Existence and uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data (Q1592602)

In this very interesting paper, the authors study a nonlinear variational wave equation of the form: \[ u_t + u v_x = - {1\over 2}v^2,\qquad v - u_x = 0 \] complemeneted by the boundary condition \(u(t,0) = 0\) and the initial condition \(v(0) = v_0 \in L^2(\mathbb{R}^+)\). The global existence and uniqueness of admissible weak solutions is proved. The admissible solutions satisfy an Oleinik type entropy condition as well as the energy inequality. There are two classes of admissible solutions considered in the paper -- the conservative ones satisfying a local version of energy equality, and the dissipative ones whose energy decays at the fastest possible rate. Besides the existence proofs, the main achievement of the paper seems to be uniqueness of both dissipative and conservative solutions for rather general initial data. The technique of mollifiers and Young measures is used.