Well-posedness and weak rotation limit for the Ostrovsky equation
Well-posedness and weak rotation limit for the Ostrovsky equation
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DOI：
10.1016/j.jde.2009.09.009
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发表时间：
2009-12
影响因子：
2.4
通讯作者：
K. Tsugawa
中科院分区：
文献类型：
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作者：
K. Tsugawa
我们考虑Ostrovsky方程的Cauchy问题。首先利用Fourier限制范数方法证明了各向异性Sobolev空间Hs，a中的时间局部适定性，其中s>−a/2−3/4，0 <$a <$−1.这个结果包括了在s>−3/4的情况下，对于正耗散和负耗散，即对于βγ>0和βγ<0，在Hs中的时间局部适定性。接下来我们考虑弱旋转极限。证明了当旋转参数γ趋于0且KdV方程的初值在L2内时，Ostrovsky方程的解收敛于KdV方程的解.为了证明这一结果，我们证明了一个关于γ是一致的双线性估计.
We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space Hs,awith s>−a/2−3/4 and 0⩽a⩽−1 by the Fourier restriction norm method. This result include the time local well-posedness in Hswith s>−3/4 for both positive and negative dissipation, namely for both βγ>0 and βγ<0. We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L2. To show this result, we prove a bilinear estimate which is uniform with respect to γ.