Logarithmic convexity and the "slow evolution" constraint in ill-posed initial value problems

This paper examines a wide class of ill-posed initial value problems for partial differential equations, and surveys logarithmic convexity results leading to Holder-continuous dependence on data for solutions satisfying prescribed bounds. The discussion includes analytic continuation in the unit disc, time-reversed parabolic equations in L-p spaces, the time-reversed Navier-Stokes equations, as well as a large class of nonlocal evolution equations that can be obtained by randomizing the time variable in abstract Cauchy problems. It is shown that in many cases, the resulting Holder-continuity is too weak to permit useful continuation from imperfect data. However, considerable reduction in the growth of errors occurs, and continuation becomes feasible, for solutions satisfying the slow evolution from the continuation boundary constraint, previously introduced by the author.