# Analysis Seminar

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*

Ancestral lineages in mutation selection equilibria with moving optimum

We investigate the evolutionary dynamics of a population structured in phenotype, subjected to trait dependent selection with a linearly moving optimum and an asexual mode of reproduction. The model consists of a non-local and non-linear parabolic PDE. Our main goal is to measure the history of traits when the population stays around an equilibrium. We define an ancestral process based on the idea of neutral fractions. It allows us to derive quantitative information upon the evolution of diversity in the population along time. First, we study the long-time asymptotics of the ancestral process. We show that the very few fittest individuals drive adaptation. We then tackle the adaptive dynamics regime, where the effect of mutations is asymptotically small. In this limit, we provide an interpretation for the minimizer of some related optimization problem, an Hamilton Jacobi equation, as the typical ancestral lineage.

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

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*Online*

Nodal count, Morse index, and all that

Nowadays it is well known that they are formed by nodal lines of eigenfunctions of an appropriate operator (in this talk, Dirichlet Laplacian), splitting the whole membrane into nodal domains. In spite of this long history, many things about nodal lines (surfaces) and domains are still not understood well and are subjects of active investigations. In this talk we will survey the status of understanding of the so-called "nodal count," i.e. the number $\nu_n$ of nodal domains of the $n$th eigenfunction. The standard Sturm ODE theorem says that in 1D $\nu_n=n$. This, however, is incorrect in higher dimensions, and all but finitely many eigenfunctions develop a positive "nodal deficiency" $n-\nu_n$. The talk will contain a survey of some open problems and results obtained in the last decade concerning the nodal deficiency.

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

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*Online*

Organizational Meeting

The Analysis, Dynamics and Applications Seminar will have an organizational meeting January 19, 2021 at 12:30pm via Zoom. If you would like to speak this semester or nominate someone to speak this semester, please attend! If you cannot attend, you can write to either of the organizers – Chris Henderson Henderson, Christopher ckhenderson@arizona.edu or Leonid Friedlander friedlan@math.arizona.edu

If you would like to subscribe to seminar announcements, please send an email to: appliedmath@math.arizona.edu

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

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*Online*