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# Uniqueness of conservative solutions to the Camassa–Holm equation via characteristics

Abstract: The paper provides a direct proof the uniqueness of solutions to the Camassa–Holm equation, based on characteristics. Given a conservative solution $u = u(t, x)$, an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities $u$ and $v = 2 \arctan u_x$ along each characteristic, it is proved that the Cauchy problem with general initial data $u_0 ∈ H^1(\mathbb{R})$ has a unique solution, globally in time.

Paper:
Available as PDF (267 Kbytes).
Author(s):
Alberto Bressan
Geng Chen
Qingtian Zhang
Submitted by:
; 2014-01-01