Ill-posedness of the Navier-Stokes equations in a critical space in 3D
Ill-posedness of the Navier-Stokes equations in a critical space in 3D
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DOI：
10.1016/j.jfa.2008.07.008
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发表时间：
2008-07
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影响因子：
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通讯作者：
J. Bourgain;Natavsa Pavlovi'c
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文献类型：
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作者：
J. Bourgain;Natavsa Pavlovi'c
我们证明了三维Navier-Stokes方程的Cauchy问题在B˙∞−1，∞上是病态的，即在有限时间内发生“范数膨胀”。更准确地说，我们证明了在B˙∞−1，∞范围内任意小的Schwartz类S的初始数据可以在任意短的时间内产生在B˙∞−1，∞范围内任意大的解。这样的结果意味着解映射本身在原点处的B˙∞−1，∞是不连续的。
We prove that the Cauchy problem for the three-dimensional Navier–Stokes equations is ill-posed in B˙∞−1,∞in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small in B˙∞−1,∞can produce solutions arbitrarily large in B˙∞−1,∞after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in B˙∞−1,∞at the origin.