Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation

By Yi A. Li, Peter J. Olver. 2000

663 citations recorded by the source corpus.

Abstract We establish local well-posedness in the Sobolev space Hs with any s> 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the Camassa–Holm equation. However, unlike the more familiar Korteweg–deVries model, we demonstrate conditions on the initial data that lead to finite time blow-up of certain solutions.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/29e0483482d9a01cbb79efc75ceeec4b2b74c29f

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Non-Uniform Dependence for the Periodic CH Equation (2010, methodology): “This result can also be proved using the method in Li and Olver [19] or Rodriguez–Blanco [26].”
Non-uniform dependence on initial data for the CH equation on the line (2009, methodology): “ween actual and approximate solutions (see Lemma 6). Finally, in section 6 we conclude with the proof of Theorem1. 2. Local well-posedness We shall need the following well-posedness result, proved in [LO] using a regularization technique. Theorem 2. [Li-Olver] Suppose that the function u 0(x) belong”
Equations with Peakon Traveling Wave Solutions (2013, background): “For related results about them, we refer the reader to [2, 5–15, 17–19, 21] and the references therein.”
Stability of the Camassa-Holm solitons (2002, background): “For initial profiles that are less regular, u0 ∈ H s(R) with s > (3)2 , it is known [16] that (1) has a unique solution in C([0, T ); H s(R)) for some T > 0 with E and F conserved.”
On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation (2005, result): “Proposition 2.1 contains as particular cases (for various choices of the constants γ and ω) the well-posedness results obtained in [13, 29 , 30]and [31].”
Well-posedness of modified Camassa-Holm equations (2009, result): “Local well-posedness in the non-periodic case was proved for the initial data in Hs(R) with s > 3/2 by Li and Olver [19] and Rodríguez-Blanco [23].”
Low-Regularity Solutions of the Periodic Camassa–Holm Equation (2007, background): “[9], [35], [8], [9], [10], [11], [14], [15], [17], [24], [30], [31], and references therein).”
Global Weak Solutions for a Shallow Water Equation (2000, background): “For initial profiles u0 2 H s .R/ with s> 3 , it is known [ 16 ,22], that (1.1) has a unique solution in C.T0 ;T/ I H s .R// for some T> 0.”
Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons (2010, methodology): “The idea of proving the existence of the local weak solution is inspired by Li and Olver in [41].”
Global weak solutions for a two-component Camassa–Holm shallow water system (2011, background): “It has been shown that this equation is locally well-posed [13,14,26,41,44] for initial data u0 ∈ H(R), s > (3)2 .”
The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation (2010, methodology): “The ideas of proving these two theorems come from those presented in Hakkaev and Kirchev [24], Li and Olver [32].”
On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey (2004, result): “This result was improved to s > 3/2 by Li and Olver [24] and Rodriguez-Blanco [27].”
Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation (2009, background): “8) where utxx is the damping term and α,b = const > 0, β = const ∈ R (see [7,8,11,20,21] and references therein).”