This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an L 2 maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the estimates from our paper [5] to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.
We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces B α of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example.
Abstract. We show that the maximal Nevanlinna counting function and theCarleson function of analytic self-maps of the unit disk are equivalent, up to constants.
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