Nonlinear partial differential equations on non-commutative Euclidean spaces

Noncommutative Euclidean spaces, otherwise known as Moyal spaces, are one of the oldest examples in noncommutative geometry. Harmonic analysis on non-commutative Euclidean spaces is similar to harmonic analysis in the familiar commutative setting but has some striking and unexpected differences, especially relating to the structure of the L_p spaces. As a demonstration of these differences, I will explain how the analysis of some nonlinear partial differential equations is radically altered in the strictly noncommutative setting. In particular, the theories of nonlinear Schrodinger equations and fluid flow equations are greatly simplified.

Studying these equations requires the development of some elementary aspects of paradifferential calculus in this setting, which can be built on the theory of double operator integrals.

 

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