Global weak solutions to the generalized mCH equation via characteristics

Abstract

In this paper, we study the generalized modi ed Camassa-Holm(gmCH) equation via characteristics. We rst change the gmCH equation forunknowns (u;m) into its Lagrangian dynamics for characteristics X( ; t), where 2 R is the Lagrangian label. When X ( ; t) > 0, we use the solutions to theLagrangian dynamics to recover the classical solutions with m( ; t) 2 Ck0 (R)(k 2 N; k 1) to the gmCH equation. The classical solutions (u;m) to thegmCH equation will blow up if inf 2R X ( ; Tmax) = 0 for some Tmax > 0.After the blow-up time Tmax, we use a double molli cation method to mollifythe Lagrangian dynamics and construct global weak solutions (with m in space-time Radon measure space) to the gmCH equation by some space-time BVcompactness arguments.