The threshold for energy conservation and uniqueness in the Hunter–Saxton equation

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.