On Periodic KP-I Type Equations
On Periodic KP-I Type Equations
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DOI：
10.1007/pl00005577
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发表时间：
2001-08
影响因子：
2.4
通讯作者：
J. Saut;N. Tzvetkov
中科院分区：
文献类型：
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作者：
J. Saut;N. Tzvetkov
我们分析与周期性 KP-I 型方程相关的双线性估计。我们证明，在 ?×ℝ 上提出的五阶 KP-I 方程的柯西问题在能量空间中是全局适定的。尽管失去了在 KdV 和 KP-II 类型方程中使用的平滑关系（这对于恢复导数损失至关重要），但由于平滑关系失效的傅里叶模式集的某些特定属性，我们仍然能够获得平滑。 KP-I 型方程背景下的反例在 ?×? 上提出被给予。这些例子表明，与 KP-II 情况相反，通过使用布尔干空间的迭代方案很难获得周期性 KP-I 型方程的局部适定性。
We analyze bilinear estimates related to periodic KP-I type equations. We prove that the Cauchy problem for the fifth order KP-I equation, posed on ?×ℝ, is globally well-posed in the energy space. Despite the loss of a smoothing relation (which is essential to recuperate the derivative loss), used in the context of the KdV and KP-II type equations, we are able to gain a moothing thanks to some specific properties of the set of Fourier modes where the smoothing relation fails. Counterexamples in the context of KP-I type equations posed on ?×? are given. These examples indicate that contrary to the KP-II case it would be difficult to obtain the local well-posedness for the periodic KP-I type equations by an iteration scheme using Bourgain spaces.