Topics:
Motivation:
Maxwell and Boltzmann were the first to propose a statistical description of the molecules in a monoatomic gas and the equation -- known as the Boltzmann equation -- governing the evolution of their velocity distribution (1866-1872). In the 20th century, the field of application of kinetic models has kept broadening: polyatomic or ionized gases, plasmas (with the Vlasov and Landau, or Balescu-Lenard equations)... In 1965, Zakharov proposed an equation similar to that of Boltzmann to describe the turbulence of waves corresponding to the interaction of Langmuir plasmons. He showed this equation has solution similar to the power law obtained by Kolmogorov in 1941 for the spectrum of the homogeneous and isotropic turbulence in the case of fluids.
Very recently, several major breakthroughs were announced regarding kinetic models: proof of the regularity of the solutions to the Landau equation by Guillen and Silvestre in 2023, justification of the validity of the Zakharov equation by Hani and Deng in 2023, then of the Boltzmann equation by Hani, Deng and Ma in 2024. The purpose of this colloquium is to take stock of various aspects of kinetic models and their connection to equations of fluid mechanics.
Claude Bardos himself contributed significantly to this theory by establishing, among other things, the regularity of global solutions to the Vlasov-Poisson system with P. Degond in 1983, the limit of the Boltzmann equation as the Navier-Stokes equation with S. Ukai in 1991 in the case of 'small' initial data... For that matter, and to celebrate his 85th birthday, this conference is dedicated to him.
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