Research
My primary research interest is the analysis of kinetic equations:
Boltzmann and related models. The Boltzmann equation is one of the
fundamental equations of statistical physics, describing the evolution
of a phase space density function for a rarefied gas. It also provides
a systematic way to derive the fluid dynamics equations (Euler,
Navier-Stokes...) from the principles of molecular dynamics.
There have been significant advances in the mathematical theory
of kinetic and fluid dynamic equations made in the recent years,
but many fundamental problems are still open and much work remains to
be done towards a rigorous theory.
I am interested
in the theory of weak solutions of kinetic equations, their possible
regularity (or singularities), and in applications of such equations
to different specific physical systems. Some of my recent work concerns
the applications of the ideas from kinetic theory to studying flows
of granular materials; I am also interested in kinetic models of wave turbulence
(which appear in statistical description of waves on the free water
surface).
Many important analytical ideas used in the theory come from
the property of the dissipation of entropy (more specifically, the
Boltzmann or the "f log f" entropy). I am more broadly interested in
the scope of methods based on the dissipation of entropy-like quantities,
which become more and more prominent in the modern PDE theory.
Recent talks:
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Regularity for the Boltzmann equation in one-dimensional spatial
geometry,
BIRS Workshop on Hyperbolic Systems of Conservation laws,
Banff, AB, Canada, Nov 1, 2006: PDF.
Papers and preprints:
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A. Biryuk, W. Craig, V. Panferov; Strong solutions of the
Boltzmann equation
in one spatial dimension, C. R. Math. Acad. Sci. Paris 342
(2006), no. 11, 843--848., preprint
PDF
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I. M. Gamba, V. Panferov and C. Villani; Upper Maxwellian bounds
for the spatially homogeneous Boltzmann equation, preprint
PDF.
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M. Herty, R. Illner, A. Klar, V. Panferov; On the qualitative
properties of the systems of Fokker-Planck equations for multilane
traffic flows, to appear in Transport Theory and Statistical
Physics, PDF
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A. V. Bobylev, I. M. Gamba and V. Panferov; Moment inequalities
and high-energy tails for the Boltzmann equations with inelastic
interactions, Journ. Stat. Phys. vol. 116, no. 5-6 pp. 1651--1682 (2004),
preprint
available from ArXiv.org;
paper
available from the journal.
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I. M. Gamba, V. Panferov and C. Villani; On the Boltzmann
equation for diffusively excited granular media;
Comm. Math. Phys. 246, no. 3, pp. 503-541 (2004);
preprint
available from ArXiv.org; paper
available from CMP.
- I. M. Gamba, V. Panferov and C. Villani;
On the inelastic Boltzmann equation with diffusive forcing.
Nonlinear problems in mathematical physics and related topics,
II, 179--192, Int. Math. Ser. (N. Y.), 2,
Kluwer/Plenum, New York, 2002.
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V. Panferov, On the interior boundary-value problem
for the stationary Povzner equation with hard and soft
interactions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 771--825.
preprint version:
Postscript, PDF.
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V. Panferov and A. Heintz; A new discrete-velocity model for
the Boltzmann equation, Math. Methods Appl. Sci. 25 (2002),
no. 7, 571--593. preprint:
Postscript, PDF;
full text of the
paper
available from Wiley Interscience (access required)