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UBC Theses and Dissertations

Study and analysis of some incompressible fluid PDEs : the Navier–Stokes equations in the half space, the MHD and the viscoelastic Navier–Stokes equations, and coupled Keller–Segel-fluid models Lai, Chen-Chih


The present dissertation is split in three parts. The first considers the (unrestricted) Green tensor of Stokes system in the half-space. We derive the first ever pointwise estimates of such tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, and explore some applications of the pointwise estimates. The second part of this dissertation considers the magnetohydrodynamics equations (MHD equations) and the viscoelastic Navier–Stokes equations with damping. We construct self-similar and discretely self-similar solutions of both the MHD equations and the viscoelastic Navier–Stokes equations with damping with large initial data in the critical weak Lebesgue space. The third part of this dissertation deals with the Patlak–Keller–Segel–Navier–Stokes system. We prove the global existence of free-energy solutions with critical and subcritical mass.

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