A Relation of the Allen–Cahn equations and the Euler equations and applications of the equipartition

IF 1.1 4区 数学 Q2 MATHEMATICS, APPLIED
Dimitrios Gazoulis
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引用次数: 1

Abstract

Abstract We will prove that solutions of the Allen–Cahn equations that satisfy the equipartition of the energy 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. Also, we will determine the structure of solutions of the Allen–Cahn system in two dimensions that satisfy the equipartition. In addition, we apply the Leray projection on the Allen–Cahn system and provide some explicit entire solutions. Finally, we obtain some examples of smooth entire solutions of the Euler equations. For specific type of initial conditions, some of these solutions can be extended to the Navier–Stokes equations. The motivation of this paper is to find a transformation that relates the solutions of the Allen–Cahn equations to solutions of the minimal surface equation of one dimension less. We prove this result for equipartitioned solutions in dimension three.
Allen-Cahn方程与Euler方程的关系及均分的应用
摘要证明了满足能量均分的Allen-Cahn方程的解可以转化为恒压条件下的Euler方程的解。因此,我们得到了De Giorgi型结果,即整个解的水平集是超平面。同时,我们将确定满足均分的二维Allen-Cahn系统的解的结构。此外,我们将Leray投影应用于Allen-Cahn系统,并给出了一些显式全解。最后,给出了欧拉方程光滑全解的一些例子。对于特定类型的初始条件,其中一些解可以推广到Navier-Stokes方程。本文的动机是寻找一种将Allen-Cahn方程的解与一维最小曲面方程的解联系起来的变换。我们在三维的等分解中证明了这个结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.70
自引率
8.30%
发文量
75
审稿时长
>12 weeks
期刊介绍: Nonlinear Differential Equations and Applications (NoDEA) provides a forum for research contributions on nonlinear differential equations motivated by application to applied sciences. The research areas of interest for NoDEA include, but are not limited to: deterministic and stochastic ordinary and partial differential equations, finite and infinite-dimensional dynamical systems, qualitative analysis of solutions, variational, topological and viscosity methods, mathematical control theory, complex dynamics and pattern formation, approximation and numerical aspects.
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