论文标题
Whitham-BoussinesQ系统的长期适应性
Long time well-posedness of Whitham-Boussinesq systems
论文作者
论文摘要
考虑了三种不同的完整分散boussinesq系统，这些系统是在浅水中弱非线性表面波的双向传播中引起的渐近模型。我们证明，在初始数据的非浪费条件下，这三个系统在订单$ \ MATHCAL {O}的时间尺度上得到很好的量，其中$ \ varepsilon $是一个小参数，测量了波浪的弱非线性。对于其中一个系统，即使在短时间内，此结果似乎是新的。另外两个系统涉及表面张力，对于其中一个系统，当表面张力很小时，必须锐化非供电条件。证明依赖于合适的对称器和双曲系统的经典理论。但是，我们必须在换向器估计中仔细跟踪小参数，以获得长时间的良好性。 最后，将我们的结果与最近的祖母绿的结果相结合，将这些系统作为水浪模型的全部理由（与经典$（a，b，c，d）$ - boussinesq Systems相比，水浪模型更大。
Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water. We prove that, under a non-cavitation condition on the initial data, these three systems are well-posed on a time scale of order $\mathcal{O}(\frac1\varepsilon)$, where $\varepsilon$ is a small parameter measuring the weak non-linearity of the waves. This result seems new for one of these systems, even for short time. The two other systems involve surface tension, and for one of them, the non-cavitation condition has to be sharpened when the surface tension is small. The proof relies on suitable symmetrizers and the classical theory of hyperbolic systems. However, we have to track the small parameters carefully in the commutator estimates to get the long time well-posedness. Finally, combining our results with the recent ones of Emerald provide a full justification of these systems as water wave models in a larger range of regimes than the classical $(a,b,c,d)$-Boussinesq systems.
