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Lipschitz metric for the Hunter-Saxton equation. (English) Zbl 1195.35046

Summary: We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation \(u_t + uu_x = \frac 14 (\int^x_{-\infty} u^2_x dx)\) with initial data \(u_{0}\). In particular, we derive a new Lipschitz metric \(d_{\mathcal D}\) with the property that for two solutions \(u\) and \(v\) of the equation we have \(d_{\mathcal D} (u(t),v(t)) \leqslant e^{C^t} d_{\mathcal D} (u_0,v_0)\).

MSC:

35B35 Stability in context of PDEs
35R09 Integro-partial differential equations
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References:

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