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Generic regularity and Lipschitz metric for the Hunter-Saxton type equations. (English) Zbl 1361.35153

One- and two-component Hunter-Saxton model describing the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal, is studied for \((x,t)\in{\mathbb R}^+\times{\mathbb R}^+\). As it is known, finite time gradient blowup may happen in this system, so the solution flow is, in general, not Lipschitz continuous with respect to natural \(H^1\) distance. The authors study generic properties of conservative solutions and construct a Finsler type metric in which the solution flow is locally uniformly Lipschitz continuous.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35L99 Hyperbolic equations and hyperbolic systems
35B35 Stability in context of PDEs
37C20 Generic properties, structural stability of dynamical systems
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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