Duan, Renjun; Sakamoto, Shota; Ueda, Yoshihiro An \(L^1_k\cap L^p_k\) approach for the non-cutoff Boltzmann equation in \(\mathbb{R}^3\). (English) Zbl 07802036 SIAM J. Math. Anal. 56, No. 1, 762-800 (2024). MSC: 35Q20 35B65 35B45 35A01 35A02 PDFBibTeX XMLCite \textit{R. Duan} et al., SIAM J. Math. Anal. 56, No. 1, 762--800 (2024; Zbl 07802036) Full Text: DOI arXiv
Cao, Chuqi; Duan, Renjun; Li, Zongguang Time-velocity decay of solutions to the non-cutoff Boltzmann equation in the whole space. arXiv:2402.02804 Preprint, arXiv:2402.02804 [math.AP] (2024). BibTeX Cite \textit{C. Cao} et al., ``Time-velocity decay of solutions to the non-cutoff Boltzmann equation in the whole space'', Preprint, arXiv:2402.02804 [math.AP] (2024) Full Text: arXiv OA License
Chen, Hongxu; Duan, Renjun Boltzmann equation with mixed boundary condition. arXiv:2401.01058 Preprint, arXiv:2401.01058 [math.AP] (2024). BibTeX Cite \textit{H. Chen} and \textit{R. Duan}, ``Boltzmann equation with mixed boundary condition'', Preprint, arXiv:2401.01058 [math.AP] (2024) Full Text: arXiv OA License
Duan, Renjun; Ko, Gyounghun; Lee, Donghyun The Boltzmann equation with a class of large-amplitude initial data and specular reflection boundary condition. (English) Zbl 07782589 J. Stat. Phys. 190, No. 12, Paper No. 189, 46 p. (2023). MSC: 35Q20 76P05 35B40 35B45 35B65 35A01 35A02 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Stat. Phys. 190, No. 12, Paper No. 189, 46 p. (2023; Zbl 07782589) Full Text: DOI arXiv
Duan, Renjun; Yang, Dongcheng; Yu, Hongjun Compressible Euler-Maxwell limit for global smooth solutions to the Vlasov-Maxwell-Boltzmann system. (English) Zbl 1522.35402 Math. Models Methods Appl. Sci. 33, No. 10, 2157-2221 (2023). MSC: 35Q35 35Q20 35Q61 76X05 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 33, No. 10, 2157--2221 (2023; Zbl 1522.35402) Full Text: DOI arXiv
Duan, Renjun; Li, Zongguang Global bounded solutions to the Boltzmann equation for a polyatomic gas. (English) Zbl 1518.35525 Int. J. Math. 34, No. 7, Article ID 2350036, 43 p. (2023). MSC: 35Q20 35Q35 35B35 35B20 35A01 35A02 35R09 76P05 76N15 PDFBibTeX XMLCite \textit{R. Duan} and \textit{Z. Li}, Int. J. Math. 34, No. 7, Article ID 2350036, 43 p. (2023; Zbl 1518.35525) Full Text: DOI arXiv
Deng, Dingqun; Duan, Renjun Low regularity solutions for the Vlasov-Poisson-Landau/Boltzmann system. (English) Zbl 1511.35252 Nonlinearity 36, No. 5, 2193-2248 (2023). MSC: 35Q20 35Q83 35J05 76X05 82C22 82C40 35B65 35A01 PDFBibTeX XMLCite \textit{D. Deng} and \textit{R. Duan}, Nonlinearity 36, No. 5, 2193--2248 (2023; Zbl 1511.35252) Full Text: DOI arXiv
Deng, Dingqun; Duan, Renjun Spectral gap formation to kinetic equations with soft potentials in bounded domain. (English) Zbl 1508.35032 Commun. Math. Phys. 397, No. 3, 1441-1489 (2023). MSC: 35Q20 82C40 35P15 76P05 PDFBibTeX XMLCite \textit{D. Deng} and \textit{R. Duan}, Commun. Math. Phys. 397, No. 3, 1441--1489 (2023; Zbl 1508.35032) Full Text: DOI arXiv
Duan, Renjun; Yang, Dongcheng; Yu, Hongjun KdV limit for the Vlasov-Poisson-Landau system. arXiv:2308.08863 Preprint, arXiv:2308.08863 [math.AP] (2023). BibTeX Cite \textit{R. Duan} et al., ``KdV limit for the Vlasov-Poisson-Landau system'', Preprint, arXiv:2308.08863 [math.AP] (2023) Full Text: arXiv OA License
