We can use partial differential equations to describe different physical phenomena. As humans, we would like to not only understand these dynamics, but we would like to control the dynamics. This begins with making sure the model used is simple enough for analytical and numerical results while being complex enough to describe the relevant phenomena.
I am currently working on a 3D-1D model that describes the relevant information of a 3D-3D model used to describe fluid flowing through a vein that is traveling through a tissue. This study involves ensuring that the 3D-3D model is well-posed, using asymptotics to derive a relevant 3D-1D model, showing this model is well-posed and completing an error analysis. This project is joint work Laurel Ohm.
For some biological models, it is important to model both the dynamics of the solid tissue and the fluid dynamics. For these types of problems, one can study fluid flow through deformable, porous media. These problems are known as fluid-solid mixtures and have been studied considerably through well-posedness analysis and numerical investigations. The applications of these studies involve optimizing the biomechanical and fluid-dynamical responses of the fluid-solid mixture. Mathematically, this translates to optimization or optimal control problems subject to a partial differential equations (PDEs) system and the associated sensitivity analysis of the solution with respect to relevant physical or biological parameters.
I have studied relevant control and optimization related problems for fluid flows through porous media with applications to biomechanics. Results of this work could aid research in preventing microstructural tissue damage and designing bioengineered tissues. Most questions related to the sensitivity analysis and control of porous media flows are open in the field. I am focused on (1) well-posedness of optimal solutions and (2) necessary optimality conditions, which characterize the optimal control. These results are crucial for subsequent sensitivity calculations, parameter estimation, and derivative-based optimization algorithms. The optimal control problems for this model was work I did with Lorena Bociu.
In connection with an application to Glaucoma, we learned that it was important to connect the dynamics of blood flow through the body to the local dynamics in the tissue. Thus, Lorena Bociu, Matthew Broussard, Giovanna Guidoboni, and I studied the well-posedness of Biot’s equations coupled with an ODE. We also completed numerics for this coupled system with Daniele Prada.
Another important PDE is the Cahn-Hilliard equation. Variations of this equation have been used in biological problems, but can also be used in a variety of other applications involving phase separation. I have been studying the derivation of Mullins-Sekerka, a PDE describing the behavior of the interface between phases governed by Cahn-Hilliard, and how that can be used on similar equations such as a Non-reciprocal Cahn-Hilliard equation. This is joint work with Daniel Gomez and Yoichiro Mori.
I am also working on a control problem for a discrete semi-linear wave equation. We show that this system can be controlled to a flock with a bounded control and are working to show that you can drive the system from one flock to another. This differs from the previous control problems I have worked on since we are not seeking an optimal control, but rather are driving the system exactly to what we want it to be. Moreover, since this is a discrete wave equation, this is an ODE control problem rather than a PDE control problem. This is joint work with Hung Tran and Minh-Binh Tran.
I am looking forward to continuing these research projects and also using the tools learned in these projects to answer questions about dynamics and our ability to control dynamics in a variety of applications.
Publications and Preprints