On the Uniqueness for the Spatially Homogeneous Boltzmann Equation with a Strong Angular Singularity
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We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules.
Our proof relies on the ideas of Tanaka (Z. Wahrscheinlichkeitstheor. Verwandte. Geb. 46(1):67–105, ) we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other.
KeywordsBoltzmann equation without cutoff Long-range interaction Uniqueness Wasserstein distance Quadratic cost
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- 4.Desvillettes, L.: Boltzmann’s Kernel and the Spatially Homogeneous Boltzmann Equation. Riv. Mat. dell’Univ. Parma 6(4), 1–22 (2001) (special issue) Google Scholar
- 6.Desvillettes, L., Mouhot, C.: Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. arXiv eprint math.AP/0606307 (2006) Google Scholar
- 8.Fontbona, J., Guérin, H., Méléard, S.: Measurability of optimal transportation and convergence rate for Landau type interacting particle systems. Preprint (2007) Google Scholar
- 12.Fournier, N., Mouhot, C.: On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Preprint (2007) Google Scholar
- 13.Horowitz, J., Karandikar, R.L.: Martingale problems associated with the Boltzmann equation. In: Seminar on Stochastic Processes, San Diego, CA, 1989. Progr. Probab., vol. 18, pp. 75–122. Birkhäuser, Boston (1990) Google Scholar
- 16.Lu, X., Mouhot, C.: About measures solutions of the spatially homogeneous Boltzmann equation. Work in progress Google Scholar
- 19.Tanaka, H.: On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules. In: Proceedings of the International Symposium on Stochastic Differential Equations. Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976, pp. 409–425. Wiley, New York (1978) Google Scholar