Well-posedness of the Cauchy problem for models of large amplitude internal waves
Well-posedness of the Cauchy problem for models of large amplitude internal waves
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大振幅内波模型柯西问题的适定性
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发表时间：
2010
期刊：
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通讯作者：
J. Saut
中科院分区：
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作者：
P. Guyenne;D. Lannes;J. Saut
本文考虑Choi和Camassa（1999 J. Fluid Mech.396 1-36）、克雷格等人（2005 Commun. Pure. 58 1587-641）（一维界面）和Bona等人（2008 J.Math.PuresAppl.89 538-66）（二维界面），用于描述不同密度的不混溶流体的两层界面处的大振幅内波。对于一维界面，这个系统是双曲型的，它的局部适定性并没有引起严重的困难，虽然其他问题（爆破，双曲性损失等）变得微妙。对于二维界面，系统是非局部的。尽管如此，我们证明了它保存的一些属性的“双曲型”，并表明相关的柯西问题是局部适定的适当的Sobolev类提供了一些自然的限制数据。这些结果说明了数值模拟的冲击波的形成为重点。
We consider in this paper the ‘shallow-water/shallow-water’ asymptotic model obtained in Choi and Camassa (1999 J. Fluid Mech. 396 1–36), Craig et al (2005 Commun. Pure. Appl. Math. 58 1587–641) (one-dimensional interface) and Bona et al (2008 J. Math. Pures Appl. 89 538–66) (two-dimensional interface) from the two-layer system with rigid lid, for the description of large amplitude internal waves at the interface of two layers of immiscible fluids of different densities. For one-dimensional interfaces, this system is of hyperbolic type and its local well-posedness does not raise serious difficulties, although other issues (blow-up, loss of hyperbolicity, etc) turn out to be delicate. For two-dimensional interfaces, the system is nonlocal. Nevertheless, we prove that it conserves some properties of ‘hyperbolic type’ and show that the associated Cauchy problem is locally well posed in suitable Sobolev classes provided some natural restrictions are imposed on the data. These results are illustrated by numerical simulations with emphasis on the formation of shock waves.