Séjour court

Piotr
BILER

Mathematical sciences

Contact details

Research topics

PROJECT

Singular solutions in chemotaxis and aggregation – a challenge for mathematicians

The project is devoted to study closely related questions on the solvability of models with irregular initial data, continuation of solutions, determination of their spatio-temporal asymptotics, and the formation of singularities eventually leading either to a blowup of solutions or concentration phenomena for the following models of current interest used in biology, astrophysics and mechanics of continuous media:

Chemotaxis, aggregation and other quasilinear parabolic problems
: local and global solvability of chemotaxis models generalizing the simplest Keller–Segel model in various ways (nonlocal diffusion, nonlinear sensitivity functions) with irregular data, determination of threshold conditions for continuation, analysis of asymptotic profiles of blowing up solutions and analysis of small diffusion regimes.

Research methodology.

The methods intended to be applied include approaches for weak, mild and classical solutions of partial differential equations involving energy and entropy estimates, moments of distribution functions of solutions describing their concentration, comparison principles for regular solutions as well as genuinely nonlocal tools of harmonic analysis and a fine analysis of diffusion.

Research project impact.

A better understanding of those issues would permit to verify the applicability of mathematical models of various physical and biological phenomena in concrete cases, use them in a more efficient way (since, e.g., some simulations show numerical artefacts only, not the genuine behaviour of solutions). Theoretical results are intended to be published in leading mathematical journals and presented at international conferences.

Activities / Resume

BIOGRAPHY

Piotr Biler, born in 1958, is the professor of mathematics (from 1996) at the Mathematical Institute, University of Wrocław and (from 2016) the corresponding member of the Polish Academy of Sciences.

His PhD thesis (in 1984, under Andrzej Krzywicki) was on Asymptotics of solutions of generalized Korteweg-de Vries-Burgers equation, and his habilitation (in 1992) was on Asymptotic behavior of solutions of nonlinear second order evolution equations. Currently, he is interested in evolution equations with fractional diffusion terms related to nonlocal models of interacting particles in physics, and collective motion models (chemotaxis) in biology.
He has held numerous visiting positions in France (universités de Paris-Sud, Nancy, Bordeaux, Val-de-Marne, Paris-Dauphine, UCB Lyon, Marne-la-Vallée), USA (Case Western Reserve University, Cleveland) and Chile (Universidad de Chile, Santiago). His academic duties included functions of the dean of the Faculty of Mathematics and Computer Science, University of Wrocław, expert of grant agencies (NCN in Poland, ANR in France), member of the national qualifying committee (CK in Poland), editor of journals (Colloquium Mathematicum, Bulletin of the Polish Academy of Sciences, Topological Methods in Nonlinear Analysis, Applicationes Mathematicae etc.), referee of many journals.

BIBLIOGRAPHY

  • Problems and Examples in Differential Equations, (with Tadeusz Nadzieja) 1-244, Pure and Applied Mathematics 164, Marcel Dekker Inc., New York, 1992, ISBN 0-8247-8637-8.
  • Singularities of solutions in chemotaxis systems, i-xxiv, 1-207; De Gruyter Series in Mathematics and Life Sciences 6, Walter de Gruyter, Berlin, 2020. ISBN 978-3-11-059789-9.
  • Attractors for the system of Schroedinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Analysis 21 (1990), 1190-1212.
  • Scattering of small solutions to generalized Benjamin-Bona-Mahony equation in several space dimensions, (with Jacek Dziubański and Waldemar Hebisch), Comm. Partial Differential Equations 17 (1992), 1737-1758.
  • Fractal Burgers equations, (with Tadahisa Funaki and Wojbor A. Woyczyński), J. Differential Equations 148 (1998), 9-46.
  • Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715-743.
  • Long time behaviour of solutions to Nernst-Planck and Debye-Hueckel drift-diffusion systems, (with Jean Dolbeault), Ann. Henri Poincaré 1 (2000), 461-472.
  • Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, (with Grzegorz Karch and Wojbor A. Woyczyński), Ann. Inst. H. Poincaré - Analyse non Linéaire 18 (2001), 613-637.
  • Global regular and singular solutions for a model of gravitating particles, (with Marco Cannone, Ignacio A. Guerra and Grzegorz Karch), Math. Ann. 330 (2004), 693-708.
  • On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis, (with Lorenzo Brandolese), Studia Math. 193 (2009), 241-261.
  • A nonlinear diffusion of dislocation density and self-similar solutions, (with Grzegorz Karch and Régis Monneau), Comm. Math. Phys. 294 (2010), 145-168.
  • Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model, (with Lucilla Corrias and Jean Dolbeault), J. Math. Biology 63 (2011), 1-32.
  • Nonlocal porous medium equation: Barenblatt profiles and other weak solutions, (with Cyril Imbert and Grzegorz Karch), Arch. Rational Mech. Anal. 215 (2015), 497-529.
  • Large global-in-time solutions to a nonlocal model of chemotaxis, (with Grzegorz Karch and Jacek Zienkiewicz), Adv. Math. 330 (2018), 834-875.
  • Concentration phenomena in a diffusive aggregation model, (with Alexander Boritchev, Grzegorz Karch and Philippe Laurencot), J. Differential Equations 271 (2021), 1092-1108.