2006/08/01 by Giuseppe Maria Coclite, Kenneth H. Karlsen · 41 citations
Mathematics · Physics and Astronomy · #A priori and a posteriori #Advanced Mathematical Physics Problems #Applied mathematics #Burgers' equation #Camassa–Holm equation #Compact space #Conservation law #Conservation of mass #Dissipative system #Limit (mathematics) #Mathematical analysis #Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Nonlinear system #Partial differential equation #Peakon #Physics #Scalar (mathematics) #Shallow water equations #Waves and shallow water
paper · open access · doi:10.1080/03605300600781600
published in Communications in Partial Differential Equations 31(8), 1253-1272 (Taylor & Francis)
openalex publication_date 2006/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
We consider a shallow water equation of the Camassa–Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.