PDE-Applied Math Archives for Academic Year 2021

On preservation of moduli of continuity by parabolic evolution

When: Thu, September 9, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Hussain Ibdah (University of Maryland College Park ) - https://www.math.umd.edu/~hibdah/
Abstract: In this talk, we will investigate the propagation of Lipschitz regularity by solutions to various nonlinear, nonlocal parabolic equations. We will locally analyze models such as the Michelson-Sivashinsky equation, incompressible Navier-Stokes system, and advection diffusion problems that include the dissipative SQG equation. Depending on the model, we will either show global well-posedness, derive new regularity criteria, or provide different proofs to and generalize previously obtained results. In particular, we will show that for abstract drift-diffusion problems, it is possible to break certain, supercritical Holder-type barriers and get regularity, which is rather surprising.

Finite time singularities for nematic liquid crystal flow and Landau-Lifshitz-Gilbert equation

When: Thu, September 23, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Yifu Zhou (Johns Hopkins University) - https://math.jhu.edu/~yzhou173/
Abstract: In this talk, we consider nematic liquid crystal flow (NLCF) and Landau-Lifshitz-Gilbert equation (LLG) in \R^2. (NLCF) is a system strongly coupling the nonhomogeneous incompressible Navier-Stokes equation and the transported harmonic map heat flow, while (LLG) serves as the basic evolution equation for the spin fields in the continuum theory of ferromagnetism. We will investigate how parabolic gluing method can be developed and applied to construct finite time blow-up solutions. This talk is based on joint works with Chen-Chih Lai, Fanghua Lin, Changyou Wang, Juncheng Wei and Qidi Zhang.

Weyl formulae for Schrodinger operators with critically singular potentials

When: Thu, October 7, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Xiaoqi Huang (University of Maryland) - https://sites.google.com/view/xiaoqi-math/
Abstract: In this talk we shall discuss generalizations of classical versions of the Weyl formula involving Schr\"odinger
operators $H_V=-\Delta_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular
potentials $V$. In particular, we extend the classical results of Avakumovi\'{c}, Levitan and H\"ormander by obtaining $O(\lambda^{n-1})$ bounds for the error term
in the Weyl formula in the universal case when we merely assume that $V$ belongs to the Kato class,
${\mathcal K}(M)$, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and
bounded from below or has favorable heat kernel bounds. We shall discuss both local point-wise and integral versions of Weyl formulae, and also improvements over the
error term under certain geometric conditions. This is based on joint work with Christopher Sogge and Cheng Zhang.

Evolution of mesoscopic interactions and scattering solutions of the Boltzmann equation

When: Thu, October 14, 2021 - 3:30pm
Where: https://umd.zoom.us/j/92329363134?pwd=KzBtRHo2dm5VUm5PU0hHWnNCSzJ1UT09
Speaker: Nima Moini (Berkley) - https://math.berkeley.edu/~nima/
Abstract: In this talk we will highlight some recent discoveries in kinetic theory. We will develop an uncertainty principle, new a-priori bounds and the concept of a blind cone with respect to an observer for the evolution particles in the mesoscopic scale solely based on conservation laws. We will show that the energy within any bounded set of the spatial variable is integrable over time. We will also discuss a generalization of these results for the 2-particle interactions and establish analogies to Morawetz and interaction Morawetz estimates for the nonlinear Schrodinger equation. We will demonstrate some applications for these estimates by showing that the total mass of the particles and interactions concentrate within a specific collection of arbitrarily acute blind cones with respect to any observer. This implies that, as uncertainty inevitably increases, particles will move away in a radial manner from any fixed observer thereby erasing the angular component of momentum. These results are independent of the specific structure of interactions, therefore they are also true for the Boltzmann equation. We will end the talk with a discussion about the Boltzmann equation. We will show the existence and uniqueness of a specific class of classical solutions to the Boltzmann equation with small initial data near vacuum. These solutions scatter to linear states in the $L^{\infty}$ norm, uniformly within any compact set of the spatial variable. Furthermore, we will discuss the asymptotic completeness of this class of solutions and establish another connection with the case of the nonlinear Schrodinger equation. Notably, this shows that solutions of the Boltzmann equation do not necessarily converge to a Maxwellian but can scatter to linear states arbitrarily close to any prescribed linear state.

Tools for multimodal sampling

When: Thu, October 21, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Michael Lindsey (Courant Institute-NYU) - https://quantumtative.github.io
Abstract: The task of sampling from a probability distribution with known density arises almost ubiquitously in the mathematical sciences, from Bayesian inference to computational chemistry. The most generic and widely-used method for this task is Markov chain Monte Carlo (MCMC), though this method typically suffers from extremely long autocorrelation times when the target density has many modes that are separated by regions of low probability. We present several new methods for sampling that can be viewed as addressing this common problem, drawing on techniques from MCMC, graphical models, and tensor networks.

