On the stationary and uniformly-rotating solutions of active scalar equations
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In this thesis, we study qualitative and quantitative properties of stationary/uniformly- rotating solutions of the 2D incompressible Euler equation and the generalized Surface Quasi- Geostrophic (SQG) equations. The main goal is to establish sufficient and necessary condi- tions for the stationary/uniformly rotating solutions to be radially symmetric. In addition, we also derive quantitative estimates for non-radial, uniformly-rotating patch solutions for the 2D Euler equation. We establish sufficient conditions for stationary/uniformly-rotating solutions for some ac- tive scalar equations to be radially symmetric. The proof is based on a variational argument that a uniformly-rotating solution can be formally thought of as a critical point of an energy functional. We apply this idea to more general active scalar equations (gSQG) and vortex sheet equation. In addition, we construct a non-radial vortex sheet with non-constant vortex strength, which is rotating with angular velocity Ω > 0. We obtain a curve of such non-radial solutions, bifurcating from trivial ones. These results come from the joint work with Javier Go ́mez– Serrano, Jia Shi and Yao Yao. We adapt the variational argument to study non-radial rotating vortex patches. It is well known that for Ω ∈ (0, 1/2 ), there are m-fold symmetric rotating patches. We derive some quantitative estimates for those patches about their angular velocities and the difference with the unit disk.