## Department of Mathematics, Purdue University PDE Seminar

SPRING 2021

Talks in SPRING 2021 will take place virtually at 4:30-5:30pm, Thursday. If you have any questions about a particular seminar, please contact the organizer at: wang2482@purdue.edu.
The Zoom meeting link is: Meeting ID: 98551029625; Meeting Passcode: 570121

Date SPEAKER and TITLE Host
February 11 Dan Spirn, University of Minnesota
TITLE: A variational method for generating cross fields and frames using higher order Q-tensors
ABSTRACT: A cross field is a locally defined orthogonal coordinate system invariant with respect to the cubic symmetry group. Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science. In this talk, I will consider the problem of generating an arbitrary n-cross field using a fourth-order Q-tensor theory that is constructed out of tensored projection matrices. Computationally, one can then use a Ginzburg-Landau relaxation towards a global projection to reliably generate n-cross fields on arbitrary Lipschitz domains. This tensor framework provides an approach to study the behavior of the singular set, i.e. the set on which the domain fails to be a cross field. In particular we can use the classical Ginzburg-Landau theory to study singularities of the associated energy. This is joint work with Dmitry Golovaty and Albert Montero.
Wang
February 18 Yu Feng, University of Wisconsin, Madison
TITLE: Phase Separation in the Advective Cahn-Hilliard Equation
ABSTRACT : The Cahn--Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn--Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection-hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection-diffusion equation.
Wang
February 25 Plamen Stefanov, Purdue University
TITLE: Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime
ABSTRACT : We study the inverse problem of recovery a compactly supported non-linearity in the semilinear wave equation $u_{tt}-\Delta u+ \alpha(x) |u|^2u=0$, in two and three dimensions. We probe the medium with complex-valued harmonic waves of wavelength $h$ and amplitude $h^{-1/2}$, then they propagate in the weakly non-linear regime; and measure the transmitted wave when it exits the support of $\alpha$. We show that one can extract the Radon transform of $\alpha$ from the phase shift of such waves, and then one can recover $\alpha$. We also show that one can probe the medium with real-valued harmonic waves and obtain uniqueness for the linearized problem. It is a joint work with Antonio Sa Barreto
Wang
March 4 Hien Nguyen, Iowa State University
TITLE: The first two eigenvalues are not social distancing in hyperbolic space
ABSTRACT: For the Laplace operator with Dirichlet boundary conditions on convex domains in $H^n, n>2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter. This property distinguishes hyperbolic spaces from Euclidean and spherical ones, where the quantity is bounded below by $3 \pi^2$. We finish by talking about horoconvex domains
Wang
March 11 Shibing Chen, University of Sciences nd Technology of China
TITLE: Free boundary regularity in optimal transport problem
ABSTRACT : Free boundaries arise in optimal transport problem when only a portion of mass is transported. The $C^1$ and $C^{1,\alpha}$ regularity of free boundaries were established by Caffarelli and McCann when the domains are disjoint, and later by Figalli, Indrei when the domains are allowed to have overlap. We will discuss our recent proof to the higher regularity of free boundary. This is based on a joint work with Jiakun Liu and Xu-Jia Wang.
Wang
March 18 Nikolas Eptaminitakis, Purdue University
TITLE: Stability for the X-Ray Transform on Asymptotically Hyperbolic Manifolds
ABSTRACT: The topic of the talk will be the geodesic X-ray transform of a function $f$, given by $If(\gamma) =\int_\gamma fds$ for each geodesic $\gamma$, in the setting of asymptotically hyperbolic manifolds. Asymptotically hyperbolic manifolds are a class of non-compact, complete Riemannian manifolds whose behavior near infinity resembles in many ways that of hyperbolic space. In part due to the behavior of geodesics and their infinite length, and the fact that those spaces have infinite volume, the study of the geodesic X-ray transform becomes particularly interesting and challenging in this setting. The specific problem that will be discussed is the one of stability of the geodesic X-ray transform, where the goal is to establish that small perturbations in the X-ray transform (the measurement) cannot originate from large perturbations of the unknown function $f$, and find appropriate spaces to measure such perturbations. We study the stability of the normal operator of the X-ray transform, which is given by its composition with a backprojection, on simple asymptotically hyperbolic manifolds. The approach is microlocal, in a similar spirit to the work of Stefanov-Uhlmann (2004) on simple compact manifolds with boundary. It turns out that a natural framework of pseudodifferential operators for studying operators on asymptotically hyperbolic manifolds is the 0-calculus of Mazzeo-Melrose. We discuss how the normal operator of the X-ray transform fits within this framework, leading to a parametrix and eventually the desired stability.
