Optimal convergence of a second-order low-regularity integrator for the KdV equation

Low regularity solution of a 5th-order KdV equation

The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravitycapillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in H(R) with s > − 4 and the local well-posedness for the modified Kawahara equation in H(...

Low regularity solutions of two fifth-order KdV type equations

The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravitycapillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in H(R) with s > − 4 and the local well-posedness for the modified Kawahara equation in H(...

The Second Order Spectrum and Optimal Convergence

The method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self-adjoint operator. We extend the method to normal operators and find optimal convergence rates for eigenvalues and eigenspaces. The convergence to eigenspaces is new, while the convergence rate for eigenvalues improves on the previous estimate by an order of magnitude.

The second order spectrum and optimal convergence

Second Order Stability for the Monge-ampère Equation and Strong Sobolev Convergence of Optimal Transport Maps

The aim of this note is to show that Alexandrov solutions of the Monge-Ampère equation, with right hand side bounded away from zero and infinity, converge strongly in W 2,1 loc if their right hand side converge strongly in Lloc. As a corollary we deduce strong W 1,1 loc stability of optimal transport maps.