Duan, Renjun; Liu, Shuangqian On smooth solutions to the thermostated Boltzmann equation with deformation. (English) Zbl 07730350 Commun. Math. Anal. Appl. 1, No. 1, 152-212 (2022). MSC: 35Q20 35B40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, Commun. Math. Anal. Appl. 1, No. 1, 152--212 (2022; Zbl 07730350) Full Text: DOI arXiv
Duan, Renjun; Liu, Shuangqian; Yang, Tong; Zhang, Zhu Heat transfer problem for the Boltzmann equation in a channel with diffusive boundary condition. (English) Zbl 1503.35156 Chin. Ann. Math., Ser. B 43, No. 6, 1071-1100 (2022). MSC: 35Q35 35Q20 35Q79 76N06 76P05 PDFBibTeX XMLCite \textit{R. Duan} et al., Chin. Ann. Math., Ser. B 43, No. 6, 1071--1100 (2022; Zbl 1503.35156) Full Text: DOI arXiv
Duan, Renjun; Yang, Dongcheng; Yu, Hongjun Asymptotics toward viscous contact waves for solutions of the Landau equation. (English) Zbl 1504.35311 Commun. Math. Phys. 394, No. 1, 471-529 (2022). MSC: 35Q35 35Q20 76P05 76N10 35B40 35A01 35A02 PDFBibTeX XMLCite \textit{R. Duan} et al., Commun. Math. Phys. 394, No. 1, 471--529 (2022; Zbl 1504.35311) Full Text: DOI arXiv
Duan, Renjun; Li, Wei-Xi; Liu, Lvqiao Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation. (English) Zbl 1481.35303 Adv. Math. 395, Article ID 108159, 75 p. (2022). MSC: 35Q20 35B65 35B20 35H10 35A01 35A02 82C40 PDFBibTeX XMLCite \textit{R. Duan} et al., Adv. Math. 395, Article ID 108159, 75 p. (2022; Zbl 1481.35303) Full Text: DOI arXiv
Duan, Renjun; Li, Zongguang; Liu, Shuangqian Global mild solution with polynomial tail for the Boltzmann equation in the whole space. arXiv:2212.04676 Preprint, arXiv:2212.04676 [math.AP] (2022). BibTeX Cite \textit{R. Duan} et al., ``Global mild solution with polynomial tail for the Boltzmann equation in the whole space'', Preprint, arXiv:2212.04676 [math.AP] (2022) Full Text: arXiv OA License
Duan, Renjun; Yang, Dongcheng; Yu, Hongjun Global quasineutral Euler limit for the Vlasov-Poisson-Landau system with rarefaction waves. arXiv:2212.07654 Preprint, arXiv:2212.07654 [math.AP] (2022). BibTeX Cite \textit{R. Duan} et al., ``Global quasineutral Euler limit for the Vlasov-Poisson-Landau system with rarefaction waves'', Preprint, arXiv:2212.07654 [math.AP] (2022) Full Text: arXiv OA License
Duan, Renjun; Liu, Shuangqian Uniform shear flow via the Boltzmann equation with hard potentials. arXiv:2210.12340 Preprint, arXiv:2210.12340 [math.AP] (2022). BibTeX Cite \textit{R. Duan} and \textit{S. Liu}, ``Uniform shear flow via the Boltzmann equation with hard potentials'', Preprint, arXiv:2210.12340 [math.AP] (2022) Full Text: arXiv OA License
Duan, Renjun; Yang, Dongcheng; Yu, Hongjun Compressible fluid limit for smooth solutions to the Landau equation. arXiv:2207.01184 Preprint, arXiv:2207.01184 [math.AP] (2022). BibTeX Cite \textit{R. Duan} et al., ``Compressible fluid limit for smooth solutions to the Landau equation'', Preprint, arXiv:2207.01184 [math.AP] (2022) Full Text: arXiv OA License