On the 2D Peskin problem

When: Thu, October 28, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Robert Strain (University of Pennsylvania) - https://www2.math.upenn.edu/~strain/
The Peskin problem models the dynamics of a closed elastic string immersed in an incompressible 2D stokes fluid. This set of equations was proposed as a simplified model to study blood flow through heart valves. The immersed boundary formulation of this problem has been widely and has proven very useful in particular giving rise to the immersed boundary method in numerical analysis. Proving the existence and uniqueness of smooth solutions is vitally useful for this system in particular to guarantee that numerical methods based upon different formulations of the problem all converge to the same solution.

In one project ``The Peskin Problem with Viscosity Contrast'', which is a joint work with Yoichiro Mori and Eduardo Garcia-Juarez, we consider the case when the inner and outer viscosities can be different. This viscosity contrast adds further non-local effects to the system through the implicit non-local relation between the net force and the free interface. We prove the first global well-posedness result for the Peskin problem in this setting. The result applies for medium size initial interfaces in critical spaces and shows instant analytic smoothing. We carefully calculate the medium size constraint on the initial data. These results are new even without viscosity contrast.

In another project ``Critical local well-posedness for the fully nonlinear Peskin problem'', which is a joint work with Stephen Cameron, we consider the case with equal viscosities but with a fully non-linear tension law. This situation has been called the fully nonlinear Peskin problem. In this case we prove local wellposedness for arbitrary initial data in the scaling critical Besov space \dot{B}^{3/2}_{2,1}. We additionally prove the high order smoothing effects for the solution. To prove this result we in particular derive a new formulation of the equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.

This talk will I explain a bit about both of these recent results and their proofs.

Asymptotic stability of kinks in 1D nonlinear scalar field theories

When: Thu, November 4, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Jonas Luhrmann (Texas A&M University) - https://www.math.tamu.edu/~luhrmann/
Abstract: Nonlinear scalar field theories on the line such as the phi^4 model or the sine-Gordon model feature soliton solutions called kinks. They are expected to form the building blocks of the long-time dynamics for these models. In this talk I will survey recent progress on the asymptotic stability problem for kinks and I will present a recent result (joint work with W. Schlag) on the asymptotic stability of the sine-Gordon kink under odd perturbations.

Linear vortex symmetrization: the spectral density function approach and Gevrey regularity

When: Thu, November 11, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Hao Jia (University of Minnesota) - https://www-users.cse.umn.edu/~jia/

Abstract: The two-dimensional incompressible Euler equation is globally well-posed but the long-time behavior is very difficult to understand due to the lack of global relaxation mechanism. Numerical simulations and physical experiments show that coherent vortices often become a dominant feature in two-dimensional fluid dynamics for a long time. The mathematical analysis of vortices, especially in connection to the so-called `vortex symmetrization` problem, has attracted a lot of attention in recent years. In this talk, after a quick review of recent developments in the study of nonlinear asymptotic stability of shear flows and the symmetrization problem for (the special case of) point vortices, we turn to the general vortex symmetrization problem and report a recent result with A. Ionescu for the linearized flow. The linearized problem has been analyzed before by Bedrossian-Coti Zelati-Vicol who obtained control on the profile of the vorticity field in Sobolev spaces with limited regularity. Our main new discovery is that in the vortex problem, unlike the shear flow case, it is no longer possible to obtain smooth control uniformly in time on a single modulated profile for the vorticity field. Rather, there are two such profiles. To address this issue (for future nonlinear applications), we propose instead to control a new object, the so-called `spectral density function`, which is naturally associated with the linearized flow and can be bounded, for the linearized flow at least, in the same Gevrey space as the initial data.

On the emergence of a quantum Boltzmann equation at the presence of a Bose-Einstein condensate

When: Thu, November 18, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Michael Hott (The University of Texas at Austin ) - https://web.ma.utexas.edu/users/mhott/
Abstract: The mathematically rigorous derivation of a nonlinear Boltzmann equation from first principles is an extremely active research area. In classical physical systems, this has been achieved in various models, based on a variety of fundamental works. In the quantum case, the problem has essentially remained open. I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics of a Bose-Einstein condensate, starting with the von Neumann equation for an interacting Boson gas. This is based on joint work with Thomas Chen.


When: Thu, December 2, 2021 - 3:30pm
Where: Kirwan Hall 3206

Traveling wave solutions to the free boundary Navier-Stokes equations

When: Thu, December 9, 2021 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Ian Tice (Carnegie Mellon Unversity ) - https://www.math.cmu.edu/~iantice/
Abstract: Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary. The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in a coordinate system moving at a constant velocity parallel to the lower boundary. The latter can model, for instance, a tube blowing air on the fluid while translating across the surface. In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system. While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity. This is joint work with Giovanni Leoni.


When: Thu, December 16, 2021 - 3:30pm
Where: https://umd.zoom.us/j/97664034062?pwd=L1dObFJMN0tQVHpKQ3JWUDBRaVFOdz09
Speaker: Yuanyuan Feng (Penn State) -