# American Institute of Mathematical Sciences

December  2011, 4(4): 919-934. doi: 10.3934/krm.2011.4.919

## Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240 2 Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501 3 17-26 Iwasaki, Hodogaya, Yokohama 240-0015 4 School of Mathematics, Wuhan University, 430072, Wuhan 5 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  May 2011 Revised  June 2011 Published  November 2011

In this paper, we consider the Cauchy problem for the non-cutoff Boltzmann equation in the soft potential case. By using a singular change of velocity variables before and after collision, we prove the uniqueness of weak solutions to the Cauchy problem in the space of functions with polynomial decay in the velocity variable.
Citation: Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential. Kinetic & Related Models, 2011, 4 (4) : 919-934. doi: 10.3934/krm.2011.4.919
##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.  Google Scholar [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Rational Mech. Anal., 198 (2010), 39-123. doi: 10.1007/s00205-010-0290-1.  Google Scholar [3] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581. doi: 10.1007/s00220-011-1242-9.  Google Scholar [4] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Rational Mech. Anal., 202 (2011) 599-661. doi: 10.1007/s00205-011-0432-0.  Google Scholar [5] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff,, to appear in Kyoto J. Math., ().   Google Scholar [6] L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Ration. Mech. Anal., 193 (2009), 227-253. doi: 10.1007/s00205-009-0233-x.  Google Scholar [7] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. (2), 130 (1989), 321-366. doi: 10.2307/1971423.  Google Scholar [8] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Maths. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.  Google Scholar [9] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584.  Google Scholar [10] T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D., 188 (2004), 178-192. doi: 10.1016/j.physd.2003.07.011.  Google Scholar [11] X. Lu, A result on uniqueness of mild solutions of Boltzmann equations, Proceedings of the Fourteenth International Conference on Transport Theory (Beijing, 1995), Transport Theory Statist. Phys., 26 (1997), 209-220.  Google Scholar [12] G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.  Google Scholar [13] S. Ukai, Solutions of the Boltzmann equation, in "Pattern and Waves" (eds. M. Mimura and T. Nishida), 37-96, Studies of Mathematics and Its Applications, 18, North-Holland, Amsterdam, 1986.  Google Scholar [14] S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications (Singap.), 4 (2006), 263-310. doi: 10.1142/S0219530506000784.  Google Scholar [15] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics" (eds. S. Friedlander and D. Serre), Vol. I, 71-305, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

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##### References:
 [1] R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.  Google Scholar [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Rational Mech. Anal., 198 (2010), 39-123. doi: 10.1007/s00205-010-0290-1.  Google Scholar [3] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581. doi: 10.1007/s00220-011-1242-9.  Google Scholar [4] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Rational Mech. Anal., 202 (2011) 599-661. doi: 10.1007/s00205-011-0432-0.  Google Scholar [5] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann Equation without angular cutoff,, to appear in Kyoto J. Math., ().   Google Scholar [6] L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Ration. Mech. Anal., 193 (2009), 227-253. doi: 10.1007/s00205-009-0233-x.  Google Scholar [7] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. (2), 130 (1989), 321-366. doi: 10.2307/1971423.  Google Scholar [8] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Maths. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.  Google Scholar [9] P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584.  Google Scholar [10] T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D., 188 (2004), 178-192. doi: 10.1016/j.physd.2003.07.011.  Google Scholar [11] X. Lu, A result on uniqueness of mild solutions of Boltzmann equations, Proceedings of the Fourteenth International Conference on Transport Theory (Beijing, 1995), Transport Theory Statist. Phys., 26 (1997), 209-220.  Google Scholar [12] G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas, J. Statist. Phys., 94 (1999), 619-637. doi: 10.1023/A:1004589506756.  Google Scholar [13] S. Ukai, Solutions of the Boltzmann equation, in "Pattern and Waves" (eds. M. Mimura and T. Nishida), 37-96, Studies of Mathematics and Its Applications, 18, North-Holland, Amsterdam, 1986.  Google Scholar [14] S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications (Singap.), 4 (2006), 263-310. doi: 10.1142/S0219530506000784.  Google Scholar [15] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Mathematical Fluid Dynamics" (eds. S. Friedlander and D. Serre), Vol. I, 71-305, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar
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