Abstract
In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well-posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability.By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.
Highlights
We consider the C-H equation with a forcing in the following form: ut − utxx + 3uux = 2uxuxx + uuxxx + ku, (1.1)
We remark that in [4], the authors provided a new characteristic method to study the existence of global conservative solutions for the C-H equation without forcing, relying heavily on the conservation of energy
We extend the arguments for the C-H equation without a forcing in [4, 6] to the forcing C-H equation (1.2) by using only the balance law (1.5)
Summary
We remark that in [4], the authors provided a new characteristic method to study the existence of global conservative solutions for the C-H equation without forcing, relying heavily on the conservation of energy There are two main difficulties to study the existence and uniqueness of global weak solutions for the forcing C-H equation (1.2): one is the loss of conservation of energy, that is,. We point out that in this paper we verified that the methods for both existence and uniqueness of weak solutions used in [4, 6] can be extended to Eq(1.2) by exploring only a balance law (1.5) instead of a conservation of energy. This idea has been used for a generalized wave equation with a higher-order nonlinearity in our forthcoming paper [9]
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