论文标题
Allen-Cahn方程以及Euler方程和应用的关系
A Relation of the Allen-Cahn equations and the Euler equations and applications of the Equipartition
论文作者
论文摘要
我们将证明，满足电位的Allen-Cahn方程的解决方案可以以恒定压力转换为Euler方程的解决方案。结果，我们获得了de giorgi型结果，即整个溶液的水平集是超平面。此外，在特定情况下，我们还获得了欧拉方程的整个解决方案的一些示例。对于特定类型的初始条件，其中一些解决方案可以扩展到Navier-Stokes方程。此外，我们将在满足平选方面的两个维度上确定艾伦-CAHN系统的解决方案的结构。最后，我们将Leray投影应用于Allen-Cahn系统，并提供一些明确的整个解决方案。
We will prove that solutions of the Allen-Cahn equations that satisfy the equipartition can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. In addition, we obtain some examples of smooth entire solutions of the Euler equations in particular cases. For specific type of initial conditions, some of these solutions can be extended to the Navier-Stokes equations. Also, we will determine the structure of solutions of the Allen-Cahn system in two dimensions that satisfy the equipartition. Finally, we apply the Leray projection on the Allen-Cahn system and provide some explicit entire solutions.
