Nonlinear Analysis ( IF 1.4 ) Pub Date : 2021-02-23 , DOI: 10.1016/j.na.2021.112292 Ricardo Grande , Kristin M. Kurianski , Gigliola Staffilani
This work is dedicated to putting on a solid analytic ground the theory of local well-posedness for the two dimensional Dysthe equation. This equation can be derived from the incompressible Navier–Stokes equation after performing an asymptotic expansion of a wavetrain modulation to the fourth order. Recently, this equation has been used to numerically study rare phenomena on large water bodies such as rogue waves. In order to study well-posedness, we use Strichartz, and improved smoothing and maximal function estimates. We follow ideas from the pioneering work of Kenig, Ponce and Vega, but since the equation is highly anisotropic, several technical challenges had to be resolved. We conclude our work by also presenting an ill-posedness result.
中文翻译:
关于非线性Dysthe方程
这项工作致力于为二维Dysthe方程的局部良好适定性理论奠定坚实的分析基础。在将波列调制渐近扩展到四阶后,可以从不可压缩的Navier-Stokes方程导出此方程。最近,该方程已被用于数值研究大型水体上的稀有现象,例如流浪。为了研究良好的姿势,我们使用了Strichartz,并改进了平滑度和最大函数估计。我们遵循Kenig,Ponce和Vega的开创性工作的想法,但是由于方程是高度各向异性的,因此必须解决一些技术难题。在总结工作时,我们还提出了不适的结果。