Abstract

Noncommutative Euclidean spaces—otherwise known as Moyal spaces or quantum Euclidean spaces—are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the analysis of some equations radically simplifies in the strictly noncommutative setting.

中文翻译：

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非交换欧几里得空间上的非线性偏微分方程

摘要

非交换欧几里得空间（也称为 Moyal 空间或量子欧几里得空间）是非紧非交换几何的标准示例。这些空间的调和分析的最新进展使我们有机会强调它们的一些独特特征。例如，非线性偏微分方程理论在这种非交换设置中具有意想不到的特性。我们开发了非交换欧几里得空间的准微分学的基本方面，并给出了非线性演化方程的一些应用。我们演示了如何在严格非交换设置下从根本上简化一些方程的分析。