On the weak solutions to a shallow water equation

By Zhouping Xin, Ping Zhang. 2000

552 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fbb8cebf3892b64c214c77b37afa0f8ebb280323

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation (2010, background): “[45] Z. Xin, P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations 27 (2002) 1815–1844.”
Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces (2012, background): “See, for example, [2], [3], [27], [28] for the construction of weak solutions arising from H1-data.”
Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation (2006, background): “On the other hand, it has global weak solutions in H 1 (R) [2, 12, 15, 46 ].”
Global weak solutions for a two-component Camassa–Holm shallow water system (2011, methodology): “On the other hand, it has global weak solutions in H 1(R) [2,3,15,22,47].”
On the Blow-up Phenomena for the Degasperis–Procesi Equation (2010, background): “As far as we know, the case of the Camassa-Holm equation is well understood by now [7, 10, 11, 12, 15, 35] and the citations therein, while the Degasperis-Procesi equation case is the subject of this paper.”
ON THE UNIQUENESS AND LARGE TIME BEHAVIOR OF THE WEAK SOLUTIONS TO A SHALLOW WATER EQUATION (2002, methodology): “[12] if u is the admissible weak solution of Eq.”
On the well-posedness of the Degasperis-Procesi equation (2006, background): “Existence and uniqueness results for global weak solutions of the Camassa–Holm equation are proved in [4,9,12,34,35,15,16].”
On the blow-up of solutions to the integrable modified Camassa–Holm equation (2014, background): “The existence of global weak solutions was investigated in [25, 26].”
Ill-posedness of the Camassa–Holm and related equations in the critical space (2018, background): “ion E(u) = Z u2 + u2 xdx. 4 ZIHUAGUO,XINGXINGLIU,LUCMOLINET,ANDZHAOYANGYIN So H1 is the most natural setting for the physically relevant weak solutions. Global weak solution in H1 was eastablished in [25]. In the H1 setting one has to associate a measure to keep track of possible singularities and o”
Initial Boundary Value Problems of the Camassa–Holm Equation (2008, background): “On the other hand, it has global weak solutions in H1 I , cf. Bressan and Constantin (2007), Constantin and Molinet (2000), Danchin (2001), Molinet (2004), Wahlén (2006), and Xin and Zhang (2000).”
Fredholm determinants and the Camassa‐Holm hierarchy (2003, background): “Xin and Zhang [15] have proved that a nice (weak) solution of CH may be obtained by introducing a dispersive term (0+) × ∂ 3v/∂3x .”
On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey (2004, background): “In [28] Xin and Zhang proved the existence of global weak solutions of (1.”