Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

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Abstract A nonlinear wave equation arises in a simplified liquid crystal model through the variational principle. The wave speed of the wave equation is a given function of the wave amplitude. In the earlier study to this equation, Hunter and Saxton have derived a simple asymptotic equation for weakly nonlinear unidirectional waves of the equation. Previous work has established the existence of weak solutions to the initial value problem for the asymptotic equation for data in the space of bounded variations. We improve the previous work to the natural space of square integrable functions, and we establish the uniqueness of weak solutions for both the dissipative and conservative types. We also have results on the full nonlinear wave equation. It has been known from joint work of the second author with Glassey and Hunter for the equation that smooth initial data may develop singularities in finite time, a sequence of weak solutions may develop concentrations, while oscillations may persist. For monotone wave speed functions in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) the smooth solutions obtained through data mollification and step (a) for not-as-smooth initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation with initial data in H 1 ( ℝ ) × L 2 ( ℝ ) . Furthermore, for initial data outside the invariant region, we can also prove the global existence of weak solution with initial Riemann invariant in L ∞ ( ℝ ) ∩ L 2 ( ℝ ) . The main tool for the weak solution is the Young measure theory and related techniques. More specifically, we will present the following results. 1. On the asymptotic equation, we have existence and uniqueness of multiple weak solutions in the weak norm L 2 of the derivative u x . 2. For the nonlinear wave equation, with monotone wave speed, we found some invariant regions and some global smooth solutions. 3. For the nonlinear wave equation, with monotone wave speed, we prove the global existence of weak solution with initial Riemann invariant in L ∞ ( ℝ ) ∩ L 2 ( ℝ ) .