Global solutions and ill-posedness for the Kaup system and related Boussinesq systems
Global solutions and ill-posedness for the Kaup system and related Boussinesq systems
复制标题
Kaup 系统和相关 Boussinesq 系统的全局解和不适定性
DOI：
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发表时间：
2019
影响因子：
1.1
通讯作者：
Timur Milgrom
中科院分区：
文献类型：
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作者：
D. Ambrose;J. Bona;Timur Milgrom
一类长峰水波的双向传播近似由Boussinesq型方程组控制。首先介绍了Boussinesq在19世纪70年代，这些方程已提出了各种形式的许多作者。这里考虑的是一类这样的系统，其中包括众所周知的第一次介绍了考普。Kaup系统特别有趣，因为它具有相关的逆散射形式主义，这意味着它的解决方案的相当详细的方面可能是触手可及的。然而，这个系统和其他类似的系统在早期的工作中受到了质疑，因为它们在静止状态周围线性化的初值问题是不适定的。它在这里表明，非线性不会消除这个问题。也就是说，Kaup系统的初值问题和某类Boussinesq型系统的初值问题在Sobolev空间中是不适定的。事实上，它表明，任意小，光滑的解决方案可以在任意短的时间内炸毁在Sobolev空间规范。这个正常的通货膨胀的结果表明，该系统是不是一个很好的候选人在实际问题中使用。
The two-way propagation of a certain class of long-crested water waves is governed approximately by systems of Boussinesq-type equations. First introduced by Boussinesq in the 1870’s, these equations have been put forward in various forms by many authors. Considered here is a class of such system which include the well known one first introduced by Kaup. The Kaup system is especially interesting since it features an associated inverse scattering formalism, which means that quite detailed aspects of its solutions may be within reach. However, this system and others like it were called into question in earlier work because the initial-value problems for their linearizations around the rest state are ill posed. It is here shown that nonlinearity does not erase this problem. That is to say, the initial-value problem for the Kaup system and others in a certain class of Boussinesq-type systems are ill posed in Sobolev spaces. Indeed, it is shown that arbitrarily small, smooth solutions can blow up in arbitrarily short time in Sobolev-space norms. This norm-inflation result indicates the system is not a good candidate for use in practical problems.