RIT on Applied PDE Archives for Fall 2023 to Spring 2024
Organizational Meeting
When: Mon, September 12, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Organizational meeting (UMD) -
Abstract: This is the first meeting of the RIT for Fall 2022 semester. We will select the speakers and discuss potential topics.
Probabilistic Data and Local Well-Posedness in Dispersive PDE
When: Mon, September 26, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker. Mickaël Latocca (UMD) - https://www.math.ens.psl.eu/~latocca/index.html
Title. Probabilistic Data and Local Well-Posedness in Dispersive PDE
Abstract. In this talk, I will go back to the basics of local well-posedness for a general PDE. Then I will show how randomness can help lower the regularity threshold for solvabilities of such PDEs and explain why it is helpful for dispersive PDEs in particular. I will try to make the exposition as self-contained as I can.
On the Local Well-Posedness of a Stochastic Vlasov-Fokker-Planck Equation
When: Mon, October 3, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker. Stavros Papathanasiou (UMD) - http://math.umd.edu/~stavrosp/
Abstract. Thedeterministic Vlasov(-Poisson)-Fokker-Planck (VPFP) equation is a well studiedkinetic PDE that serves as a simplified model for the distribution function ofa cloud of electrons which interact electrostatically and experience collisionsand friction against a spatially uniform Maxwellian background of positiveions. We perturb VPFP by an external stochastic electric field and discuss aPicard iteration scheme for the local well-posedness of the resulting SPDE.
Entropy Measure-Valued Solutions to Conservation Laws Equations
When: Mon, October 10, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Jingcheng Lu (UMD) -
Abstract: Entropy solutions are widely accepted as the framework to distinguish 'physical' solutions to conservation laws. However, De Lellis et al and Chiodaroli et al showed that the entropy solutions to multidimensional conservation systems may not be unique. To help resolve the problem of non-uniqueness, the idea of entropy measure-valued (EMV) solutions is to define the physical solution as a Young measure (random field). In my talk I will give an introduction to the concepts of EMV solutions and review the related well-posedness results. Numerical experiments will be presented to reinforce analytical discussions.
Bose Condensation Described by the Kompaneets Equation
When: Mon, October 17, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Charles D. Levermore (UMD) - https://www.math.umd.edu/~lvrmr/
Abstract: The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long-time convergence of solutions to Bose--Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose--Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as t→∞, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in L1. None of these results impose a boundary condition at the zero-energy boundary.
Circulation and Energy Theorem Preserving Stochastic Fluids
When: Mon, October 24, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Ethan Dudley (UMD) -
Abstract: Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved, i.e. Kelvin's theorem. Noisy flows can be used to model turbulent flows, but the natural question is whether smooth solutions mirror their deterministic counterparts. More specifically, are smooth solutions for noisy flows characterized by a generalized Kelvin's Theorem and do they obey pathwise energy conservation/dissipation. Results from Drivas and Holm (2020) will be presented that this is not true in general.
Congested crowd motion, chemotaxis and free boundary problems.
When: Mon, October 31, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Antoine Mellet (UMD) - https://www.math.umd.edu/~mellet/
Abstract: I will present several recent results concerning the well-posedness and the asymptotic behavior of some drift-diffusion equations under incompressibility constraint. Example includes simple drift equation with upper bound on the density (congested crowd motion) or Keller-Segel type models with an incompressibility condition. The goal is to connect these classical PDE to some (equally classical) free boundary problems (Hele-Shaw and Hele-Shaw with surface tension).
: Decay for Solutions of Schrödinger Equations with Small Time Independent Potentials in R^3
When: Mon, November 7, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Ben Goldschlager (UMD) -
The classical Schrödinger equation governs the wave function of a quantummechanical system. The wave function, u , has the interpretation that |u|^2 represents the probability density offinding a particle at a certain location at a given time. The solution is known to satisfy the decay estimate: \|u(t)\|_{L^{\infty}} \leq t^{-n/2} \|f\|_{L^1} where n is the number of spatial dimensions, and, thus, the solution decays in time. More generally, the Schrödinger equation with a potential V governs the wave function of a quantum mechanical system with a potential V. We will show, under certain assumptionson, that, in three spatial dimensions, the wavefunction still satisfies a similar decay estimate.
Global well-posedness for the Landau equation near Maxwellians
When: Mon, November 14, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Chi-Hao Wu (UMD) -
Title: Global well-posedness for the Landau equation near Maxwellians
Abstract: We talk about the existence and uniqueness of the global in time solution to the Landau equation. The main reference for this presentation is "The Landau Equation in a Periodic Box" by Yan Guo.
An introduction to branched transport
When: Mon, November 21, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Antonio De Rosa (UMD) - https://sites.google.com/view/antonioderosa/home
Abstract:I will provide an introduction to optimal branched transport, using geometricmeasure theory tools to model the transport of particles subject to groupdynamics.
