臺灣博碩士論文加值系統

1988年時，Golse、Lions、Perthame和Sentis證明了速度平均會增加函數正則性，這個現象後來被稱作「速度平均引理」，速度平均引理在DiPerna和Lions的波茲曼方程之柯西問題整體解存在性理論有重要的應用，而其自身也有許多富有意義的延伸，在此論文中，我們首先回顧經典的速度平均引理，再透過速度平均效應導出我們關於穩態線性化波茲曼方程正則性的新結果。

In 1988, Golse, Lions, Perthame and Sentis jointly proved that velocity averaging has regularizing effects. This phenomenon was later called “Velocity Averaging Lemmas.” The Velocity Averaging Lemmas have significant applications in the global existence theory of the Cauchy problem for Boltzmann equations by DiPerna and Lions and also have many meaningful extensions themselves. In this thesis, we first review classical Velocity Averaging Lemmas, and then we present our new regularity results for the stationary linearized Boltzmann equation by velocity averaging effects.

誌謝 i
中文摘要 ii
Abstract iii
目錄 iv
1 Introduction 1
2 Preliminaries 3
3 Velocity Averaging Lemmas 5
4 Regularity of the Stationary Linearized Boltzmann Equation 22
References 30

[1] R. A. Adams and J. J. Fournier. Sobolev spaces, volume 140. Elsevier, 2003.
[2] V. I. Agoshkov. Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation. In Dokl. Akad. Nauk SSSR, volume 276, pages 1289–1293, 1984.
[3] R. E. Caflisch. The boltzmann equation with a soft potential. Communications in Mathematical Physics, 74(1):71–95, 1980.
[4] C. Cercignani, R. Illner, and M. Pulvirenti. The mathematical theory of dilute gases, volume 106. Springer Science & Business Media, 2013.
[5] M. Cessenat. Th´eoremes de trace lp pour des espaces de fonctions de la neutronique. Comptes rendus des s´eances de l’Acad´emie des sciences. S´erie 1, Math´ematique, 299(16):831–834, 1984.
[6] M. Cessenat and R. DAUTRAY. Th´eoremes de trace pour des espaces de fonctions de la neutronique. Comptes rendus des s´eances de l’Acad´emie des sciences. S´erie 1, Math´ematique, 300(3):89–92, 1985.
[7] I.-K. Chen. Regularity of stationary solutions to the linearized boltzmann equations. SIAM Journal on Mathematical Analysis, 50(1):138–161, 2018.
[8] I.-K. Chen, C.-H. Hsia, and D. Kawagoe. Regularity for diffuse reflection boundary problem to the stationary linearized boltzmann equation in a convex domain. In Annales de l’Institut Henri Poincar´e C, Analyse non lin´eaire. Elsevier, 2018.
[9] L. Desvillettes. About the use of the fourier transform for the boltzmann equation. Riv. Mat. Univ. Parma, 7(2):1–99, 2003.
[10] R. DeVore and G. Petrova. The averaging lemma. Journal of the American Mathematical Society, 14(2):279–296, 2001.
[11] E. DI NEZZA, G. PALATUCCI, and E. VALDINOCI. Hitchhiker’s guide to the fractional sobolev spaces. arXiv preprint arXiv:1104.4345, 2011.
[12] R. J. DiPerna and P.-L. Lions. On the cauchy problem for boltzmann equations: global existence and weak stability. Annals of Mathematics, pages 321–366, 1989.
[13] R. J. DiPerna, P.-L. Lions, and Y. Meyer. Lp regularity of velocity averages. In Annales de l’Institut Henri Poincare (C) Non Linear Analysis, volume 8, pages 271–287. Elsevier, 1991.
[14] L. C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010.
[15] F. Golse. Un r´esultat pour les ´equations de transport et application au calcul de la limite de la valeur propre principale d’un op´erateur de transport. Note CR Acad. Sci. Paris, 301:341–344, 1985.
[16] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. Journal of functional analysis, 76(1):110–125, 1988.
[17] H. Grad. Asymptotic theory of the boltzmann equation. The physics of Fluids, 6(2):147–181, 1963.
[18] P.-E. Jabin. Averaging lemmas and dispersion estimates for kinetic equations. Rivista di Matematica della Universita di Parma, contribution to the special issue devoted to the Summer School, 2008.
