LOCAL WELL-POSEDNESS IN LOW REGULARITY OF THE MKDV EQUATION WITH PERIODIC BOUNDARY CONDITION
LOCAL WELL-POSEDNESS IN LOW REGULARITY OF THE MKDV EQUATION WITH PERIODIC BOUNDARY CONDITION
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DOI：
10.3934/dcds.2010.28.1635
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发表时间：
2010-12-01
影响因子：
1.1
通讯作者：
Tsutsumi, Yoshio
中科院分区：
文献类型：
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作者：
Nakanishi, Kenji;Takaoka, Hideo;Tsutsumi, Yoshio
通过改进Bourgain的Fourier限制方法，研究了一维环面上mKdV方程Cauchy问题在低正则性下的局部适定性.我们证明了H（s）中的局部适定性，s > 1/3。当s > 1/4时，在对初值作一定的附加假设下，证明了解在H（s）中的局部存在性，并进一步证明了其在H（s）中的适定性.为了证明，我们修改了傅立叶限制范数，以考虑到非线性相互作用引起的解的相位的振荡。
We study the local well-posedness in low regularity of the Cauchy problem for the mKdV equation on one-dimensional torus by modifying the Fourier restriction method due to Bourgain. We show the local well-posedness in H(s), s > 1/3. In the case s > 1/4, we prove the local existence of solution in H(s) and more over the well-posedness in H(s) under a certain additional assumption on initial data. For the proof, we modify the Fourier restriction norm to take into account the oscillation of the phase of solution, which is caused by the nonlinear interaction.