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Applied Mathematics Symposium

Applied Mathematics Symposium: Applied Analysis of PDE



The Departments of Mathematics at IIT Guwahati, IIT Hyderabad and VIT Vellore are jointly organizing an online symposium titled “Applied Mathematics Symposium: Artificial Intelligence meets Fluid Dynamics” on April 19, 2024 (Friday) during 14:00-18:45 IST. The symposium will be held online via Microsoft Teams.

Date: April 19, 2024 (Friday)

Mode: Online Platform (Microsoft Teams)

Microsoft Teams Link: Click here to join the meeting

About the symposium: This series of symposia are meant to introduce the latest cutting-edge research in an area to Indian researchers, to improve the overall state of applied mathematics research in India, and to foster greater interaction between Indian and foreign researchers. The theme for this symposium is "Applied Analysis of PDE". The targeted audience are researchers at all levels, not necessarily experts in the thematic area of the symposium, but also those who aspire to work in this area.

Speakers


  • José Carrillo, Mathematical Insitute, University of Oxford, UK (Plenary speaker)

  • Ch. Srinivasa Rao, IIT Madras, India (Invited speaker)

  • Harsha Hutridurga, IIT Bombay, India (Invited speaker)

  • Bhakti Bhusan Manna, IIT Hyderabad, India (Invited speaker)

  • Nibedita Ghosh, RWTH Achen, Germany (Invited speaker)

    • Program Schedule


    14:00-14:10 IST                                Welcome Address

    14:10-15:10 IST                                Plenary Lecture

    Speaker: José Carrillo, Mathematical Insitute, University of Oxford, UK

    Title: Nonlocal Aggregation-Diffusion Equations: fast diffusion and partial concentration

    Abstract: We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L, Vázquez.

    15:15-16:00 IST                                Invited Lecture 1

    Speaker: Harsha Hutridurga, IIT Bombay, India

    Title: Entropy method: a tool for studying reaction-diffusion systems

    Abstract: The objective of this talk is to illustrate how the entropy method enables us to study, in an elementary fashion, the long-time asymptotic of solutions to reactiondiffusion systems corresponding to reversible chemistry. Loosely speaking, the entropy method looks for a nonnegative Lyapunov functional (also called entropy) and its nonnegative dissipation along the flowof the given evolution equation. This is then followed by proving a certain functional inequality referred to as the entropy-entropy dissipation inequality. The method concludes by using techniques from the theory of ordinary differential equations to derive decay estimates with explicit rates of convergence. The novelty of our work is to adapt the entropy method to address degenerate reactiondiffusion systems where one of the chemical species is non-diffusive.

    16:15-17:00 IST                                Invited Lecture 2

    Speaker: Ch. Sriivas Rao, IIT Madras, India

    Title: Large time asymptotic solutions of some nonlinear partial differential equations

    Abstract: In this lecture we shall concentrate on the large time asymptotic solutions of a class of nonlinear partial differential equations, which are generalizations of the most celebrated Burgers equation. In the first instance we shall discuss the construction of solutions of a nonhomogeneous Burgers equation on the whole real line. The solutions of the nonhomogeneous Burgers equation subject to certain unbounded initial profiles are constucted in terms of the self-similar solutions of a variable coefficient linear partial differential equation. One can easily get the large time asymptotic behaviour of the solutions of the nonhomogeneous Burgers equation from the representation written in terms of these self-similar solutions. We shall also discuss the construction of large time asymptotics to anti-symmetric solutions and periodic solutions of the nonhomogeneous Burgers equation. A brief outline of the importance of separable solutions and traveling wave solutions of some generalized Burgers equations will also be presented.

    17:05-17:50 IST                                Invited Lecture 3

    Speaker: Bhakti Bhusan Manna, IIT Hyderabad, India

    Title: Singularly Perturbed Elliptic Problems: Some recent developments

    Abstract: In this talk, we will discuss recent results related to the perturbation theory for elliptic problems. We shall consider the following type of equations ε2 L u - u + f(u) = 0 in A ⊂ Rn, where L is an elliptic operator, A is any domain in R n, n ≤ 3, ε > 0 is the perturbation parameter, and f is a nonlinearity of some special type. Solutions to these perturbed problems generally show concentrating behavior for small ε . We will focus on different types of concentration profiles for these types of equations.

    17:55-18:40 IST                                Invited Lecture 4

    Speaker: Nibedita Ghosh, RWTH Achen, Germany

    Title: Domain decomposition for COSMO models

    Abstract: The conductor-like screening model or COSMO model has wide applications for computing the electrostatic interaction between a solvent and a particular solute molecule. However, mathematically this involves solving a Laplace boundary value problem on a domain with a number of intersecting balls. We proposed to solve this problem more efficiently by employing a numerical method based on Schwarz’s domain decomposition method. The purpose of our work is to detail the methodology, set up a theoretical foundation of the approach, and study the convergence behavior for the inexact ddCOSMO method.

    18:40-18:45 IST                                Vote of Thanks