# Niclas Bernhoff

## Research

My research interests are in kinetic theory and related topics. Especially, I study the Boltzmann equation or Boltzmann-like equations.

Previously, I have mainly studied these equations by considering discrete velocity models (DVMs) - i.e., models where the velocity variable (or a corresponding variable) only takes a finite number of values - for them. It is a well-known fact that DVMs can approximate the Boltzmann equation up to any order. I have studied existence and uniqueness of boundary layers for DVMs of the Boltzmann equation for single species, binary mixtures (of two vapors or one vapor and one noncondensable gas), multicomponent mixtures, polyatomic molecules (with different internal energies), and chemical reactions, but also discrete versions of the quantum Boltzmann equation (Nordheim-Boltzmann or Uehling-Uhlenbeck equation), including an extension for anyons, and an equation for excitations in a Bose gas interacting with a Bose condensate. I have also studied existence and uniqueness of shock profiles for DVMs of the Boltzmann equation for single species, mixtures, polyatomic molecules (with different internal energies), and chemical reactions.

More recently, I have studied existence and uniqueness of boundary layers for the full Boltzmann equation - in form of joint work, or, to a large extent based on discussions, with Prof. Francois Golse at Ecole Polytechnique - including for regime transitions. Studies for corresponding linearized problems have been made also in a general setting, i.e., applicable for models for multicomponent and polyatomic gases. Essential mathematical properties of the (full) linearized collision operator for multicomponent and polyatomic gases have been studied in a series of recent papers.

## Bio

Niclas was born in 1971 in Lund. He got his B.Sc. (M.Sc.) in Mathematics at Uppsala University in 1995 (1998) and a diploma as Upper Secondary School Teacher in Mathematics from Stockholm Institute of Education in 1996.

In July 1996 he become a lecturer at Karlstad University, where he began his PhD studies in 2000, presented his licentiate thesis "Discrete Velocity Models and Half-Space Problems" in 2003, and awarded his PhD in Mathematics in 2005. The title of his PhD thesis was "On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation". In 2005 he become promoted to senior lecturer/assistant professor, and in 2015 he become a Docent (associate professor) in Mathematics at Karlstad University.

Niclas has done research visits at the Mathematical, Physical and Computer Sciences at Parma University, Italy, (2 weeks May 2012, 1 week November 2013/October 2019/September 2022 - supported by SVeFUM in 2012/2022), at the Department of Mechanical Engineering and Science, Graduate School of Engineering at Kyoto University, Japan (1 month, October 2009, 2 weeks November 2015 - the visits were supported by the Japan Society for the Promotion of Science: No. PE 09549 and BR 150501), at the Department of Mathematics at Bordeaux University (1 week September 2016/October 2017 - supported by French Institute in Sweden in 2016 and by SVeFUM in 2017), Department of Mathematics at Ecole Polytechnique, Paris (1 week September 2016/October 2017/May 2019/November 2021/2022 visiting also Paris Diderot University in 2016/2017, Paris Dauphine University in 2016, Paris Descartes University in 2019 and Sorbonne University - Pierre and Marie Curie Campus - in 2019/2022 - supported by French Institute in Sweden in 2016, by SVeFUM in 2017/2019/2021, and by the Royal Swedish Academy of Sciences: MA2020-0007 in 2022), in Hong Kong at Chinese University of Hong Kong and City University of Hong Kong (1 week October 2017), and at Institute for Mathematics at TU Ilmenau (1 week October 2018 - supported by DFG-project 393468618).

Niclas was second reviewer for the doctoral thesis of Thomas Sasse at TU Ilmenau in 2022.

Niclas was main organizer of the "7th International Workshop on Kinetic Theory & Applications" at Karlstad University in June 19-21, 2017 and (together with Bernt Wennberg, Chalmers University of Technology) of the "6th International Workshop on Kinetic Theory & Applications" at Karlstad University in May 12-14, 2014, and gave himself several talks in international conferences.

Niclas also participated in teacher exchanges inside the Erasmus/Erasmus+ Programmes at the Department of Mathematics at Lucian Blaga, University of Sibiu (3 weeks May 2009, 2 weeks May 2010, 2 weeks May 2011, 2 weeks April-May 2013, 1 week March 2016, 1 week May-June 2017), the Department of Mathematics at University of Catania (1 week October 2014), and the Department of Mathematical, Physical and Computer Sciences at Parma University (1 week May 2018).

## Related info

The Boltzmann equation: in the trail of Torsten Carleman May 16-22, 2022, Stockholm

The Legacy of Carlo Cercignani: from Kinetic Theory to Turbulence Modeling May 24-28, 2021, Milan/Online, Italy

International Congress on Industrial and Applied Mathematics (ICIAM 2019) July 15-19, 2019, Valencia, Spain

Workshop on Kinetic and Related models, October 25-26, 2017, Hong Kong

7th International Workshop on Kinetic Theory and Applications, June 19-21, 2017, Karlstad

Recent Advances on Particle Systems in Kinetic Theory, May 8-12, 2017, Austin, Texas

## Selected publications

N. Bernhoff,* *Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures*, *Acta Appl. Math. 183, 3:1-45 (2023)

N. Bernhoff,* *Half-Space Problems for the Boltzmann Equation of Multicomponent Mixtures*, *in From Kinetic Theory to Turbulence Modeling, Springer (2023) pp. 37-49

N. Bernhoff and F. Golse,* *On the Boundary Layer Equations with Phase Transition in the Kinetic Theory of Gases*, *Arch. Ration. Mech. Anal. 240, 51-98 (2021)

N. Bernhoff, Discrete velocity models for polyatomic molecules without nonphysical collision invariants, J. Stat. Phys. 172, 742-761 (2018)

N. Bernhoff, Boundary layers for discrete kinetic models: multicomponent mixtures, polyatomic molecules, bimolecular chemical reactions, and quantum kinetic equations, Kinet. Relat. Models 10, 925-955 (2017)

N. Bernhoff and M. Vinerean, Discrete velocity models for mixtures without nonphysical collision invariants, J. Stat. Phys. 165, 434-453 (2016)

N. Bernhoff and A.V. Bobylev, Weak shock waves for the general discrete velocity model of the Boltzmann equation, Comm. Math. Sci. 5, 815-832 (2007)

( see also publications)

## Publications

- Niclas Bernhoff, 2023
- Niclas Bernhoff, Francois Golse, 2021
- Niclas Bernhoff, 2019
- Niclas Bernhoff, 2018
- Niclas Bernhoff, 2017
- Niclas Bernhoff, 2016
- Niclas Bernhoff, Mirela Vinerean, 2016
- Niclas Bernhoff, 2016
- Niclas Bernhoff, 2015
- Niclas Bernhoff, 2012
- Niclas Bernhoff, 2012
- Niclas Bernhoff, 2012
- Niclas Bernhoff, 2010
- Niclas Bernhoff, 2010
- Niclas Bernhoff, 2008
- Niclas Bernhoff, Alexander Bobylev, 2007
- Niclas Bernhoff, Alexander Bobylev, 2007
- Niclas Bernhoff, 2005
- Alexander Bobylev, Niclas Bernhoff, 2003
- Niclas Bernhoff, 2003