Title: Hölder regularity of the 2D dual semigeostrophic equations via analysis of linearized Monge-Ampère equations,
Abstract: We obtain the Hölder regularity of time derivative of solutions to the dual
semigeostrophic equations in two dimensions when the initial potential density
is bounded away from zero and infinity. Our main tool is an interior Hölder
estimate in two dimensions for an inhomogeneous linearized Monge-Ampère
equation with right hand side being the divergence of a bounded vector field.
As a further application of our Hölder estimate, we prove the Hölder
regularity of the polar factorization for time-dependent maps in two dimensions
with densities bounded away from zero and infinity. Our applications improve
previous work by G. Loeper who considered the cases of densities sufficiently
close to a positive constant.