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