Analytical and numerical issues for non-conservative non-linear Boltzmann transport equation

1. Analytical and numerical issues • for non-conservative • non-linear Boltzmann transport equation • Irene M. Gamba • Department of Mathematics and ICES • The University of Texas at Austin • In collaboration with: • Alexandre Bobylev , Karlstad University, Sweden, and • Carlo Cercignani, Politecnico di Milano, Italy, on selfsimilar asymptotics • and decay rates to generalized models for multiplicative stochastic interactions. • Sri Harsha Tharkabhushanam , ICES- UT Austin, on Deterministic-Spectral solvers for non-conservative, non-linear Boltzmann transport equation • MAMOS workshop – UT Austin – October 07

3. Rarefied ideal gases-elastic: conservativeBoltzmann Transport eq. • Energy dissipative phenomena: Gas of elastic or inelastic interacting systems in the presence of a thermostat with a fixed background temperature өb or Rapid granular flow dynamics: (inelastic hard sphere interactions): homogeneous cooling states, randomly heated states, shear flows, shockwaves past wedges, etc. • (Soft) condensed matter at nano scale: Bose-Einstein condensates models and charge transport in solids: current/voltage transport modeling semiconductor. • Emerging applications from stochastic dynamics for multi-linear Maxwell type interactions : Multiplicatively Interactive Stochastic Processes: • Pareto tails for wealth distribution, non-conservative dynamics: opinion dynamic models, particle swarms in population dynamics, etc (Fujihara, Ohtsuki, Yamamoto’ 06,Toscani, Pareschi, Caceres 05-06…). • Goals: • Understanding of analytical properties: large energy tails • long time asymptotics and characterization of asymptotics states • A unified approach for Maxwell type interactions. • Development of deterministic schemes: spectral-Lagrangian methods

4. A general form for Boltzmann equation for binary interactions with external ‘heating’ sources

6. For a Maxwell type model: a linear equation for the kinetic energy

7. Time irreversibility is expressed in this inequality stability In addition: The Boltzmann Theorem:there are only N+2 collision invariants

11. ( )

12. asymptotics

13. An important application: The homogeneous BTE in Fourier space

16. Boltzmann Spectrum

19. A benchmark case:

22. Deterministic numerical method: Spectral Lagrangian solvers

31. Numerical simulations

35. Comparisons of energy conservation vs dissipation For a same initial state, we test the energy Conservative scheme and the scheme for the energy dissipative Maxwell-Boltzmann Eq.

38. Numerical simulations

41. Moments calculations: Thank you very much for your attention !!