Determine whether analyticity yields control over larger times

Determine whether analytic regularity can provide control of the nonlinear Vlasov-HMF dynamics near compactly supported inhomogeneous stationary states over times larger than the finite time scale established for finite Sobolev regularity.

Background

The paper proves nonlinear stability and decay estimates for perturbations of compactly supported, inhomogeneous stationary states of the Vlasov-HMF model, but only on large finite time intervals whose length depends on the perturbation size. The analysis is carried out at finite Sobolev regularity and uses modulated action-angle coordinates.

The authors note that analyticity may not straightforwardly extend the time interval because the nonlinearity loses the convolution structure in action-angle coordinates, preventing the usual propagation of echoes to high frequencies. Whether analytic regularity can nevertheless yield control over larger times is left unresolved.