Global convergence of a sticky particle method for the modified Camassa-Holm equation

In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) - uxx(•, t) is bounded by the total variation of the initial data m0. We also obtain W^1,1(ℝ)-stability of weak solutions when solutions are in L^∞ (0, ∞; W^1,2(ℝ)). (Notice that peakon weak solutions are not in W^1,2(ℝ).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.

Duke Scholars

Author Jian-Guo Liu Physics