Local Well and Ill Posedness for the Modified KdV Equations in Subcritical Modulation Spaces
| Autor: | Chen, Mingjuan, Guo, Boling |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q}(\mathbb{{R}})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map fails to be $C^3$ continuous in $M^{s}_{2,q}(\mathbb{{R}})$ when $s<1/4$. It is well-known that $H^{1/4}$ is a critical Sobolev space of mKdV so that it is well-posedness in $H^s$ for $s\geq 1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s'}$ with $s'<1/4$. Noticing that $M^{1/4}_{2,q} \subset B^{1/q-1/4}_{2,q}$ is a sharp embedding and $H^{-1/4}\subset B^{-1/4}_{2,\infty}$, our results contains all of the subcritical data in $M^{1/4}_{2,q}$, which contains a class of functions in $H^{-1/4}\setminus H^{1/4}$. Comment: The paper was completed and submitted on 1st May, 2018 |
| Databáze: | arXiv |
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