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Existence and ill-posedness for fluid PDEs with rough data

University of British Columbia

It has been of great interest in recent decades to know whether the incompressible Euler equations are well-posed in the borderline spaces. In order to understand the behavior of solutions in these spaces, the logarithmically regularized 2D Euler equations were introduced. In the borderline Sobolev space, the local wellposedness was proved by Chae-Wu when the regularization is sufficiently strong, while strong ill-posedness of the unregularized case was established by Bourgain-Li. The first part of the dissertation closes the gap between the two results, by establishing the strong ill-posedness in the remaining intermediate regime of the regularization. The second part of the thesis considers the Cauchy problem of incompressible 3D Navier-Stokes equations with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data are either for perturbations of constants or when the velocity gradients are in L^p with finite p. This work presents how to construct global weak solutions for non-decaying initial data whose local oscillations decay, no matter how slowly.

Text

2019-04-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69981

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/