CT2: Topics in PDE Analysis


Session A: Oct.1, 5:00pm-6:20pm, Classroom Building 122
Session B: Oct.2, 10:30am-11:50am, Classroom Building 122


5:00pm-5:20pm (Oct 1)
CLB 122

Himanshu Singh,
University of South Florida

Higher order Liouville weighted composition operators over the Fock space


In recent times, the Liouville operator theory has gradually became as one of the dominant candidates for machine learning of dynamical processes for data driven methods. We intend to augment and study the Liouville operators theory and its interactions over reproducing kernel Hilbert spaces, in particular the Fock Space of analytic functions. This presentation continues the study of the Liouville weighted composition operators to higher dimension over the Fock space and hence the name: Higher order Liouville weighted composition operators. This discussion provides the basic definition of it in connection with the dynamical systems. It also provides various important operator theoretic properties such as closability, boundedness and compactness, as well as essential norm estimates of the operator over the Fock space.

5:20pm-5:40am (Oct 1)
CLB 122
Khoi Vo,
University of California, Riverside

Preliminary report on symmetric and asymmetric cell division and modeling of interacting cell populations in the colonic crypt
  Mathematical modeling can be used to describe the behavior of cells within the colonic crypt. The colon is made up of nearly 10 million crypts which are responsible for producing the epithelial cells within the colon. Symmetric and asymmetric stem cells and cycling cells produce the cells within the crypt and when this behavior becomes dysregulated it can lead to the development of colorectal cancer. This model aims to make a simple spatial and time-dependent model to describe the behavior of two types of cells within the colon. Both analytic and numerical solutions are presented for a range of initial conditions and time points. The model is then expanded for stochastic analysis to further examine the spatial relationships among the cell types.

5:40pm-6:00pm (Oct 1)
CLB 122
Nicholas Fisher,
Minnesota State University, Mankato

Quasi-interpolation for the Helmholtz–Hodge decomposition
  We propose a computationally efficient and stable quasi-interpolation based method for numerically computing the Helmholtz-Hodge decomposition of a vector field. To this end, we first explicitly construct a matrix kernel in a general form from polyharmonic splines such that it includes divergence-free/curl-free/harmonic matrix kernels as special cases. Then we apply the matrix kernel to vector decomposition via a convolution technique together with the Helmholtz-Hodge decomposition. More precisely, we show that if we convolve a vector field with a scaled divergence-free (curl-free) matrix kernel, then the resulting divergence-free (curl-free) convolution sequence converges to the corresponding divergence-free (curl-free) part of the Helmholtz-Hodge decomposition of the field as the scale parameter tends to zero. Finally, by discretizing the convolution sequence via certain quadrature rule, we construct a family of (divergence-free/curl-free) quasi-interpolants (defined both in the whole space and over a bounded domain) for approximating divergence-free/curl-free part corresponding to the Helmholtz-Hodge decomposition of the field, respectively.

6:00pm-6:20pm (Oct 1)
CLB 122

Taige Wang,
University of Cincinnati

Forced oscillations of incompressible Navier-Stokes equation in a 2D bounded domain
  We establish the existence of time-periodic solutions for incompressible Navier-Stokes equation (NSE) posed in 2D domain mainly in Sobolev space Hs(Ω), s = 1. In this situation, fluid is motivated by a time-periodic force in the domain, and generates the forced oscillation (periodic solutions) satisfying a Burgers equation. Further, local and global stability are achieved for this periodic solution.

CT2-B-1 (canceled)
10:30am-10:50am (Oct 2)
CLB 122
Wojciech Ozanski,
Florida State University

Global well-posedness and exponential decay of a model of fluid-structure interaction
  In the talk we will discuss the problem of a fluid-structure interaction, which consists of a incompressible, viscous fluid, described by the 3D Navier-Stokes equations, with a homogeneous Dirichlet boundary condition, and an elastic structure, described in Lagrangian coordinates by the wave equation with linear damping, and equipped with a homogeneous Dirichlet boundary condition. The elastic structure and the fluid interact via a common free boundary, on which we assume continuity of the displacement as well as continuity of the stresses. The most notable feature of the model is the lack of any artificial stabilization terms, which are often used to deduce local well-posedness of the system. It turns out that removing the stabilization terms can in principle cause the system not to decay to zero as time passes. We will discuss how one can analyze various notions of energies of the system to avoid the stability issues related to the lack of the stabilization terms. We will show that the system is globally well-posed for small data, and decays exponentially to a final state which is zero only thanks to the preservation of the volume of the system.

10:50am-11:10am (Oct 2)
CLB 122
John Kyei,
University of South Florida

Densely Defined Multiplication Operators in a NARMAX-type Identification Scheme
  In this presentation, we shall discuss a parsimonious signal approximation technique based on the multiplication operator on a reproducing kernel Hilbert space. Valid trajectories of a dynamical system shall be encoded with occupation kernels in the domain of the adjoint multiplication operator. We shall explore an interaction between the adjoint operator and the above kernel that motivates an alternative formulation of the NARMAX system identification scheme.
This research is conducted in collaboration with Himanshu Singh, Drs. Joel A. Rosenfeld and Benjamin P. Russo and is funded by AFOSR Award FA 9550-20-1-0127 and NSF award ECCS-2027976.

11:10am-11:30am (Oct 2)
CLB 122
Bakhyt Alipova,
University of Kentucky, International IT University

Mathematical modeling of the process of movement of arterial blood in the arteries for angioplasty and stenting of the coronary arteries

Based on the general Navier-Stokes equations and convective heat conduction, a specific boundary value problem (BVP) of CFD is formulated, having determined the calculation area D with the initial and boundary conditions. For a 2-D problem of the flow of a viscous heat-conducting fluid in a channel with internal obstacles (blood particles), the momentum equations, the continuity equation, and the energy equation are considered.

The boundary of the region D is multiply connected, the channel (artery) walls are thermally insulated; To simplify the formulation of the BVP, reduce numerical calculations, and reduce the number of problem parameters, the problem is assumed to be dimensionless.

The numerical implementation of the problem is supposed to be performed by the finite difference method (FDM). It is supposed to build homogeneous and denser grids in time and space. For the selected grid function, a detailed discretization is performed, and the resulting system of linear algebraic equations is solved using the tridiagonal matrix method.

It is expected to develop an virtual reality simulator for angioplasty and coronary artery stenting: Develop an algorithm for the operation of tools, Develop an algorithm for the behavior of organic tissues etc.