Bypassing Holder super-criticality barriers in the incompressible Navier-Stokes equations
When: Mon, November 28, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Hussain Ibdah (University of Maryland) - https://www.math.umd.edu/~hibdah/
Abstract: We will go over the main ideas used in proving that $L^1_tC_x^{0,\beta}$ solutions to the incompressible Navier-Stokes equations in any dimension (in the absence of physical boundaries) emanating from smooth initial data are regular, no matter how small $\beta$ is (as long as it is positive). In comparison, the a-priori bounds we have are at the $L^1_tL^{\infty}_x$ level. To our knowledge, this is the very first genuinely supercritical regularity criterion for this system of equations.
Traveling surface waves in porous media
When: Mon, December 5, 2022 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Huy Nguyen (UMD) - https://www.math.umd.edu/~hnguye90/
Abstract: We will discuss the existence and stability of traveling surface waves in porous media.
These waves are solutions to the one-phase Muskat problem, which concerns the dynamics of the free boundary of a fluid flow modeled by Darcy’s law.
Organizational Meeting
When: Mon, January 30, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Hussain, Huy, and Antoine (UMD) -
Abstract: This is the organizational meeting for the RIT on PDE during the spring semester.
Compactness Criteria: Arzela-Ascoli to velocity averaging
When: Mon, February 6, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Charles D. Levermore (UMD) - https://www.math.umd.edu/~lvrmr/
Enhanced Regularity-Part 1
When: Mon, February 20, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Hussain Ibdah (University of Maryland) - https://www.math.umd.edu/~hibdah/
Abstract: We will explain how certain drift velocities may enhance the regularizing effect of the Laplacian in the context of parabolic evolution and help absorb singular lower order terms. Such phenomenon allows us to bypass certain super-criticality barriers in drift-diffusion systems.
Chemotaxis and a JKO-type scheme for the congested parabolic-parabolic Keller-Segel system
When: Mon, February 27, 2023 - 3:00pmWhere: Kirwan Hall 1313
Speaker: Michael Rozowski (University of Maryland) -
Abstract: We will review the phenomena of chemotaxis and classical models for its description, including the parabolic-parabolic and parabolic-elliptic Keller-Segel systems. After a brief review of blow-up results for these systems, we will describe some methods for adjusting the model to prevent blow-up. In particular, we present a novel method, inspired by a model for congested crowd motion due to Santambrogio and collaborators. We then describe a JKO-type scheme that we use for proving the existence of solutions to the congested parabolic-parabolic system
p-alignment with pressure
When: Mon, March 13, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Eitan Tadmor (University of Maryland ) - https://www.math.umd.edu/~tadmor/index.htm
Abstract: We discuss the swarming behavior of hydrodynamic p-alignment, based on 2p-graph Laplacians and weighted by a general family of symmetric communication kernels. This extends the classical alignment model corresponding to p=1. The main new aspect here is the long time emergence behavior for a general class of pressure tensors without a closure assumption, beyond the mere requirement that they form an energy dissipative process. We refer to such pressure laws as `entropic', and prove the flocking of p-alignment hydrodynamics, driven by singular kernels with general class of entropic pressure tensors.
De-Giogri iterations
When: Mon, March 27, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Antoine Mellet (University of Maryland) - https://www.math.umd.edu/~mellet/
K41-Turbulence
When: Mon, April 3, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Ethan Dudley (University of Maryland) -
Alfvén wave damping
When: Mon, April 10, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Chi-Hao Wu (University of Maryland) -
Mesoscopic averaging of the two-dimensional KPZ equation
When: Mon, April 17, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Ran Tao (University of Maryland) -
Abstract: We study the limit of a local average of the KPZ equation in dimension d=2 with general initial data in the subcritical regime. Our result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equation and a Gaussian random variable that depends solely on the scale of averaging. This shows a unique mesoscopic averaging phenomenon that is only present in dimension two.
On a family of nonlinear Fourier transforms
When: Mon, April 24, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Shiferaw Berhanu (University of Maryland) -
Abstract: We will discuss a family of nonlinear Fourier trans- forms that characterizer smoothness, real analyticity, and Gevrey regularity. An application to the microlocal regularity of first order nonlear pdes with complex coefficients will be presented.
On a quantum theory for plasmons via the Schroedinger-Poisson system
When: Mon, May 1, 2023 - 3:00pmWhere: Kirwan Hall 1311
Speaker: Dionisios Margetis (UMD) - http://www.math.umd.edu/~diom/
Abstract: The plasmon is a collective excitation of the electron charge caused by the presence of the electromagnetic field. In most theoretical treatments to date, the respective dispersion relation (plasmon energy versus its wavelength) is derived via familiar approaches of classical physics, by use of Maxwell’s equations with phenomenological material parameters. In this talk, I will address the derivation of the plasmon dispersion relation in a 2D material by invoking notions of non-relativistic quantum theory. To this end, I use a toy model in which the particle motion is described by the Schroedinger equation having a potential that consists of: (i) an external trap (binding potential) that keeps the particle close to a fixed plane; and (ii) a term induced by the electron excess charge, ie, the deviation of the local electron charge density from a steady-state density. (This model forms a crude but intriguing attempt to bypass difficulties of quantum electrodynamics which is the natural framework for the plasmon problem). The model is expressed by a Schroedinger-Poisson system. I study a particular asymptotic solution of this system, and show how the plasmon dispersion can emerge formally by an appropriate scale separation.