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Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
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 |
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. |
[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. |
[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. |
[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. |
[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., ().
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[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. |
[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. |
[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. |
[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., ().
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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