Abstracts

Jonathan Stanfill, Baylor University (February 1, 2022)

Title: Spectral zeta functions and zeta regularized functional determinants for singular Sturm-Liouville operators

Abstract: We employ a recently developed unified approach to the computation of traces of resolvents and zeta functions to compute spectral zeta functions associated with singular (three-coefficient) self-adjoint Sturm-Liouville differential expressions. We then discuss the zeta regularized functional determinant, illustrating what extends from a recent treatment of regular expressions and what remains open. As an application, we consider the generalized Bessel equation on a finite interval and a regularized singular problem.

Jun-Yi Guo, School of Earth Sciences, The Ohio State University (March 10, 2022)

Title: Solution of the PDEs governing the Earth's infinitesimal elastic deformation based on spherical harmonic expansions

Abstract: The Earth responds to several external forcing mainly as an elastic body; the effect of the deviation from pure elasticity is extremely small or even negligible. The most known such forcings include the tidal generating force (i.e., the difference of gravitational attraction of extraterrestrial celestial bodies on different parts of the Earth) and the surface loading (i.e., the variation of pressure on the Earth's surface and that of the gravitational attraction to the interior of the Earth related to the mass transport in the atmosphere-hydrosphere-cryosphere system). For global geophysical problems such as those mentioned above, the change of self-gravitation of the Earth related to the deformation should be considered. Hence, the governing PDEs are Navier's equation of elasticity and Poisson's equation of gravitational potential combined. The focus of this talk is on elastic surface loading.

The geodetic data are mainly surface displacement measured using Global Navigation Satellite System (GNSS) and variation of gravity measured by satellite or surface gravimetry. An important application of these data is to infer the mass transport in the atmosphere-hydrosphere-cryosphere system. Therefore, the more accurate the response of the solid Earth to the external forcing is modeled, the more accurate the mass transport could be inferred, provided that the data are accurate enough.

Up to present, the most used is the spherically symmetric non-rotating elastic isotropic (SNREI) Earth model, where all material parameters are functions of the distance to the center of the sphere, and the effect of rotation (Coriolis force and centrifugal force) is neglected. There are surfaces of discontinuity of the material parameters, which are also spherical. In the solution of the PDEs, all unknown functions are expressed in the form of spherical harmonic series with coefficients as functions of the distance to the center of the sphere. The mass of load on the Earth's surface is also expressed as a spherical harmonic series. Upon substitution into the PDEs and boundary conditions (including the pressure on the Earth's surface), all terms of different degree or order are decoupled, and a system of ODEs is obtained for the coefficients of each degree and order. In fact, the ODEs are independent of the order, and the solution is the same for all orders of the same degree. As a result, the surface displacement and the change of gravitational potential of the Earth are represented using a set of Love numbers defined based on the solution of the ODEs for each degree. In other words, the Love numbers relate the deformation of the Earth to the mass of load on the surface of the Earth in the form of spherical harmonic series.

In this talk, a generalization of the spherical harmonic approach for the SNREI Earth model is to be presented. An Earth model with rotation and lateral heterogeneity superimposed on the SNREI Earth model is to be studied based on the perturbation method. The lateral heterogeneity includes not only the dependence of the Earth's material parameters on latitude and longitude in the bulk (including the effect of the Earth's ellipticity), but also the deviations of surfaces of discontinuity of material parameters from a sphere (including the Earth's surface where the mass of load is located). These make the governing PDEs much more complicated. The formulation includes not only the derivation of the PDEs for the perturbation in the bulk, but also the projection of boundary conditions on each undulating boundary onto a sphere. The PDEs of the perturbation, together with the boundary conditions, are again converted to ODEs by expressing the unknown functions in the form of spherical harmonic series with coefficients as functions of the distance to the center of the sphere. However, the perturbation part of the Earth's deformation caused by any degree and order forcing spreads to the whole spectrum of the spherical harmonic expansion, I.e., for any degree/order mass of load, a system of ODEs is to be solved for all degree/order terms of the perturbation. Therefore, although Love numbers are also defined based on the solutions of the ODEs to represent the surface displacement and the change of gravitational potential of the Earth, a lot more of them are required; they depend not only on the degree and order of the mass of load, but also the degree and order of the deformation. Nevertheless, this is the most straightforward approach for such a complicated Earth model to be adopted by geodesists who are traditionally using the Love numbers of the SNREI Earth model.

Lorena Bociu, North Carolina State University (March 24, 2022)

Title: Analysis and Control in Fluid Flows through Deformable Porous Media

Abstract: Fluid flows through deformable porous media are relevant for many applications in biology, medicine and bio-engineering, including tissue perfusion and fluid flow inside cartilages and bones. We are interested in perfusion inside the eye and its connection to the development of glaucoma. Mathematically, the problem translates into the study of a quasi-static nonlinear poroelastic system, which is a system of PDEs of mixed parabolic-elliptic type. We answer questions related to ocular tissue biomechanics via well-posedness, sensitivity analysis, and optimal control for the PDE coupled system applied to the eye.

Yunbai Cao, Rutgers University (April 14, 2022)

Title: Lipschitz continuous solutions of the Vlasov-Maxwell systems with a conductor boundary condition

Abstract: In this talk we consider relativistic plasma particles subjected to an external gravitation force in a 3D half-space whose boundary is a perfect conductor. When the mean free path is much bigger than the variation of electromagnetic fields, the collision effect is negligible. As an effective PDE, we study the relativistic Vlasov-Maxwell system and its local-in-time unique solvability in the space-time locally Lipschitz space, for several basic mesoscopic (kinetic) boundary conditions: the inflow, diffuse, and specular reflection boundary conditions. This is joint work with Chanwoo Kim.

Tien Khai Nguyen, North Carolina State University (April 21, 2022)

Title: Shocks interaction for the Burgers-Hilbert Equation

Abstract: In 2009 J. Biello and J. Hunter derived a balance law modeling nonlinear waves with constant frequency, obtained from Burgers' equation by adding the Hilbert transform as a source term. For general L^2(R) initial data, the global existence of entropy weak solutions was proved by Bressan and Nguyen in 2014, together with a partial uniqueness result. Recently, unique piecewise continuous solutions with a single shock and the shock formation have been recently studied. This talk will describe a further type of local generic singularities for solutions, namely, points where two shocks interact.

Jiaxin Jin, The Ohio State University (April 28, 2022)

Title: Damping of kinetic transport equation with diffuse boundary condition

Abstract: We first prove that exponential moments of a fluctuation of the pure transport equation decay pointwisely almost as fast as $t^{-3}$ when the domain is any general strictly convex subset of $\R^3$ with the smooth boundary of the diffuse boundary condition. The key of the proof is to establish a novel $L^1$-$L^\infty$ framework via stochastic cycles. Next we consider the model in an upper half space $T^2 \times \R_{+}$ subjected to gravitation-dominant conservative force field. At the boundary, the molecules bounces back following the non-isothermal diffusive reflection boundary condition. Then we show the moments of a fluctuation of the pure transport equation has exponential decay.

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