Existence, uniqueness, and regularity for the Kuramoto--Sakaguchi equation with unboundedly supported frequency distribution

Abstract

The Kuramoto-Sakaguchi (or simply Kuramoto) equation is considered when the "frequency distribution", the frequency being an independent variable in the model equation, has unbounded support. This equation is a nonlinear, Fokker-Planck-type, parabolic integro-differential equation, and arises from the statistical description of the dynamical behavior of populations of infinitely many nonlinearly coupled random oscillators. The space-integral term in the equation accounts for mean-field interaction occurring among these oscillators. Existence, uniqueness, and regularity of solutions are established here, taking suitable limits in the formulation of the previously studied problem, where the aforementioned support was assumed to be bounded.