Abstract
Using the Kato theorem for abstract differential equations, the local well‐posedness of the solution for a nonlinear dissipative Camassa‐Holm equation is established in space C([0, T), Hs(R))∩C1([0, T), Hs−1(R)) with s > 3/2. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1 ≤ s ≤ 3/2 is developed.
Highlights
Camassa and Holm 1 used the Hamiltonian method to derive a completely integrable wave equation ut − uxxt 2kux 3uux 2uxuxx uuxxx, 1.1 by retaining two terms that are usually neglected in the small amplitude, shallow water limit
The ideas of proving the second result come from those presented in Li and Olver 25
For any real number s, Hs Hs R denotes the Sobolev space with the norm defined by h Hs
Summary
Camassa and Holm 1 used the Hamiltonian method to derive a completely integrable wave equation ut − uxxt 2kux 3uux 2uxuxx uuxxx, 1.1 by retaining two terms that are usually neglected in the small amplitude, shallow water limit. Li and Olver established the local well-posedness in the Sobolev space Hs R with s > 3/2 for 1.1 and gave conditions on the initial data that lead to finite time blowup of certain solutions. It was shown in Constantin and Escher that the blowup occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. By using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for 1.2 with any β and arbitrary positive integer N in space C 0, T , Hs R C1 0, T , Hs−1 R with s > 3/2. The ideas of proving the second result come from those presented in Li and Olver 25
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