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LOCAL WELL-POSEDNESS IN LOW REGULARITY OF THE MKDV EQUATION WITH PERIODIC BOUNDARY CONDITION

机译:具有周期边界条件的MKDV方程低正则性的局部适定性

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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 moreover 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.
机译:我们通过修正由于布尔加因(Bourgain)引起的傅立叶限制方法,研究了一维圆环上的mKdV方程的柯西问题的低规则性的局部适定性。我们以H〜s表示当地的适定性,s> 1/3。在s> 1/4的情况下,我们证明了H〜s中解的局部存在,并且在对初始数据的某些附加假设下,证明了H〜s中的适定性。为了证明这一点,我们修改了傅立叶约束范数,以考虑到由非线性相互作用引起的解的相位振荡。

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