# Diego Ayala

### Research

The main area of my research lies in the intersection of Analysis applied to Partial Differential Equations, Optimization and Numerical Analysis. My current PhD project is related to the saturation of energy estimates in the context of Navier-Stokes equation in 2 and 3 dimensions, with periodic boundary conditions.

The Navier-Stokes equation is a set of partial differential equations describing the dynamics of fluids in motion. The physical quantities involved in the equations are the velocity field $$\mathbf{u}$$, the pressure $$p$$, the density $$\rho$$ and the coefficient of kinematic viscosity $$\nu$$.

Depending on the physical context, boundary conditions are imposed on both the velocity field and the pressure. From a mathematical perspective, the use of periodic boundary conditions simplifies the treatment of the problem. Moreover, under the assumption of an incompressible, homogeneous fluid, the Initial-Value Problem can be written as: $\begin{array}{rcll} \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} & = & -\frac{\nabla p}{\rho} + \nu\triangle\mathbf{u} & \quad\mbox{in}\,\Omega\times(0,T] \\ \nabla\cdot\mathbf{u} & = & 0 & \quad\mbox{in}\,\Omega\times(0,T] \\ \mathbf{u}(\mathbf{x},0) & = & \mathbf{u}_0(\mathbf{x}) & \quad\mathbf{x}\in\Omega \end{array}$

For 3D flows, the proof of existence of smooth global solutions for arbitrary initial data is still an open problem (see, The Millenium Problems, from the Clay Mathematics Institute). In 2D, it is known that smooth solutions exist for $$t>0$$ and all initial data. However, one can still use 2D models to gain valuable understanding of 3D phenomena.

Some videos are available in the corresponding section.

### Publications

• D. Ayala, B. Protas. Maximum Palinstrophy Growth in 2D Incompressible Flows.
Journal of Fluid Mechanics, 742:340-367, 2014. [ arXiv:1305.7259 ]
• D. Ayala, B. Protas. Vortices, Maximum Growth and the Problem of Finite Time Singularity Formation.
Fluid Dynamics Research, 46 (2014) 031404. [ arXiv:1307.3589 ]
• D. Ayala, B. Protas. On Maximum Enstrophy Growth in a Hydrodynamic System.
Physica D, 2011. [PDF file]

### Videos

The following videos were generated using a numerical solver based on Krylov Subspaces. The initial conditions are given as the fields that maximize the instantaneous generation of Palinstrophy P(t).

 $$\mathcal{P}(0) = 10^7$$ $$T = 0.3$$ $$\omega_0(x)=\arg\max \frac{d\mathcal{P}}{dt}$$ fixed $$\mathcal{P}_0$$ $$\mathcal{P}(0) = 1.7\times 10^5$$, $$\mathcal{K}(0) = 1$$ T = 1.0 $$\omega_0(x)=\arg\max \frac{d\mathcal{P}}{dt}$$ fixed $$\mathcal{P}_0$$, $$\mathcal{K}_0$$ $$\mathcal{P}(0) = 1.7\times 10^6$$, $$\mathcal{K}(0) = 1$$ T = 1.0 $$\omega_0(x)=\arg\max \frac{d\mathcal{P}}{dt}$$ fixed $$\mathcal{P}_0$$, $$\mathcal{K}_0$$