The Cauchy problem for an integrable shallow-water equation

By A. Alexandrou Himonas, G. Misiołek. Differential and Integral Equations, 2001

129 citations recorded by the source corpus.

Subfields: Mathematical Physics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b17a7b82fd509f442e5a83c6c8573b83e1569831

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Carlos E. Kenig
2. A. Alexandrou Himonas
3. Luc Molinet
4. Yong Zhou
5. Lorenzo Brandolese
6. Gustavo Ponce
7. Giuseppe Maria Coclite
8. Kenneth Hvistendahl Karlsen

In Numerical Analysis

1. Luc Molinet
2. Yong Zhou
3. A. Alexandrou Himonas
4. Lorenzo Brandolese
5. Yongsheng Li
6. Yimin Zhang
7. Wei Yan
8. Kenneth Hvistendahl Karlsen

In Statistical and Nonlinear Physics

1. Lorenzo Brandolese
2. A. Alexandrou Himonas
3. Carlos E. Kenig
4. Yong Zhou
5. Helge Holden
6. G. Misiołek
7. Luc Molinet
8. Xavier Raynaud

Citations in the corpus

• Non-Uniform Dependence for the Periodic CH Equation (2010, background): “The proof of (i) and (ii) can be found in [11, 23].”
• 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.”
• A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation (2006, background): “Local well-posedness results are proved in Constantin and Escher (1998a), Himonas and Misiołek (2001), Li and Olver (2000), Rodríguez-Blanco (2001).”
• The Cauchy problem for the Novikov equation (2012, methodology): “Ill-posedness of the Novikov equation in H( ) with s < 3/2 In this section, motivated by the idea of [8], we will use the solitary wave solution uc(x, t) = √ ce−|x−x0−ct|, where c > 0 and x0 is a constant, to prove the following Theorem 3.”
• On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey (2004, methodology): “In this direction, in [19], by using the specific form of the solitons, it is shown that the flow-map associated to (1.”
• Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces (2012, background): “= −∞. This phenomenon is also referred as wave breaking. For s &lt; 3/2, the well-posedness of equation (1.1) in Hs is not known (only counterexamples to the uniform well-posedness are available, see [19]). However, existence and uniqueness results of weak solutions have been recently established al”
• A convergent numerical scheme for the Camassa--Holm equation based on multipeakons (2005, background): “Local and global well-posedness results as well as results concerning breakdown are proved in [8, 12, 16, 17].”
• Analyticity of the Cauchy problem for an integrable evolution equation (2003, background): “In another contrast to KdV, local well-posedness for CH is known merely in C 1 or in the Sobolev space H s with s> 3/2 (for a detailed proof of this see [M2] and also Danchin [D] in the periodic case, and Li and Olver [LO] in the nonperiodic case, see also [ HM1 ]).”
• Local well‐posedness and blow‐up criteria of solutions for a rod equation (2005, background): “On the other hand, Himonas and Misiole [10] showed that the Camassa–Holm equation is not well-posed for initial data in H for s < 3/2.”
• Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation (2006, methodology): “For well-posedness results in the periodic case we refer to [ HM1 ], [Mi], and [DKT], where the equation is studied in its integral-dierent ial form (see (1.2) below) as an ODE on the space of diffeomorphisms of the circle.”
• Blow-up of solutions to a nonlinear dispersive rod equation (2006, background): “On the other hand, Himonas and Misiole [10] showed that the Camassa-Holm equation is not well-posed for initial data in Hs for s 3/2.”