On the ill-posedness of some canonical dispersive equations

By Carlos E. Kenig, Gustavo Ponce, Luis Vega. 2001

429 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/9c9e7406e66e6d9ad97c4086e9b906b4c1e1e8df

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Non-uniform dependence on initial data for the CH equation on the line (2009, methodology): “. For s2 non-uniform continuity of the CH solution map in the periodic case was established in [HM] using high frequency traveling wave solutions and following an approach similar to the one used in [KPV] by Kenig, Ponce and Vega. We mention that this method does not work in the non-periodic case be”
Two singular dynamics of the nonlinear Schrödinger equation on a plane domain (2003, background): “3. See Lemma 2.1 below for the precise notion of local well-posedness with uniformly continuous flow map (see also [ KePV ], [BGT2]).”
Sharp well-posedness results for the BBM equation (2008, background): “Bekiranov [1], Kenig, Ponce and Vega [18, 19], Bona, Sun and Zhang [10] and Zhang [23, 24, 25] to name but a few).”
Long time behavior of solutions to the mKdV (2014, background): “Below s = 1 4 the solution map fails to be uniformly continuous [1, 26] although weaker forms of well-posedness hold [2].”
Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr (2005, background): “In order to construct spaces which work all the way down to s ≥ −1, we have to modify the Xs,b spaces somewhat.”
Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems (2016, neutral): “properties for CH type and related nonlinear evolution equations, we refer the reader to [12], [13], [11], [15], [16], [6], [23], [49], [53], [50], [17], [45], [8], [42], [43], [44], [4], [3], [64], [18], [5], [47], [48], [10], [9], [70], and the references therein.”
Non-Uniform Dependence for the Periodic CH Equation (2010, methodology): “The approach followed is similar to the one used in [21] by Kenig et al.”
Invariance of the White Noise for KdV (2009, background): “More recently, Kappeler-Topalov [13] proved the global well-posedness of the KdV in H−1(T), using the complete integrability of the equation.”
Nonlinear wave interactions for the Benjamin-Ono equation. (2004, background): “Moreover, essentially all papers [6, 7, 8, 12] deal with the self interaction of a single high frequency wave which naturally lead to upper bounds on the Sobolev regularity s which is consistent with the smooth dependence on the initial data for large s holding in the cases considered in [6, 7, 8, 1”