Global solutions for the modified Camassa–Holm equation

Abstract

This paper is concerned with the global solutions for the modified Camassa-Holm equation. We first obtain the global existence and uniqueness of strong solutions for the initial datum 𝑢 0 (𝑥) ∈ 𝐻 𝑠 (ℝ) with 𝑠 > 3∕2, and then give some norm estimates of strong solutions and its momentum density. Finally, by using the compensated compactness method and regularization technique, we establish the global existence and uniqueness of weak solutions for the initial datum 𝑢 0 (𝑥) ∈ 𝐻 1 (ℝ).