Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations
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DOI：
10.1353/ajm.2003.0040
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发表时间：
2002-03
影响因子：
1.7
通讯作者：
Michael Christ;J. Colliander;Terrence Tao
中科院分区：
文献类型：
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作者：
Michael Christ;J. Colliander;Terrence Tao
在最近的一篇论文中，Kenig，Ponce和Vega研究了聚焦非线性薛定谔（NLS），聚焦修正的Korteweg-de弗里斯（mKdV）和复杂的Korteweg-de弗里斯（KdV）方程的低正则性行为。使用孤立子和呼吸器的解决方案，他们证明了缺乏当地的适定性，这些方程低于各自的端点？在本文中，我们研究了这些方程的散焦类似物，即散焦NLS，散焦mKdV，和真实的KdV，都在一维空间中，合适的孤子和呼吸器的解决方案是不可用的。我们为每个这些方程类的修改后的散射解决方案，存在于全球的时间，并渐近相应的线性方程组的解决方案，明确的相移。这些解决方案是用来证明缺乏当地的适定性在某些Sobolev空间，在这个意义上说，依赖于初始数据的解决方案未能一致连续。特别是，我们表明，mKdV流是不一致连续的L2拓扑结构，尽管存在的全球弱解在这个规律性。最后，我们研究了KdV方程在端点正则性H -1/2处的解，并构造了真实的和复KdV方程的解。该构造提供了一个非平凡的时间区间[- T，T ]和一个局部Lipschitz连续映射，该映射将H-H2中的初始数据取为分布解u ∈ C 0（[- T，T ]; H-H2），该分布解u ∈ C 0（[- T，T ]; H-H2）是对所有光滑数据唯一定义的。证明使用广义Miura变换转移现有的端点正则性理论mKdV KdV。
In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrodinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity H -¾ , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [- T, T ] and a locally Lipschitz continuous map taking the initial data in H -¾ to a distributional solution u ∈ C 0 ([- T, T ]; H -¾ ) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.