Abstract
We study stability of solutions of the Cauchy problem on the line for the Camassa--Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_D(u(t),v(t))\le e^{Ct} d_D(u_0,v_0)$. The relationship between this metric and the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f''(u)(u_x)^2)_x=0$ (for $f$ without inflection points).
Highlights
The Cauchy problem for the Camassa–Holm (CH) equation [3, 4], (1.1)ut − uxxt + κux + 3uux − 2uxuxx − uuxxx = 0, where κ ∈ R is a constant, has attracted much attention due to the fact that it serves as a model for shallow water waves [8] and its rich mathematical structure
Ut − uxxt + κux + 3uux − 2uxuxx − uuxxx = 0, where κ ∈ R is a constant, has attracted much attention due to the fact that it serves as a model for shallow water waves [8] and its rich mathematical structure
We here focus on the construction of the Lipschitz metric for the semigroup of conservative solutions on the real line
Summary
We introduce a pseudometric in Lagrangian coordinates which does not distinguish between elements of the same equivalence class and which, at the same time, leaves the semigroup St locally Lipschitz continuous. This strategy has been used in [2] for the Hunter–Saxton equation and in [12] for the Camassa–Holm equation in the periodic case. Identifying elements belonging to the same equivalence class, the pseudometric d turns into a metric on the set of equivalence classes By bijection, it yields a metric in D which makes the semigroup of conservative solutions Lipschitz continuous.
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