Global solutions of the Hunter-Saxton equation
Alberto Bressan, Adrian Constantin
TL;DR
The paper studies global behavior of the Hunter-Saxton equation, a nonlinear PDE modeling nematic liquid crystals. The authors construct a continuous semigroup of weak, dissipative solutions to this equation, establishing a global solution framework. They introduce a new distance functional based on an optimal transport problem, which yields sharp estimates for how solutions depend on the initial data. This approach provides a robust stability theory and practical tool for analyzing nematic liquid crystal models through globally defined dynamics.
Abstract
We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.