Local Well-Posedness and a Continuation Criterion for a Camassa–Holm Equation with a Gentle Bottom

We study a Camassa–Holm-type equation with a prescribed, time-independent bottom profile, mt+(u+h(x))mx+2uxm+12hx(x)m=0,m=(1−∂x2)u. The model is considered here as a mathematically motivated bottom-modified Camassa–Holm equation. The bottom modifies the transport velocity and the lower-order term is chosen so that the basic momentum balance keeps the same cancellation structure as in the flat-bottom case. We clarify the meaning of a gentle bottom in terms of bounded multiplier norms of the prescribed profile and do not claim a complete asymptotic derivation from the Euler equations. Under suitable regularity assumptions on h, we prove local well-posedness in Sobolev spaces by verifying the hypotheses of Kato’s quasilinear semigroup theorem. We also derive an L2 momentum identity and a continuation criterion based on the integrability of ∥ux∥L∞. The proof of the continuation criterion is strengthened by combining the momentum bound with a high-order Hs energy estimate.