Schedule, Notes/Handouts & Videos

Various models for the flow of a slightly compressible fluid through a saturated deformable porous medium are described. The classical porous medium equation can model the evolution of porosity of the medium and the pressure of the fluid in one dimension. This primitive model has been extended to reflect detailed mechanics of the medium by the Biot system that consists of a diffusion equation for the fluid flow coupled to a momentum equation for the deformable porous solid consolidation. The small-strain constitutive laws may include nonlinear or degenerate relations of elasto-visco-plastic type or variational inequalities of contact mechaincs. We shall describe the non-linear evolution equations in Hilbert space that characterize these problems for which the existence or uniqueness of a global weak or strong solution is proved by means of monotonicity methods. In order to exploit the available data for the variation of solid porosity with fluid pressure, the simple porous medium model in $\Re$ can be extended to a two-phase system in $\Re^3$ that contains general visco-elastic media with hysteresis arising from irreversible damage. This leads to a mathematical theory in $L^1$ for compaction in sedimentary basins.

A stent is a mesh that is placed in a narrowed or closed part of a blood vessel. They play a crucial role in the treatment of various cardiovascular diseases by providing mechanical support to arteries and facilitating blood flow. The mechanical properties of vascular stents, such as radial strength, flexibility, and conformability, are critical for successful deployment and stability within the arterial environment. Thus optimal design of these medical devices is very important. In this study, we address a series of optimal design problems with parameters related to stent strut thickness and the geometry of struts. The optimization algorithm is of the gradient type and efficient since it is grounded on the elastic stent model with struts modeled by the one-dimensional curved rod model. This research contributes to advancing the precision and efficacy of cardiovascular interventions by refining the design parameters critical to stent performance.

In this presentation, we discuss recent progress and ongoing open questions in space-time modeling, their discretization and numerical solution of coupled problems. Under the terminology coupled problems, we understand nonstationary, nonlinear, coupled PDE system and variational inequalities (CVIS). First, we have made progress in goal-oriented a posteriori error control and adaptivity with the dual-weighted residual method for incompressible flow (Navier-Stokes equations) and fluid-structure interaction. Second, we extended those concepts to space-time model order reduction with application to porous media problems, so far prototype benchmarks such as Mandel's problem. Third, ongoing discussions with engineering colleagues yield new concepts in space-time variational material modeling, satisfying thermodynamical principles and internal variables, based on Hamilton's principle, in which open questions, specifically to us, the mathematical community, persist.

In this talk, we discuss our recent work concerning a nonstandard implementation of the Babuska-Brezzi Theorem, by way of ascertaining strongly continuous semigroup generation for a certain coupled compressible flow -- incompressible fluid dynamics. The coupling between these two disparate dynamics is enacted via a boundary interface. The modus operandi, entailed in this (continuous) mixed variational formulation, allows for the derivation of a (discrete) finite element method (FEM) with which to numerically approximate the fluid-flow solution variables. This is joint work with Paula Egging.

We establish the maximal hyperbolic development of Cauchy data for the multi-dimensional compressible Euler equations. For an open set of compressive and generic H7 initial data, we construct unique H7 solutions to the Euler equations in the maximal spacetime region such that at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-2 surface of “first singularities” called the pre-shock; second, a downstream hypersurface emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. We develop a new geometric framework for the description of the acoustic characteristic surfaces which is based on the Arbitrary Lagrangian Eulerian (ALE) framework, and combine this with a new type of differentiated Riemann-type variables which are linear combinations of gradients of velocity and sound speed and the curvature of the fast acoustic characteristic surfaces. With these new variables, we establish uniform H7 Sobolev bounds for solutions to the Euler equations without derivative loss and with optimal regularity. This is the first result on the maximal hyperbolic development of compressive Cauchy data in all regions of spacetime.   This is joint work with Vlad Vicol.

We consider the interaction between an incompressible, viscous fluid and a poroelastic medium modelled by a Biot equation. Both linear and nonlinear cases are considered. In the linear case, the fluid flow and the poroelastic structure are coupled across a fixed interface, while in the nonlinear case, they are coupled via the moving interface. We discuss the existence of weak solutions for both cases and highlight the primary mathematical challenges associated with the analysis of fluid-poroelastic structure interactions (FPSI), particularly focusing on the difficulties arising from studying the moving boundary problem. Furthermore, we explore numerical methods for solving FPSI problems. The presented results are joint work with L. Bociu, M. Bukač, S. Čanić, J. Kuan and J. Webster.

In the lecture I will survey recent progress on regularity estimates for solutions of Navier-Stokes equations interacting with an elastic shell. The shell is assumed to be perfectly elastic, which means that it is governed by a hyperbolic evolution. The deformation of the shell prescribes a part of the fluid domain, which makes the problem inherently non-linear. It will be shown how the ''parabolic effect'' of the fluid suffices to show results for weak solutions to the coupled fluid-structure interactions previously known for the Navier-Stokes equations in fixed domains. In particular I will discuss the validity of the so-called Ladyzhenskaya-Prodi-Serrin condition for regularity and weak-strong uniqueness for 3D Navier Stokes in the context of fluid-structure interactions. The lecture is based on works in collaboration with D. Breit, P. R. Mensah, B. Muha, M. Sroczinski and P. Su.

Circulating tumor cells (CTCs) are malignant cells that detach from the primary tumor and enter the bloodstream. Early detection of CTCs is vital for diagnosis, yet it poses a challenge due to their low frequency in blood samples. Microfluidic devices emerge as a promising detection method, actively enriching CTCs through external fields or passively separating them based on physical properties. Our collaborator at Texas Tech University has proposed a microfluidic device to isolate CTCs, experimenting with various micro-post sizes and layouts for optimal capture efficiency. However, the intricate transport and adhesion behaviors of CTCs in blood cell suspensions remain not fully understood. In this study, we introduce a cell-based numerical approach, employing the Lattice Boltzmann method, to assess the binding behavior and trajectories of CTCs under diverse flow conditions. This includes variations in cell size, coating density, microfluidic design, and cell collisions. Validated results from our approach contribute to the enhancement of the microfluidic device design. Furthermore, we plan to integrate the Long Short-Term Memory (LSTM) approach to analyze the trajectory and capture location of each CTC, aiming to achieve subphenotype classification.