The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H 1
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RESEARCH
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The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H (R) Shaoyong Lai1* , Nan Li1 and Yonghong Wu2 *
Correspondence: [email protected] 1 Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China Full list of author information is available at the end of the article
Abstract The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space C([0, ∞) × R) ∩ L∞ ([0, ∞); H1 (R)) under the assumption that the initial value u0 (x) only belongs to the space H1 (R). The limit of viscous approximations, a one-sided super bound estimate and a space-time higher-norm estimate for the equation are established to prove the existence of the global weak solution. MSC: 35G25; 35L05 Keywords: global weak solution; Camassa-Holm type equation; existence
1 Introduction In this work, we investigate the Cauchy problem for the nonlinear model ut – utxx + ∂x f (u) = ux uxx + uuxxx – λuN+ + βum uxx ,
()
where λ ≥ , β ≥ , f (u) is a polynomial with order n, N and m are nonnegative integers. When f (u) = ku + u , λ = , β = , Eq. () is the standard Camassa-Holm equation [–]. In fact, the nonlinear term –λuN+ + βum uxx can be regarded as a weakly dissipative term for the Camassa-Holm model (see [, ]). Here we coin () a weakly dissipative CamassaHolm equation. To link with previous works, we review several works on global weak solutions for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions of the standard Camassa-Holm equation have been proved by Constantin and Escher [], Constantin and Molinet [] and Danchin [, ] under the sign condition imposing on the initial value. Xin and Zhang [] established the global existence of a weak solution for the Camassa-Holm equation in the energy space H (R) without imposing the sign conditions on the initial value, and the uniqueness of the weak solution was obtained under certain conditions on the solution []. Under the sign condition for the initial value, Yin and Lai [] proved the existence and uniqueness results of a global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. Lai and Wu [] obtained the existence of a local weak solution for Eq. () in the lower-order Sobolev space H s (R) with ≤ s ≤ . For other meaningful methods to handle the problems relating to dy© 2013 Lai et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Lai et al. Boundary Value Problems 2013, 2013:26 http://www.boundaryvalueproblems.com/content/2013/1/26
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namic properties of the Camassa-Holm equation
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