Current contact: Irina Mitrea
The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.
Monday January 22, 2024 at 14:30, Wachman 617
On elliptic and parabolic PDEs in double divergence form
Seick Kim, Yonsei University
Abstract: We consider an elliptic, double divergence form operator L*, which is the formal adjoint of the elliptic operator in non-divergence form L. An important example of a double divergence form equation is the stationary Kolmogorov equation for invariant measures of a diffusion process. We are concerned with the regularity of weak solutions of L*u=0 and show that Schauder type estimates are available when the coefficients are of Dini mean oscillation and belong to certain function spaces. We will also discuss some applications and parabolic counterparts.
Monday February 12, 2024 at 14:30, Wachman 617
Multiplier Weak-Type Inequalities for Maximal Operators and Singular Integrals
Brandon Sweeting, University of Alabama
Abstract: We discuss a kind of weak-type inequality for the Hardy-Littlewood maximal operator and Calderon-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior; in particular, as shown by Muckenhoupt and Wheeden, the class of weights characterizing such inequalities is strictly larger than $A_p$. In this talk, I will discuss quantitative estimates obtained for $A_p$ weights, $p > 1$, that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Rios for $p = 1$, both in the scalar and matrix weighted setting. I will also discuss recent work on the characterization of those weights for which these inequalities hold for the maximal operator.Monday February 26, 2024 at 14:30, Wachman 617
Two recent homogenization results for dielectric elastomer composites.
Silvia Jimenez Bolanos, Colgate University
Abstract: First, we will discuss the periodic homogenization for a weakly coupled electroelastic system of a nonlinear electrostatic equation with an elastic equation enriched with electrostriction. Such coupling is employed to describe dielectric elastomers or deformable (elastic) dielectrics. We will show that the effective response of the system consists of a homogeneous dielectric elastomer described by a nonlinear weakly coupled system of PDEs whose coefficients depend on the coefficients of the original heterogeneous material, the geometry of the composite and the periodicity of the original microstructure. The approach developed here for this nonlinear problem allows obtaining an explicit corrector result for the homogenization of monotone operators with minimal regularity assumptions. Next, we will discuss the homogenization of high-contrast dielectric elastomer composites, The considered heterogeneous material consisting of an ambient material with inserted particles is described by a weakly coupled system of an electrostatic equation with an elastic equation enriched with electrostriction. It is assumed that particles gradually become rigid as the fine-scale parameter approaches zero. We will see that the effective response of this system entails a homogeneous dielectric elastomer, described by a weakly coupled system of PDEs. The coefficients of the homogenized equations are dependent on various factors, including the composite’s geometry, the original microstructure’s periodicity, and the coefficients characterizing the initial heterogeneous material. Particularly, these coefficients are significantly influenced by the high-contrast nature of the fine-scale problem’s coefficients. Consequently, as anticipated, the high-contrast coefficients of the original yield non-local effects in the homogenized response.Monday March 11, 2024 at 14:30, 617 Wachman Hall
Wave propagation on rotating cosmic strings - Cancelled
Katrina Morgan, Temple University
Abstract: A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch.
Monday March 25, 2024 at 14:30, 617 Wachman Hall
Homogenization of nonconvex Hamilton-Jacobi equations in stationary ergodic media
Atilla Yilmaz, Temple University
Abstract: After giving a self-contained introduction to the homogenization of Hamilton-Jacobi (HJ) equations in stationary ergodic media in any dimension, I will focus on the case where the Hamiltonian is nonconvex, and highlight some interesting differences between: (i) periodic vs. truly random media; (ii) dimension one vs. higher; and (iii) inviscid vs. viscous HJ equations. In particular, I will present a recent result (from joint work with E. Kosygina) on the loss of quasiconvexity which can happen only in the viscous case.
Monday April 1, 2024 at 14:30, Wachman Hall 617
The Moser method for infinitely degenerate equations with non-vanishing right hand side
Cristian Rios, University of Calgary
Abstract: We will present an implementation of the Moser iteration method in a non-doubling geometry with applications to boundedness and continuity of solutions to elliptic equations in which the ellipticity degenerates to infinite order. This is the first implementation of the Moser iteration in such a degenerate setting, allowing us to obtain continuity of solutions when the right hand side is non-vanishing and admissible. This work is continuation of a project in which continuity was previously established via a De Giorgi iteration but only for vanishing right hand sides. The work is done in collaboration with Luda Korobenko, Eric Sawyer, and Ruipeng Shen.
