The modified camassa-holm equation in lagrangian coordinates
© 2018 American Institute of Mathematical Sciences. All Rights Reserved. In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data m0, we show that classical solutions to this equation blow up in finite time Tmax. Before Tmax, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: Tmax ≥||m0||L∞ ||m0||L1 . And there is a unique solution 1 X(ξ, t) to the Lagrange dynamics which is a strictly monotonic function of ξ for any t ∈ [0, Tmax): Xξ(·, t) > 0. As t approaching Tmax, we prove that the classical solution m(·, t) in Eulerian coordinates has a unique limit m(·, Tmax) in Radon measure space and there is a point ξ0 such that Xξ(ξ0, Tmax) = 0 which means Tmax is an onset time of collisions of characteristics. We also show that in some cases peakons are formed at Tmax. After Tmax, we regularize the Lagrange dynamics to prove global existence of weak solutions m in Radon measure space.
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