The Kuramoto–Sivashinsky equations (KSE) arise in many diverse scientific areas, and are of much mathematical interest due in part to their chaotic behavior, and their similarity to the Navier–Stokes equations. However, very little is known about their global well-posedness in the 2D case. Moreover, regularizations of the system (e.g., adding large diffusion, etc.) do not seem to help, due to the lack of any control over the $L^2$ norm. In this work, we propose a new “reduced” 2D model that modifies only the linear part of (the vector form of) the 2D KSE in only one component. This new model shares much in common with the 2D KSE: it is 4th-order in space, it has an identical nonlinearity which does not vanish in energy estimates, it has low-mode instability, and it lacks a maximum principle. However, we prove that our reduced model is globally well-posed. We also examine its dynamics computationally. Moreover, while its solutions do not appear to be close approximations of solutions to the KSE, the solutions do seem to hold many qualitative similarities with those of the KSE. We examine these properties via computational simulations comparing solutions of the new model to solutions of the 2D KSE.

Kuramoto-Sivashinsky方程（KSE）出现在许多不同的科学领域，并且由于其混沌行为以及与Navier-Stokes方程的相似性而引起了人们的极大数学兴趣。但是，对于它们在2D情况下的整体适定性知之甚少。此外，由于缺乏对系统的任何控制，因此系统的正则化（例如，添加较大的扩散等）似乎无济于事。${\mathrm{大号}}^2$规范。在这项工作中，我们提出了一个新的“精简” 2D模型，该模型仅在一个分量中修改了2D KSE（矢量形式）的线性部分。这个新模型与2D KSE有很多共同点：它在空间上是4阶的，具有相同的非线性，其能量估计不会消失，具有低模不稳定性，并且缺乏最大原理。但是，我们证明了我们的简化模型在全球范围内是正确的。我们还通过计算来检查其动力学。此外，尽管其解决方案似乎与KSE的解决方案并不十分接近，但这些解决方案似乎确实与KSE拥有许多定性相似之处。我们通过比较新模型的解决方案与2D KSE解决方案的计算仿真来检查这些属性。