Wang
March 25 Xiang Xu, Old Dominion University
TITLE: Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals
ABSTRACT : The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential. More specifically, we provide precise estimates of both this singular potential and its gradient as the Q-tensor approaches its physical boundary.
Wang
April 1 Fengbo Hang, New York University
TITLE: Improved functional inequalities under constraints and quadrature formulas
ABSTRACT: Aubin's improvement of Sobolev inequality and Moser-Trudinger inequality under the vanishing of first order moments are useful in the study of scalar curvature equation and Gauss curvature equations respectively. We will discuss the corresponding inequalities under the vanishing of higher order moments. Classical and new open problems concerning quadrature (cubature) formulas on the sphere appear in this study. Our problem is partly motivated from a sequence of inequalities following from Szego limit theorem on the unit circle (as observed by Widom).
Wang
April 8 Raghav Venkatraman, Carnegie Mellon University
TITLE: Geometry and Dynamics of Phase Transitions in Heterogeneous Media
ABSTRACT : In this talk we discuss recent progress on homogenization of the Allen-Cahn equation to Brakke's formulation of anisotropic mean curvature flow. Along the way, our analysis also provides resolution to a long-standing open problem in geometry: what do distance functions to planes 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. Joint work with Irene Fonseca, Jessica Lin and Rustum Choksi.
Wang
April 15 Dan Phillips, Purdue University
TITLE: Defects in thin nematic liquid-crystalline films
ABSTRACT : We investigate the structure of nematic liquid crystal thin films described by the Ball-Majumdar, tensor-valued order parameter model with Dirichlet boundary conditions on the sides of nonzero degree. We prove that as the elastic modulus goes to zero in the energy a limiting uniaxial nematic texture forms, locally minimizing a planar Frank energy trapping a finite number of defects, all of degree 1/2 or all of degree -1/2, corresponding to vertical disclination lines at those locations. This joint work with Patti Bauman.
Wang
April 22 Qi S. Zhang, University of California, Riverside
TITLE: Time analyticity and reversibility of some parabolic equations
ABSTRACT :We describe a concise way to prove time analyticity for solutions of parabolic equations including the heat and Navier Stokes equations. In some cases, results under sharp conditions are obtained. An application is a necessary and sufficient condition for the solvability of the backward heat equation which is ill-posed, helping to remove an old obstacle in control theory. Part of the work is joint with Hongjie Dong.
Wang
April 29 Hung Tran, University of Wisconsin, Madison
TITLE: Large time behavior and large time profile of viscous Hamilton-Jacobi equations
ABSTRACT : I will describe our recent results on large time behavior and large time profile of viscous Hamilton-Jacobi equations in the periodic setting. Here, the diffusion matrix might be degenerate, which makes the problem more difficult and challenging. Based on joint works with Cagnetti, Gomes, Mitake.
Wang
May 6 Simon Schulz, Cambridge University, UK
TITLE: Liouville theorems for the stationary MHD equations
ABSTRACT : What boundedness assumptions on the solution of an equation cause it to vanish everywhere? This is the question that a Liouville type theorem seeks to answer. For reasons that will become apparent, these results are intimately related to problems of regularity and uniqueness. In this talk, we will consider Liouville type theorems for equations arising in the study of incompressible fluid dynamics. In particular, we will give a criterion for smooth solutions of the stationary equations of magneto-hydrodynamics (MHD) to be identically zero. The result in question is a refinement of previous ones, all of which systematically required stronger integrability and the additional assumption of finite Dirichlet integral, and builds on the earlier works of Seregin, Chae and Weng, and Chae and Wolf. This is joint work with Nicola De Nitti and Francis Hounkpe.
Wang