A local sensitivity analysis in Landau damping for the kinetic Kuramoto equation with random inputs

Abstract

We present a local sensitivity analysis in Landau damping for the kinetic Kuramoto equation with random inputs. The kinetic Kuramoto equation governs the temporal-phase dynamics of the one-oscillator distribution function for an infinite ensemble of Kuramoto oscillators. When random inputs are absent in the coupling strength and initial data, it is well known that the incoherent state is nonlinearly stable in a subscritical regime where the coupling strength is below the critical coupling strength which is determined by the geometric shape of the distribution function for natural frequency. More precisely, the Kuramoto order parameter measuring the fluctuations around the incoherent state tends to zero asymptotically and its decay mode depends on the regularity(smoothness) of natural frequency distribution function and initial datum. This phenomenon is called Landau damping in the Kuramoto model in analogy with Landau damping arising from plasma physics. Our analytical results show that Landau damping is structurally robust with respect to random inputs at least in a subcritical regime. As in the deterministic setting, the decay mode for the derivatives of the order parameter in the random space can be either algebraic or exponential depending on the regularities of the initial datum and natural frequency distribution, respectively, and the smoothness for the order parameter in the random space is determined by the smoothness of the coupling strength.