ASYMPTOTIC STABILITY OF THE PHASE-HOMOGENEOUS SOLUTION TO THE KURAMOTO-SAKAGUCHI EQUATION WITH INERTIA

Abstract

We present global-in-time existence and uniqueness of strong solutions around a phase-homogeneous solution, and its large-time behavior for the Kuramoto-Sakaguchi equation with inertia. Our governing equation describes the evolution of the probability density function for a large ensemble of Kuramoto oscillators under the effects of inertia and stochastic noises. In this paper, we take a perturbative framework around the Maxwellian type equilibrium and use the classical energy method together with careful analysis based on the decomposition of the perturbation. We establish the global-in-time existence and uniqueness of strong solutions with large initial data when the noise strength is large enough. For the large-time behavior, we show the exponential decay of solutions toward the equilibrium under the same assumptions as those for the global solutions.