Non-uniform dependence on initial data for the CH equation on the line

By A. Alexandrou Himonas, Carlos E. Kenig. Differential and Integral Equations, 2009

179 citations recorded by the source corpus.

For s > 3=2 two sequences of CH solutions living in a bounded set of the Sobolev space H s (R) are constructed, whose distance at the initial time is converging to zero while at any later time is bounded below by a positive constant. This implies that the solution map of the CH equation is not uniformly continuous in H s (R).

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

View Paper: https://www.semanticscholar.org/paper/d8f2403a7f457faa6da32f7fe648882f8c124a99

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Non-Uniform Dependence for the Periodic CH Equation (2010, result): “In the non-periodic case the analogous result to Theorem 1 has been proved recently in [10] for s > 1.”
Well-posedness and Continuity Properties of the Fornberg-Whitham Equation in Besov Spaces (2016, unknown): “This was motivated by the works of Himonas and Holliman [14, 15, 16], Himonas and Kenig [17], Himonas, Kenig and Misio lek [18],”
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.”
Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems (2016, background): “While the solution map of these equations is continuous, it is not uniformly continuous (see [37] and [38] for CH, [35] for DP, [34] NE, and [39] for FORQ).”
Non-uniform dependence on initial data for the Camassa-Holm equation in Besov spaces (2020, methodology): “Himonas–Misio lek [26] obtained the first result on the non-uniform dependence for the CH equation in H(T) with s ≥ 2 using explicitly constructed travelling wave solutions, which was sharpened to s > 3 2 by Himonas–Kenig [23] on the real-line and Himonas–Kenig–Misio lek [24] on the circle.”