News & Events

Seminar

Date/Time: 
Tuesday, October 12, 2021 - 15:00 to 16:00
Venue: 
Online (GoogleMeet)
Speaker: 
Debopriya Mukherjee
Affiliation: 
Montan University
Title: 
Stochastic analysis of certain constrained physical and biological models

 

In my talk, I will address several mathematical questions about constrained Stochastic Partial Differential Equations (SPDEs) arising from the dynamics of (I) ferromagnetism, (II) viscoelastic fluid models, and (III) mechanism for spatial pattern formation analysing chemotaxis patterns under random perturbation. Standard or recent techniques for the analysis of parabolic, semilinear and non-constrained SPDEs do not apply in a straightforward way to the model problem. Here, we take this opportunity to study the solvability of such problems. Due to the environmental structure of such models, it is natural to follow that incorporation of noise in the usual Itˆo (or Itˆo-L ́evy) sense does not work in this context.

To start with, we analyse the theory of magnetisation of ferromagnetic materials below a certain temperature. Since the magnetisation has a unit length at initial time, therefore the question of how to incorporate a suitable perturbation modelling the constrained structure without destroying its invariance property is a delicate one; [4, 5, 9, 10].

In the later part of the talk, we will focus on the viscoelastic fluid model where the stress tensor is invariant under coordinate transformation. One needs to take into account this property in order to have a full understanding of the effect of fluctuating forcing field; [1, 2, 3, 6, 7].

Finally, to study the oriented movement of cells (or an organism) in response to a chemical gradient, we are careful with the environmental structure and the fluctuation of parameters. An appropriate mathematical approach to establish such realistic models with the incorporation of randomness in the system is exquisite; [8, 11, 12, 13].

References

  1. [1]  U. Manna and D. Mukherjee‘Strong Solutions of Stochastic Models for Viscoelastic Flows of Oldroyd Type,’ Nonlinear Analysis. 165, 198—242 (2017).

  2. [2]  P. Agarwal, U. Manna and D. Mukherjee‘Stochastic Control of Tidal Dynamics Equation with L ́evy Noise,’ Applied Mathematics and Optimization. 76(2), 1–70 (2017).

  3. [3]  U. Manna and D. Mukherjee‘Optimal relaxed control of stochastic hereditary evolution equations with L ́evy noise,’ ESAIM: Control, Optimisation and Calculus of Variations. 25(61), 48 pp. (2019).

  4. [4]  Z. Brze ́zniak, U. Manna and D. Mukherjee‘Existence of solution for Stochastic Landau-Lifshitz-Gilbert equation via Wong-Zakai approximation,’ Journal of Differential Equations. 267(2), 776-825 (2019).

  5. [5]  Akash A. Panda, U. Manna and D. Mukherjee‘Wong-Zakai approximation for the stochastic Landau- Lifshitz-Gilbert equations with anisotropy energy.’ J. Math. Anal. Appl. 480(1), 13 pp. (2019).

  6. [6]  U. Manna and D. Mukherjee‘Weak solutions and invariant measures of stochastic Oldroyd-B type model driven by jump noise.’ Journal of Differential Equations. 272, 760–818 (2021).

  7. [7]  U. Manna and D. Mukherjee‘Weak martingale solution of stochastic critical Oldroyd-B type models perturbed by pure jump noise.’ Stochastic Analysis and Applications, DOI:10.1080/07362994.2021.1947855; (2021).

  8. [8]  E. Hausenblas, D. Mukherjee, and T. Tran‘The one-dimensional stochastic Keller–Segel model with time- homogeneous spatial Wiener processes,’ to appear in Journal of Differential Equations, arXiv:2009.13789v1.

  9. [9]  E. Hausenblas, D. Mukherjee and Kitsolil FahimWong–Zakai approximation for Landau–Lifshitz-–Gilbert

    equations driven by geometric rough paths. Applied Mathematics and Optimization, DOI:10.1007/s 00245-021-

    09808-1; (2021).

  10. [10]  M. Biswas, E. Hausenblas, and D. MukherjeeLandau–Lifshitz–Gilbert Equation: Controllability by low

    modes forcing for deterministic version and Support Theorems for stochastic version, (Preprint available),

    (2021).

  11. [11]  E. Hausenblas, D. Mukherjee and Johannes Lankeit‘Existence of a local solution to the two dimensional

    stochastic Keller Segel Model.’ (Submitted, preprint available), (2020).

  12. [12]  E. Hausenblas, D. Mukherjee and Ali Zakaria‘Strong solution to a stochastic chemotaxis system with

    porous medium diffusion.’ (Submitted, preprint available), (2020).

  13. [13]  E. Hausenblas, and D. Mukherjee‘Pathwise uniqueness to the stochastic Keller-Segel systems.’ (Submitted,

preprint available), (2020).

Contact us

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