Anomalous Dissipation and Non-uniqueness of Fluid Equations

We introduction some pathological behaviors of the weak solutions to several fluid equations, including Navier-Stokes equations, Euler equations, Electron Magneto Hydrodynamics equations and Surface Quasi-geostrophic equations. Chapter I provides the background for the fluid equations in discussion. In Chapter II, the proof of the main theorem regarding to the anomalous dissipation of Euler and Navier-Stokes equations is given. Chapter III proved the existence of the weak solutions to the stationary EMHD equations via convex integration. Finally, in Chapter IV, using convex integration a non-uniqueness result of the forced SQG equations is proven.