Low Regularity Solutions for Gravity Water Waves II: The 2D Case
Low Regularity Solutions for Gravity Water Waves II: The 2D Case
复制标题
重力水波 II 的低规律性解决方案：2D 案例
作者：
Albert Ai
重力水波方程描述了不可压缩、不可旋转的流体在重力作用下表面的演化。这个系统适定性的经典正则性阈值要求初始速度场在$$H^S$$H S，$$S>；裂缝{1}{2}+1$$S>；1 2+1中，并且可以通过证明标准能量估计来获得。这一门槛通过额外证明Strichartz估计有损失而得到改善。本文建立了$$S>；分式{1}{2}+1-分式{1}{8}$S>；1 2+1-1 8的适定性结果，相应地证明了无损的Strichartz估计。对于将Strichartz估计与能量估计相结合的方法，这提供了清晰的正规性阈值。
The gravity water waves equations describe the evolution of the surface of an incompressible, irrotational fluid in the presence of gravity. The classical regularity threshold for the well-posedness of this system requires initial velocity field in $$H^s$$ H s , with $$s > frac{1}{2}+ 1$$ s > 1 2 + 1 , and can be obtained by proving standard energy estimates. This threshold was improved by additionally proving Strichartz estimates with loss. In this article, we establish the well-posedness result for $$s > frac{1}{2}+ 1 - frac{1}{8}$$ s > 1 2 + 1 - 1 8 , corresponding to proving lossless Strichartz estimates. This provides the sharp regularity threshold with respect to the approach of combining Strichartz estimates with energy estimates.