Global Convergence of a Sticky Particle Method for the Modified Camassa--Holm Equation

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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 $u_x$ are space-time BV functions. The total variation of $m(\cdot ,t)=u(\cdot, t)-u_{xx}(\cdot,t)$ is bounded by the total variation of the initial data $m_0$. We also obtain $W^{1,1}(\mathbb{R})$-stability of weak solutions when solutions are in $ L^\infty(0,\infty;W^{2,1}(\mathbb{R}))$. (Notice that peakon weak solutions are not in $W^{2,1}(\mathbb{R})$.) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.