Sharp regularizing estimates for the gain term of the Boltzmann collision operator

Jin-Cheng Jiang

Department of Mathematics

National Tsing Hua University

jcjiang@math.nthu.edu.tw

We prove the sharp regularizing estimates for the gain term of the Boltzmann collision operator including hard sphere, hard potential and Maxwell molecule models. Our new estimates characterize both regularization and convolution properties of the gain term which were studied by Lions [4], Wennberg [6], Bouchut & Desvillettes [2], Mouhot & Villani [5] etc. and Duduchava & Kirsch & Rjasanow [3], Alonso & Carneiro & Gamba [1] etc. respectively. The new estimates have the following features. The regularizing exponent is sharp both in the `L^2` based inhomogeneous and homogeneous Sobolev spaces which is exact the exponent of the kinetic part of collision kernel. The functions in these estimates belong to a wider scope of (weighted) Lebesgue spaces than the previous regularizing estimates. Furthermore, for the estimates in homogeneous Sobolev spaces, we only need functions lying in Lebesgue spaces instead of weighted Lebesgue spaces, i.e., no loss of weight occurs in this case.Keyword: Boltzmann collision operator, Gain term, Regularizing, hard sphere, hard potential, Maxwell molecule, Fourier integral operator

References

[1] R. Alonso, E. Carneiro and I.M. Gamba, Convolution inequalities for the Boltzmann collision operator, Comm. Math. Physics, 298 (2010), pp. 293-322.[2] F. Bouchut and L. Desvillettes, A proof of the smoothing properties of the positive part of Boltzmann's kernel, Rev. Mat. Iberoamericana, 14 (1998), pp. 47-61.

[3] R. Duduchava, R. Kirsch and S. Rjasanow, On estimates of the Boltzmann collision operator with cutoff, Comm. Math. Physics, 298 (2010), pp. 293-322.

[4] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, J. Math. Kyoto Univ., 34 (1994), pp. 391-427, pp. 429-461.

[5] C. Mouhot and C. Villani, Regularity Theory for the Spatially Homogeneous Boltzmann Equation with Cut-Off, Arch. Rational Mech. Anal., 173 (2004), pp. 169-212.

[6] B. Wennberg, Regularity in the Boltzmann equation and the radon transform, Comm. Partial Differential Equations, 19 (1994), pp. 2057-2074.