Sharp ill-posedness results for the KdV and mKdV equations on the torus

By Luc Molinet. 2011

73 citations recorded by the source corpus.

Abstract We establish a new a priori bound for L 2 -bounded sequences of solutions to the mKdV equations on the torus. This first enables us to construct weak solutions in L 2 for this equation and to check that the “solutions” constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-maps associated with the mKdV and the KdV equations are discontinuous for the H s ( T ) topology for respectively, s 0 and s − 1 . These last results are sharp.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b93a7363be31c656610c48dad3c1861bf64861f9

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In Mathematical Physics

1. Carlos E. Kenig
2. Rowan Killip
3. Monica Visan
4. B. Dodson
5. Aynur Bulut
6. Didier Pilod
7. Terence Tao
8. F. Merle

In Statistical and Nonlinear Physics

1. Carlos E. Kenig
2. Rowan Killip
3. Monica Visan
4. Didier Pilod
5. B. Dodson
6. F. Merle
7. Shuanglin Shao
8. Justin Holmer

Citations in the corpus

• Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces (2018, unknown): “On the one hand, Molinet [41] applied the short-time Fourier restriction norm method and proved that the solutions constructed in [31] are indeed distributional solutions.”
• Well-posedness for the fifth-order KdV equation in the energy space (2012, methodology): “Following [31], we use that (ηk)k≥0 is dyadic partition of the unity and the Cauchy-Schwarz inequality in τ ′ to get that I := 2 αl 2 ∥∥η≤l̃(τ − w(ξ)) ∫”
• KdV is well-posed in H–1 (2018, background): “The papers [51, 52] show that well-posedness cannot persist (in either geometry) in H for any s < −1.”