This paper shows a Liouville-type law for the incompressible Stokes system and a trace principle over the uniform domains for the BMO-Q spaces $(-\Delta)^{-\frac{\alpha}{2}}\mathscr{L}^{2,2\alpha} (\mathbb R^{n})$ with $(\alpha,n-1)\in [0,1)\times\mathbb N$.

We consider a Mean Field Games model where the dynamics of the agents is subdiffusive. According to the optimal control interpretation of the problem, we get a system involving fractional time-derivatives for the Hamilton-Jacobi-Bellman and the Fokker-Planck equations. We discuss separately the well-posedness for each of the two equations and then we prove existence and uniqueness of the solution to the Mean Field Games system.

We consider the Cauchy problem of third order Benjamin-Ono type equations on the torus. Nonlinear terms may yield derivative losses, which prevents us from using the classical energy method. In order to overcome that difficulty, we add a correction term into the energy. We also use the Bona-Smith type argument to show the continuous dependence.

Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the long-time behavior of some geometric flows of closed plane curves without a convexity assumption.