Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces

TL;DR

This paper extends classical Sobolev inequalities to noncommutative Euclidean spaces, deriving key inequalities and applying them to establish well-posedness and decay properties of nonlinear PDEs in this setting.

Contribution

It introduces Sobolev, Gagliardo-Nirenberg, and Nash inequalities in noncommutative Euclidean spaces and explores their implications for nonlinear PDEs and heat equation decay.

Findings

Sobolev inequality established in noncommutative Euclidean spaces

Global well-posedness of nonlinear PDEs shown

Decay rates for heat equation solutions derived

Abstract

In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo-Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the noncommutative Euclidean space. Moreover, we show that the logarithmic Sobolev inequality is equivalent to the Nash inequality for possibly different constants in this noncommutative setting by completing the list in noncommutative Varopoulos's theorem in [37]. Finally, we present a direct application of the Nash inequality to compute the time decay for solutions of the heat equation in the noncommutative setting.