Monday April 8, 2024 at 14:30, Wachman 617
Anna Zemlyanova, Kansas State University
Abstract: In this talk, we study a problem concerning a nano-sized material surface attached to the boundary of an elastic isotropic semi-plane. The material surface is modeled using the Steigmann-Ogden form of surface energy. The study of stationary points of the total elastic energy functional produces a boundary-value problem with non-classical boundary conditions. This problem is solved by using integral representations of stresses and displacements through certain unknown functions. With the help of these functions, the problem can be reduced to either a system of two singular integral equations or a single singular integral equation. The numerical solution of the system of singular integral equations is obtained by expanding each unknown function into a series based on Chebyshev polynomials. The accuracy of the numerical procedure is studied, and various numerical examples for different values of the surface energy parameters are considered.
Monday April 15, 2024 at 14:30, Wachman 617
Analysis of a simple model of passive intermodulation
Erich Stachura, Kennesaw State University
Abstract: I will discuss a basic model of passive intermodulation (PIM). PIM occurs when multiple signals are active in a passive device that exhibits a nonlinear response. It is known that certain nonlinearities (e.g. the electro-thermal effect) which are fundamental to electromagnetic wave interaction with matter should be accounted for. In this talk, I will discuss existence, uniqueness, and regularity of solutions to a simple model for PIM. This in particular includes a temperature dependent conductivity in Maxwell's equations, which themselves are coupled to a nonlinear heat equation. I will also discuss challenges related to a similar problem when the permittivity $\varepsilon$ also depends on temperature. This is joint work with Niklas Wellander and Elena Cherkaev.
Monday April 22, 2024 at 14:30, Wachman 617
Local energy decay for the acoustic wave equation in low regularity
Jacob Shapiro, University of Dayton
Abstract: We discuss recent results and work in progress on local energy decay for the acoustic wave equation in low regularity. The main challenge is to establish suitable control over the resolvent of the associated Helmholtz operator at both large and small frequencies. For large frequencies, we employ (after rescaling) a semiclassical Carleman estimate. Near zero frequency we obtain a resolvent expansion by perturbative methods. Both tools are sensitive to the decay of the coefficients near infinity.Monday April 29, 2024 at 14:30, Wachman 617
Stein-Weiss type inequality in $L^1$ norm for vector fields and applications.
Tiago Henrique Picon, University of São Paulo, Ribeirão Preto
Abstract: In this talk, we investigate the limit case p = 1 of the Stein–Weiss inequality for the Riesz potential. Our main result is a characterization of this inequality for a special class of vector fields associated to cocanceling operators. As applications, we recovered some classical div-curl inequalities and obtained new solvability results for equations associated to canceling and elliptic differential operators on measures. This is joint work with Pablo De Nápoli (Universidad de Buenos Aires - Argentina) and Victor Biliatto (UFSCar - Brazil).Monday September 9, 2024 at 14:30, Wachman Hall 617
Long Time Behavior of Electroconvection Models
Mihaela Ignatova, Temple University
We present two electroconvection models describing the interaction between a surface charge density and a fluid in a two-dimensional situation. These are nonlinear partial differential equations with nonlocal terms. We compare these models with another well-known nonlocal nonlinear equation, the surface quasi-geostrophic equation in bounded domains, give some background on the subject and recall some recent results. For the first electroconvection model, we describe global existence results in bounded domains and show that the long-time asymptotic state of the system is finite dimensional, if body forces are applied to the fluid, and a singleton solution in the absence of fluid body forces. In the whole space, in the absence of forcing, we obtain optimal decay rates. For the more challenging second model, corresponding to electroconvection through porous media, we describe global existence for subcritical cases and for small data in the critical case.
Monday September 16, 2024 at 14:30, Wachman Hall 617
On global regularity theory for the Peskin problem
Susanna Haziot, Princeton University
The Peskin problem describes the flow of a Stokes fluid through the heart valves. We begin by presenting the simpler 2D model and investigate its small data critical regularity theory, with initial data possibly containing small corners. We then present the 3D problem and describe the challenges that arise to proving global well-posedness.
The first part is joint work with Eduardo Garcia-Juarez, and the second is on-going work with Eduardo Garcia-Juarez and Yoichiro Mori.
