The threshold for energy conservation and uniqueness in the Hunter–Saxton equation
Abstract
Abstract We investigate the Cauchy problem for the Hunter–Saxton equation with a general continuous forcing term . For weak solutions satisfying , we determine as the sharp critical exponent governing energy conservation and uniqueness. When , every weak solution conserves the ‐energy and is unique for arbitrary continuous . Conversely, for , we construct continuous forcings that admit non‐conservative weak solutions and exhibit failure of uniqueness. Our counterexample further shows that dissipative solutions with identical initial data may be non‐unique, thereby distinguishing the conservative and dissipative frameworks. The analysis relies on Littlewood–Paley theory, Besov embeddings, and delicate commutator estimates.
Fields of Study & Subjects
Nonlinear Waves and Solitons
Fractional Differential Equations Solutions
Navier-Stokes equation solutions
Publication details
- Journal / source
- Bulletin of the London Mathematical Society
- Publisher
- Wiley
- Volume
- 58
- Issue
- 5
- Publication date
- 1 May 2026
- Language
- EN