[19] P.-E. Jabin and B. Perthame. Regularity in kinetic formulations via averaging lemmas. ESAIM: Control, Optimisation and Calculus of Variations, 8:761–774, 2002.
[20] D. Kawagoe. Regularity of solutions to the stationary transport equation with the incoming boundary data. 2018.
[21] T.-P. Liu and S.-H. Yu. The green’s function and large-time behavior of solutions for the one-dimensional boltzmann equation. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 57(12):1543–1608, 2004. [1] R. A. Adams and J. J. Fournier. Sobolev spaces, volume 140. Elsevier, 2003.
[2] V. I. Agoshkov. Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation. In Dokl. Akad. Nauk SSSR, volume 276, pages 1289–1293, 1984.
[3] R. E. Caflisch. The boltzmann equation with a soft potential. Communications in Mathematical Physics, 74(1):71–95, 1980.
[4] C. Cercignani, R. Illner, and M. Pulvirenti. The mathematical theory of dilute gases, volume 106. Springer Science & Business Media, 2013.
[5] M. Cessenat. Th´eoremes de trace lp pour des espaces de fonctions de la neutronique. Comptes rendus des s´eances de l’Acad´emie des sciences. S´erie 1, Math´ematique, 299(16):831–834, 1984.
[6] M. Cessenat and R. DAUTRAY. Th´eoremes de trace pour des espaces de fonctions de la neutronique. Comptes rendus des s´eances de l’Acad´emie des sciences. S´erie 1, Math´ematique, 300(3):89–92, 1985.
[7] I.-K. Chen. Regularity of stationary solutions to the linearized boltzmann equations. SIAM Journal on Mathematical Analysis, 50(1):138–161, 2018.
[8] I.-K. Chen, C.-H. Hsia, and D. Kawagoe. Regularity for diffuse reflection boundary problem to the stationary linearized boltzmann equation in a convex domain. In Annales de l’Institut Henri Poincar´e C, Analyse non lin´eaire. Elsevier, 2018.
[9] L. Desvillettes. About the use of the fourier transform for the boltzmann equation. Riv. Mat. Univ. Parma, 7(2):1–99, 2003.
[10] R. DeVore and G. Petrova. The averaging lemma. Journal of the American Mathematical Society, 14(2):279–296, 2001.
[11] E. DI NEZZA, G. PALATUCCI, and E. VALDINOCI. Hitchhiker’s guide to the fractional sobolev spaces. arXiv preprint arXiv:1104.4345, 2011.
[12] R. J. DiPerna and P.-L. Lions. On the cauchy problem for boltzmann equations: global existence and weak stability. Annals of Mathematics, pages 321–366, 1989.
[13] R. J. DiPerna, P.-L. Lions, and Y. Meyer. Lp regularity of velocity averages. In Annales de l’Institut Henri Poincare (C) Non Linear Analysis, volume 8, pages 271–287. Elsevier, 1991.
[14] L. C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010.
[15] F. Golse. Un r´esultat pour les ´equations de transport et application au calcul de la limite de la valeur propre principale d’un op´erateur de transport. Note CR Acad. Sci. Paris, 301:341–344, 1985.
[16] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. Journal of functional analysis, 76(1):110–125, 1988.
[17] H. Grad. Asymptotic theory of the boltzmann equation. The physics of Fluids, 6(2):147–181, 1963.
[18] P.-E. Jabin. Averaging lemmas and dispersion estimates for kinetic equations. Rivista di Matematica della Universita di Parma, contribution to the special issue devoted to the Summer School, 2008.
[19] P.-E. Jabin and B. Perthame. Regularity in kinetic formulations via averaging lemmas. ESAIM: Control, Optimisation and Calculus of Variations, 8:761–774, 2002.
[20] D. Kawagoe. Regularity of solutions to the stationary transport equation with the incoming boundary data. 2018.
[21] T.-P. Liu and S.-H. Yu. The green’s function and large-time behavior of solutions for the one-dimensional boltzmann equation. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 57(12):1543–1608, 2004.