Problems of Regularity in Models Arising from Fluid Dynamics

This work expands regularity results for equations related to fluid motion. First, we improve previously known lower bounds for Sobolev norms of potential blow-up solutions to the three-dimensional Navier-Stokes equations in the homogeneous Sobolev space $\dot{H}^{3/2}$. Next, we study the inviscid dyadic model of the Euler equations and prove some regularizing properties of the nonlinear term that occur due to forward energy cascade. We show every solution must have $\frac{3}{5}$ $L^2$-based regularity for all positive time. We conjecture this holds up to Onsager's scaling, where the $L^2$-based exponent is $\frac{5}{6}$. Finally, we prove that a solution $u$ to the three-dimensional Boussinesq equations does not blow-up at time T if $\| u_{\le Q}\|_{B^1_{\infty, \infty}}$ is integrable on $(0, T)$, where $u_{\le Q }$ represents the low modes of Littlewood-Paley projection of the velocity $u$.