Dr. Justin Webster awarded a new NSF grant

Scope of the supported project

Dr. Justin Webster's most recent NSF grant is on the mathematical foundations of the motion of a fluid through a deformable medium.  An integral part of the award is the training of doctoral students and postdoctoral collaborators.

Research

Fluid flow can deform a material through which it moves, while deformation of that material can in turn change the flow. This interaction occurs, for example, when groundwater moves through saturated soil or rock, when fluids are injected or extracted from subsurface regions, and when blood move through and around soft biological tissues. These systems are complex, because the fluid and the surrounding material respond to one another in tandem.

This NSF-supported research develops and studies the partial differential equations (PDEs) used to describe such interactions, with the goal of understanding when such mathematical models can be used reliably for prediction, computation, and control of poroelastic filtrations. In particular, Webster and collaborator George Avalos at the University of Nebraska–Lincoln study coupled fluid-poroelastic interaction systems in which a freely moving fluid interfaces with a deformable, fluid-saturated material. Their work develops mathematical foundations for these models, including weak and strong solutions, regularity, spectral properties, and stability, across several physically-motivated regimes and configurations. These results help connect rigorous PDE analysis with computational methods for multiphysics systems arising in engineering, geoscience, and biomechanics.

Training

The collaborative project also supports the training of the next generation of applied mathematicians. At UMBC, the award will support a Ph.D. student, while Webster and Avalos will jointly mentor two doctoral students and a postdoctoral researcher working on problems related to fluid-poroelastic interaction, PDE analysis, and scientific computation.

Related work

• P. Lavagnino, A. Lee, and J. T. Webster, Weak Solutions and Inertial Limits for Quasi-static Filtrations (2026). This paper studies the quasi-static regime and establishes weak solutions through a singular inertial limit.
  Preprint
• G. Avalos, G. Richard, and J. T. Webster, Semigroup Generation for Multilayered Filtrations, Contemporary Mathematics, accepted 2026. The work develops a semigroup framework for a multilayer system coupling free flow, bulk poroelasticity, and a poroelastic plate interface.
  Preprint
• G. Avalos and J. T. Webster, Uniqueness of Weak Solutions for Biot-Stokes Interactions, Pure and Applied Analysis 7 (2025), 1111–1139.
  Journal article · Preprint
• G. Avalos, E. Gurvich, and J. T. Webster, Weak and Strong Solutions for a Fluid-Poroelastic-Structure Interaction via a Semigroup Approach, Mathematical Methods in the Applied Sciences 48 (2024), 4057–4089.
  Journal article · Preprint