogero–Moser model. These two particle systems yield the same Lagrangian dynamics in the continuum limit.
 Publication Date:
 NSFPAR ID:
 10355154
 Journal Name:
 Quarterly of Applied Mathematics
 Volume:
 79
 Issue:
 1
 Page Range or eLocationID:
 55 to 102
 ISSN:
 0033569X
 Sponsoring Org:
 National Science Foundation
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