Illposedness for dispersive equations: Degenerate dispersion and Takeuchi--Mizohata condition
| Autor: | Jeong, In-Jee, Oh, Sung-Jin |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We provide a unified viewpoint on two illposedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi--Mizohata condition. Our approach is based on a robust energy- and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydynamics. Concretely, the main results in this paper concern strong illposedness of the Cauchy problem (e.g., non-existence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schr\"odinger- and KdV-type equations, including the Hunter--Smothers equation, $K(m, n)$ models of Rosenau--Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of combination of two effects: degenerate dispersion -- which is a property of the principal term in the presence of degenerating coefficients -- and the evolution of the amplitude governed by the Takeuchi--Mizohata condition -- which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical $L^{2}$-illposedness result by Mizohata for linear variable-coefficient Schr\"odinger equations with failed Takeuchi--Mizohata condition. Comment: 54 pages |
| Databáze: | arXiv |
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