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1

Existence of Solutions for the Keller–Segel Model of Chemotaxis with Measures as Initial Data

Biler P., Zienkiewicz J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2015

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Vol. 63, no. 1

41--51

EN

A simple proof of the existence of solutions for the two-dimensional Keller-Segel model with measures with all the atoms less than 8π as the initial data is given. This result was obtained by Senba and Suzuki (2002) and Bedrossian and Masmoudi (2014) using different arguments. Moreover, we show a uniform bound for the existence time of solutions as well as an optimal hypercontractivity estimate.

2

Równania różniczkowe cząstkowe na przełomie XX i XXI wieku

Biler P.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

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2003

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[Z] 39

65-84

3

Asymptotics and high dimensional approximations for nonlinear pseudodifferential equations involving Levy generators

Biler P., Karch G., Woyczyński W.

Demonstratio Mathematica

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2001

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Vol. 34, nr 2

403-413

EN

Nonlinear pseudodifferential equations involving Levy semigroup generators are used in physical models where the diffusive behavior is affected by hopping and trapping phenomena. In this paper we present several results concerning asymptotics and high dimensional Monte Carlo-type approximations via interacting particle systems for two classes of such equations.

4

Interacting particle approximation for nonlocal quadratic evolution problems

Biler P., Funaki T., Woyczynski W. A.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 2

267--286

EN

The existence of McKean's nonlinear jump Markov processes and related Monte Carlo type approximation schemes by interacting particle systems (propagation of chaos) are studied for a class of multidimensional doubly nonlocal evolution problems with a fractional power of the Laplacian and a quadratic nonlinearity involving an integral operator. Asymptotically, these equations model the evolution of density of mutually interacting particles with anomalous (fractal) Lévy diffusion.