双向Whitham方程的一个条件适定性结果
A conditional well-posedness result for the bidirectional Whitham equation
摘要：
我们考虑双向Whitham方程的初值问题，该方程是一个将不可压缩Euler方程的完全双向色散关系与典型浅水非线性相结合的系统.我们证明了局部适定性在经典的索伯列夫空间在本地化以及周期的情况下，使用平方根型变换对称的系统。存在性理论需要一个非零的表面高程，表明问题是不适定的更一般的初始数据。
英文摘要：
We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local well-posedness in classical Sobolev spaces in the localised as well as the periodic case, using a square-root type transformation to symmetrise the system. The existence theory requires a non-vanishing surface elevation, indicating that the problem is ill-posed for more general initial data.
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