RIT on Applied PDE Archives for Fall 2024 to Spring 2025
Organizational Meeting
When: Mon, September 18, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Organizers (UMD) -
Abstract: This is the organizational meeting for the RIT on PDE for the Fall 2023 semester.
Stochastic fluid-structure interaction
When: Mon, September 25, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Jeffrey Kuan (UMD) - https://math.berkeley.edu/~jkuan/
Abstract: In this talk, I present recent work on stochastic fluid-structure interaction systems involving the coupled dynamics of fluids interacting with elastic structures under the additional influence of stochastic (random) effects. Fluid-structure interaction arises in real-life applications to biomedical, civil, and mechanical engineering, and there has been recent interest in quantifying and understanding the impact of stochastic effects on coupled fluid-structure dynamics. I will focus on a well-posedness result for a stochastic fully coupled fluid-structure system describing the dynamics of a Stokes flow through a channel with elastic walls, under the additional influence of stochastic forcing in time. We will discuss a constructive existence proof which employs an operator splitting scheme to semi-discretize the full problem in order to construct approximate solutions. We then discuss how to use methods from both fluid-structure interaction and stochastic PDEs in order to pass to the limit in the approximate solutions. This methodology provides a robust mathematical framework for analyzing a variety of complex fully coupled stochastic systems of interest in fluid-structure interaction. This is joint work with Sunčica Čanić at University of California, Berkeley.
Supercritical incompressible NSE regularity-Part I
When: Mon, October 2, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Hussain Ibdah (UMD) - https://www.math.umd.edu/~hibdah/
Abstract: This is the first of a series of 3 (possibly more) lectures where I will discuss how to show that $L^1_tC_x^{0,\beta}$ solutions to the incompressible NSE are regular (https://arxiv.org/abs/2305.17882). In this first talk, I will give an overview of the key ideas involved.
Supercritical incompressible NSE regularity-Part II
When: Mon, October 9, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Hussain Ibdah (UMD) - https://www.math.umd.edu/~hibdah/
Abstract: We will continue discussing the paper https://arxiv.org/abs/2305.17882
Supercritical incompressible NSE regularity-Part III
When: Mon, October 16, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Hussain Ibdah (UMD) - https://www.math.umd.edu/~hibdah/
Abstract: We will continue discussing the paper https://arxiv.org/abs/2305.17882
K41 Theory in Bounded Domains
When: Mon, October 30, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Ethan Dudley (UMD) -
Abstract: Kolmogorov's Theory of Turbulence, otherwise known as K41 Theory, can be characterized as the persistence of energy dissipation in the inviscid limit. Recent work has rigorously connected these ideas to the movement of energy to smaller and smaller length scales over the Torus in $\R^3$. I consider the case of a bounded domain where I identify the emergence of small scale structures and the persistence of dissipation to the blow up of the drag forces along the boundary and the creation of a global energy equality for Euler.
History of the Equations of Fluid Dynamics I: Foundations
When: Mon, November 13, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Charles D. Levermore (UMD) - https://www.math.umd.edu/~lvrmr/
Abstract: TBD
History of the Equations of Fluid Dynamics II: Edifices
When: Mon, November 27, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Charles D. Levermore (UMD) - https://www.math.umd.edu/~lvrmr/
Organizational Meeting
When: Mon, January 29, 2024 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Hussain Ibdah, Jeffrey Kuan, and Huy Nguyen (UMD) -
Abstract: This is the organizational meeting for the Spring 2024 RIT on PDE
How Symmetries relate to Conservation Laws: An Introduction to Noether Theory"
When: Mon, February 12, 2024 - 3:00pmWhere: MTH1311
Speaker: Charles D. Levermore
Non-self-adjointness and nonlinear stability in a free-boundary model of cell motion
When: Mon, February 26, 2024 - 3:00pmWhere: MTH1311
Speaker: Clarke Alexander Safsten
Abstract: Steady cell motion may be modeled as traveling wave solutions to PDE models in which the cell membrane constitutes a free boundary. I present 1D and 2D models for this motion which combine a Keller-Segel PDE, a Hele-Shaw boundary condition, and the Young-Laplace law with a nonlocal regularizing term which precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. The free boundary and the active and nonlocal terms in this model lead to non-self-adjoint operators when linearizing about traveling waves. Since traditional linear stability analysis using eigenvalues is insufficient for non-self-adjoint operators in the infinite dimensional case, I will demonstrate an alternative technique using spectral analysis. Finally, I will show that traveling waves are not only linearly but nonlinearly stable.
On Moffat's Magnatic Relaxation Equations
When: Mon, March 4, 2024 - 3:00pmWhere: MTH1311
Speaker: Sepher Mohammadkhani
Traveling wave solutions to the free boundary incompressible Navier-Stokes equations
When: Mon, March 11, 2024 - 3:00pmWhere: MTH1311
Speaker: Seyed Abdolhamid Banihashemi
An Introduction to Kinetic Equations: Conservation, Dissipation and Equilibration
When: Mon, March 25, 2024 - 3:00pmWhere: MTH1311
Speaker: Charles D. Levermore
An introduction to kinetic models of flocking
When: Mon, April 1, 2024 - 3:00pmWhere: MTH1311
Speaker: Jeffrey Kuan
Mechanical Models of Tumor Growth
When: Mon, April 15, 2024 - 3:00pmWhere: MTH1311
Speaker: Maeve Wildes
Abstract: I will present a paper by Perthame, Quiros and Vazquez in which they present two tumor growth models and show that the solutions of one model are an asymptotic limit of the solutions of the other model.
Quantitative stability and error estimates for optimal transport plans
When: Mon, April 29, 2024 - 3:00pmWhere: MTH1311
Speaker: Gonzalo Alejandro Benavides
Volume-preserving mean-curvature flow as a singular limit of a Keller-Segel-type system
When: Mon, May 6, 2024 - 3:00pmWhere: MTH1311
Speaker: Michael Rozowski
Abstract: We study the well-posedness of weak solutions and a singular limit of a class of elliptic-parabolic Keller-Segel systems. The system arises as a model for the aggregation of some organisms following a chemical signal that they themselves produce. The organism density is described by an elliptic equation featuring a degenerate, nonlinear diffusion (modeling a volume-exclusion principle obeyed by the organisms) and an advective term (modeling an attractive chemotaxis effect). The chemical signal towards which the organisms are attracted obeys a simple parabolic equation that is forced by the organism density. By recognizing the parabolic equation obeyed by the chemoattractant as an Allen-Cahn-type equation with a nonlocal reaction term, existence of weak solutions to the overall system is established by exploiting the underlying gradient flow structure and applying a minimizing movements scheme. Uniqueness is established using Lipschitz continuity of the mapping from the chemoattractant concentration to the organism density and a Gronwall argument. We then study the singular limit of these weak solutions in the regime of slow chemoattractant diffusion and fast chemical reaction. We establish a phase separation property, where both the chemoattractant concentration and the organism density converge to (scalar multiples of) characteristic functions of the same set of finite perimeter. It is then shown that these limits evolve by volume-preserving mean-curvature flow, under an additional assumption that the time-integrated energy the elliptic-parabolic system converges to the time-integrated energy of the limiting system.
This is a joint work with my advisor, Antoine Mellet.