PHASE MIXING FOR SOLUTIONS TO 1D TRANSPORT EQUATION IN A CONFINING POTENTIAL

Consider the linear transport equation in 1D under an external confining potential Phi: partial derivative(t)f + v partial derivative(x)f - partial derivative(x)Phi partial derivative(v)f = 0. For Phi = x(2)/2 + epsilon x(4)/2 (with epsilon > 0 small), we prove phase mixing and quantitative decay estimates for partial derivative(t)phi := -Delta(-1) integral(R) partial derivative(t) f dv, with an inverse polynomial decay rate O(< t > (-2)). In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov-Poisson system in 1D under the external potential Phi.