Existence and Uniqueness of Solutions to the Generalized Hydrodynamics Equation

TL;DR

This paper proves the existence and uniqueness of solutions to the generalized hydrodynamics equation for integrable models, ensuring well-posedness and absence of shocks under broad initial conditions.

Contribution

It establishes the first rigorous proof of existence and uniqueness for the full GHD equation, including differentiable and weak solutions, using a novel fixed-point approach.

Findings

Solutions exist and are unique for a large class of initial conditions.

Differentiable initial conditions lead to differentiable solutions at all times.

Weak initial conditions like the Riemann problem have unique weak solutions that preserve entropy.

Abstract

The generalized hydrodynamics (GHD) equation is the equivalent of the Euler equations of hydrodynamics for integrable models. Systems of hyperbolic equations such as the Euler equations usually develop shocks and are plagued by problems of uniqueness. We establish for the first time the existence and uniqueness of solutions to the full GHD equation and the absence of shocks, from a large class of initial conditions with bounded occupation function. We assume only absolute integrability of the two-body scattering shift. In applications to quantum models of fermionic type, this includes all commonly used physical initial states, such as locally thermal states and zero-entropy states. We show in particular that differentiable initial conditions give differentiable solutions at all times and that weak initial conditions such as the Riemann problem have unique weak solutions which preserve entropy. For this purpose, we write the GHD equation as a new fixed-point problem (announced in a companion paper). We show that the fixed point exists, is unique, and is approached, under an iterative solution procedure, in the Banach topology on functions of momenta.