Abstract

We consider a two dimensional electroconvection model which consists of a nonlinear and nonlocal system coupling the evolutions of a charge distribution and a fluid. We show that the solutions decay in time in \(L^2({{\mathbb {R}}}^2)\) at the same sharp rate as the linear uncoupled system. This is achieved by proving that the difference between the nonlinear and linear evolution decays at a faster rate than the linear evolution. In order to prove the sharp \(L^2\) decay we establish bounds for decay in \(H^2({{\mathbb {R}}}^2)\) and a logarithmic growth in time of a quadratic moment of the charge density.

中文翻译：

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$${\mathbb {R}}^2$$ 中电对流模型解的长时间行为

摘要

我们考虑一个二维电对流模型，它由耦合电荷分布和流体演化的非线性和非局部系统组成。我们证明解在\(L^2({{\mathbb {R}}}^2)\)中随时间衰减，其速率与线性非耦合系统相同。这是通过证明非线性和线性演化之间的差异以比线性演化更快的速度衰减来实现的。为了证明\(L^2\)急剧衰减，我们在\(H^2({{\mathbb {R}}}^2)\)中建立衰减界限以及二次矩时间的对数增长的电荷密度。