Monday September 23, 2024 at 14:30, Wachman Hall 617
On the inviscid limit for the Navier-Stokes equations
Igor Kukavica, USC
The question of whether the solution of the Navier-Stokes equation converges to the solution of the Euler equation as the viscosity vanishes is an important one in fluid dynamics. In the talk, we will review current results on this problem. We will also present a result, joint with V. Vicol and F. 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 Sobolev regularity in the interior. We will also discuss the Prandtl expansions of solutions of the Navier-Stokes equations.
Monday September 30, 2024 at 14:30, Wachman Hall 617
On the feasibility of extrapolation of completely monotone functions
Henry Brown, Temple University
Completely monotone functions are Laplace transforms of positive Borel-regular measures. Given two completely monotone functions which agree to a given relative precision on an interval, how large can their relative difference be away from the interval? We show to the left of the interval, the answer is infinity, but to the right, the maximum relative difference obeys a power law, which we derive explicitly. In this talk, I will show our method of proof, which can be broken down into two stages: (1) The introduction of an auxiliary problem over functions in a Hardy space which, via tools from linear programming, is reduced to an integral equation and solved explicitly, and (2) the introduction of a family of intermediate Hardy-like spaces which bridges the gap between the auxiliary problem and the original problem. This is based on a joint work with my advisor, Yury Grabovsky.
Monday October 7, 2024 at 14:30, Wachman Hall 617
Distinguished Coefficient Tensors for Second Order Elliptic Differential Operators and Applications to Boundary Value Problems
Irina Mitrea, Temple University
In this talk I will discuss how the choice of coefficient tensor for a second order weakly elliptic constant coefficient differential operator $L$ affects the Fredholm properties of the boundary layer potential operators associated with $L$ considered on domains $\Omega$ with compact boundaries. While all these integral operators share many common properties (such as nontangential maximal function estimates, boundedness properties, jump-relations, etc.), some more specialized functional analytic features are heavily dependent on the nature of the coefficient tensor involved. This is joint work with Dorina Mitrea and Marius Mitrea from Baylor University and is part of our recent five volume Springer series Geometric Harmonic Analysis.
Monday October 21, 2024 at 14:30, Wachman Hall 617
Anomalous diffusion in a critical advection-diffusion equation
Peter Morfe, Max Planck Institute
I will describe an example of anomalous diffusion in an advection-diffusion equation in two dimensions. In this example, the advection is a random divergence-free vector field, and I will explain how techniques from homogenization theory can be used to study the long-time, large-scale asymptotic behavior of solutions. This is joint work with G. Chatzigeorgiou, F. Otto, C. Wagner, and L. Wang.
Monday October 28, 2024 at 14:30, Wachman Hall 617
On the locally self-similar blowup for the generalized SQG equation
Cecilia Mondaini, Drexel University
The question of either global regularity or formation of a finite-time singularity (blowup) is still not settled for the SQG equation, a model used to describe the evolution of surface temperature or buoyancy in certain large-scale atmospheric or oceanic flows. An extension is provided by the generalized SQG equation, which provides a more general constitutive relation between the velocity and the active scalar (e.g. surface temperature). Depending on the value fixed for such parameter, the velocity field can be either more regular or more singular in comparison to the SQG. However, the same question regarding global regularity or finite-time singularity also remains open for the generalized SQG equation under all typical values of this parameter. Motivated by numerical simulations from several previous works, we analyze the possible formation of finite-time blowup for the generalized SQG equation of locally self-similar type. Under an L^q growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero (no blowup), or its L^p asymptotic behavior can be characterized, for suitable q, p. This is a joint work with A. Bronzi (Unicamp, Brazil) and R. Guimaraes (Unicamp, Brazil).
Monday November 18, 2024 at 14:30, Wachman Hall 617
Some non-decaying, non-periodic existence theory for fluid equations
David Ambrose, Drexel
We consider the irrotational Euler equations and the surface quasi-geostrophic equation in the case that the unknowns do not decay and are not spatially periodic. In such settings, constitutive laws of convolution type (such as the Biot-Savart law) do not apply directly, as the convolution integral does not converge. These can be replaced with identities of Serfati type, which separate the integrals into near-field and far-field pieces, with the far field contribution being able to be manipulated for better convergence properties. We use these identities to find existence of solutions for the 2D Euler equations with bounded velocity and vorticity (generalizing a result of Serfati), for the 3D Euler equations in uniformly local Sobolev spaces, and for SQG in Holder spaces and in uniformly local Sobolev spaces. This includes joint work with Elaine Cozzi, Daniel Erickson, James Kelliher, Milton Lopes Filho, and Helena Nussenzveig Lopes.