# Analysis Seminar

Boundary Integral Formulations in Linear Thermo-Elasto Dynamics

In this theoretical work, we study boundary integral formulations for an interior/exterior initial boundary value problem arising from the thermo-elasto-dynamic equations in a homogeneous and isotropic domain. The time dependence is handled through a passage to the Laplace domain. In the Laplace domain, combined single- and double-layer potential boundary integral operators are introduced and proven to be coercive . Based on the Laplace domain estimates, it is possible to prove the existence, uniqueness and regularity properties of solutions in the time domain, as well as regularity requirements for problem data. This analysis may serve as the mathematical foundation for discretization schemes based on the combined use of the boundary element method and convolution quadrature.

Place: Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*

Some tumor growth models and connections between them

This talk concerns PDEs modeling tumor growth. We show that a novel free boundary problem arises via the stiff-pressure limit of a certain model. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. We also discuss connections between these problems and other PDEs arising in tumor growth modeling. This is joint work with Inwon Kim.

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*

Low degree spline approximation of surfaces and functions

Splines (piecewise polynomial or, more generally, rational functions) are commonly used for numerical approximation of (arbitrary) functions in a variety of contexts. For example, the celebrated Bezier/NURBS curves and surfaces are ubiquitous in geometric design and computer graphics. Some of the most widely used frameworks involve tensor products of 1d cubic polynomials which lead to degree 9 polynomials in 3d. This high degree leads to excessive amounts of data and long evaluation times, which makes the utility of such approximations in real-time environments quite limited, even with the state of the art hardware. One of the ways to deal with this issue is to develop methods for low degree approximation of geometric data. There are, however, some geometric and topological obstructions to this. In my talk I will review the history of this subject, highlight some specific problems and ideas of how to address them.

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*

The Boltzmann equation near and far from equilibrium

If an ideal gas is in thermodynamic equilibrium, the distribution of particle velocities is given explicitly by the Maxwellian (a.k.a. Maxwell-Boltzmann) distribution. If the gas is out of equilibrium, modeling the dynamics is naturally much more difficult. The evolution equation satisfied by the particle density, known as the Boltzmann equation, lacks a suitable well-posedness theory despite its long history and widespread use in statistical physics. So far, global solutions have only been constructed for initial data that is sufficiently close to an equilibrium state. In this talk, I will describe progress from the last few years on the "large-data" or "far-from-equilibrium" regime, including joint work with C. Henderson and A. Tarfulea on local existence, instantaneous filling of vacuum regions, and continuation criteria.

Zoom: https://arizona.zoom.us/j/99410014231 Passworappappd: “arizona” (all lower case)

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*Online*

Geometry and Dynamics of Phase Transitions in Periodic Media

In this talk we discuss recent progress on homogenization of the Allen-Cahn equation in a periodic medium to Brakke's formulation of anisotropic mean curvature flow. The starting point of our work is the recent Gamma-convergence result of Cristoferi-Fonseca-Hagerty-Popovici, and we combine it with the Sandier-Serfaty scheme for the convergence of gradient flows. Along the way, our analysis also provides resolution to a long-standing open problem in geometry: what do distance functions to planes look like, in periodic Riemannian metrics on $R^n$ that are conformal to the Euclidean one? The corresponding question about distance functions to points, has a long history that goes back to Gromov and Burago. This represents joint work with Irene Fonseca (CMU), Rustum Choksi and Jessica Lin (McGill).

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*

Numerical approximation of statistical solutions of hyperbolic systems of conservation laws

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions for multi-dimensional hyperbolic systems of conservation laws. We present a numerical algorithm to approximate statistical solutions of conservation laws and show that under the assumption of ‘weak statistical scaling’, which is inspired by Kolmogorov’s 1941 turbulence theory, the approximations converge in an appropriate topology to statistical solutions. We will show numerical experiments which indicate that the assumption might hold true.

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*

On the inviscid problem for the Navier-Stokes equations

The question of whether the solution of the Navier-Stokes equation converges to the solution of the Euler equation as the viscosity vanishes is one of the fundamental problems in fluid dynamics. In the talk, we will review current results on this problem. We will also present a recent result, joint with Vlad Vicol and Fei Wang, which shows that the inviscid limit holds for the initial data that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the interior.

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*

The effect of surface tension on thin elastic rods

A recent experiment by Mora et al. showed that surface tension can cause a thin elastic rod to spontaneously bend when immersed in a fluid [PRL 111, 114301]. This experiment is remarkable in that it defies common intuition: the typical examples of large deformations caused by surface tension involve objects floating on the surface of a fluid, and the deformations are driven by a difference in surface tension between the two sides of the object. We use nonlinear elasticity and dimension reduction to derive an equation for the energy of a thin rod with surface tension, correctly predicting which rods bent when immersed in fluid and which rods did not.

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*

The impact of Gårding inequalities on uniqueness

In this talk we discuss the question of uniqueness of minimizers of integral quasiconvex functionals. It is known that, in this setting, no uniqueness is to be expected in general. However, the situation is different if we impose suitable smallness conditions on the Dirichlet boundary datum. Employing strategies typically arising in regularity theory, the quasiconvexity condition allows us to establish a Gårding inequality that, in turn, leads to our uniqueness result and other interesting observations regarding uniqueness of minimizers. The talk is based on a joint work with Jan Kristensen.

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Zoom*

Nonexistence of subcritical solitary waves

We consider waves on the surface of an incompressible fluid, in particular localized solitary waves which travel at a constant speed. It is a long-standing conjecture that such waves must be "supercritical", traveling faster than infinitesimal periodic waves. While there are physical grounds for expecting subcritical solitary waves to be extremely rare, it seems impossible to turn these ideas into a rigorous proof. I will outline a surprisingly simple proof of this conjecture, which hinges instead on the properties of an auxiliary function related to momentum conservation. This is joint work with Vladimir Kozlov and Evgeniy Lokharu.

Zoom: https://arizona.zoom.us/j/99410014231 Password: “arizona” (all lower case)

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*Online*