Well-posedness and derivative blow-up for a dispersionless regularized shallow water system. (4th October 2019)

• Record Type: Journal Article
• Authors: Liu, Jian-Guo; Pego, Robert L; Pu, Yue
• Is Part Of: Nonlinearity. Volume 32:Number 11(2019)
• Page Start: 4346
• Page End: 4376
• Publication Date: 2019-10-04
• Subjects: Saint-Venant equations -- Green–Naghdi equations -- weak solutions -- nonlocal hyperbolic system -- long waves -- breakdown; 35B44 -- 35B60 -- 35Q35 -- 76B15 -- 35L67; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab2cf1
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We study local-time well-posedness and breakdown for solutions of regularized Saint-Venant equations (regularized classical shallow water equations) recently introduced by Clamond and Dutykh. The system is linearly non-dispersive, and smooth solutions conserve an H 1 -equivalent energy. No shock discontinuities can occur, but the system is known to admit weakly singular shock-profile solutions that dissipate energy. We identify a class of small-energy smooth solutions that develop singularities in the first derivatives in finite time.