Duan, Renjun; Liu, Shuangqian The Boltzmann equation for uniform shear flow. (English) Zbl 1477.76075 Arch. Ration. Mech. Anal. 242, No. 3, 1947-2002 (2021). MSC: 76P05 35Q20 35Q35 82C40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, Arch. Ration. Mech. Anal. 242, No. 3, 1947--2002 (2021; Zbl 1477.76075) Full Text: DOI arXiv
Duan, Renjun; Liu, Shuangqian; Yang, Tong Global classical solutions for the Vlasov-Nordström-Fokker-Planck system. (English) Zbl 1478.35206 SIAM J. Math. Anal. 53, No. 5, 6164-6190 (2021). MSC: 35Q84 35Q20 35Q83 35A09 35A01 PDFBibTeX XMLCite \textit{R. Duan} et al., SIAM J. Math. Anal. 53, No. 5, 6164--6190 (2021; Zbl 1478.35206) Full Text: DOI
Duan, Renjun; Yin, Haiyan; Zhu, Changjiang A half-space problem on the full Euler-Poisson system. (English) Zbl 1476.35032 SIAM J. Math. Anal. 53, No. 5, 6094-6121 (2021). MSC: 35B35 35B40 35L50 35L60 PDFBibTeX XMLCite \textit{R. Duan} et al., SIAM J. Math. Anal. 53, No. 5, 6094--6121 (2021; Zbl 1476.35032) Full Text: DOI arXiv
Duan, Renjun; Liu, Shuangqian Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains. (English) Zbl 1479.35661 Trans. Am. Math. Soc. 374, No. 11, 7867-7924 (2021). MSC: 35Q35 35Q31 35Q20 76P05 76N06 82C40 35B65 35A01 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, Trans. Am. Math. Soc. 374, No. 11, 7867--7924 (2021; Zbl 1479.35661) Full Text: DOI arXiv
Duan, Renjun; Liu, Shuangqian; Sakamoto, Shota; Strain, Robert M. Global mild solutions of the Landau and non-cutoff Boltzmann equations. (English) Zbl 1476.35161 Commun. Pure Appl. Math. 74, No. 5, 932-1020 (2021). Reviewer: Eugene Postnikov (Kursk) MSC: 35Q20 35Q82 35B65 35A01 35A02 82C22 82C40 PDFBibTeX XMLCite \textit{R. Duan} et al., Commun. Pure Appl. Math. 74, No. 5, 932--1020 (2021; Zbl 1476.35161) Full Text: DOI arXiv
Duan, Renjun; Yang, Dongcheng; Yu, Hongjun Small Knudsen rate of convergence to rarefaction wave for the Landau equation. (English) Zbl 1467.82074 Arch. Ration. Mech. Anal. 240, No. 3, 1535-1592 (2021). MSC: 82C40 35Q20 76P05 35Q31 PDFBibTeX XMLCite \textit{R. Duan} et al., Arch. Ration. Mech. Anal. 240, No. 3, 1535--1592 (2021; Zbl 1467.82074) Full Text: DOI arXiv
Duan, Renjun; Yu, Hongjun The 3D Vlasov-Poisson-Landau system near 1D local Maxwellians. (English) Zbl 1462.35396 J. Stat. Phys. 182, No. 2, Paper No. 33, 101 p. (2021). MSC: 35Q83 35Q31 35A01 35B35 35B40 76P05 76X05 PDFBibTeX XMLCite \textit{R. Duan} and \textit{H. Yu}, J. Stat. Phys. 182, No. 2, Paper No. 33, 101 p. (2021; Zbl 1462.35396) Full Text: DOI
Duan, Renjun; Liu, Shuangqian; Yang, Tong The Boltzmann equation for plane Couette flow. arXiv:2107.02458 Preprint, arXiv:2107.02458 [math.AP] (2021). BibTeX Cite \textit{R. Duan} et al., ``The Boltzmann equation for plane Couette flow'', Preprint, arXiv:2107.02458 [math.AP] (2021) Full Text: arXiv OA License
Duan, Renjun; Liu, Shuangqian; Sakamoto, Shota; Strain, Robert M. Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential. (English) Zbl 1455.35166 RIMS Kôkyûroku Bessatsu B82, 29-46 (2020). MSC: 35Q20 82D10 82C40 35A01 PDFBibTeX XMLCite \textit{R. Duan} et al., RIMS Kôkyûroku Bessatsu B82, 29--46 (2020; Zbl 1455.35166) Full Text: Link
Duan, Renjun; Liu, Shuangqian; Zhang, Zhu Ion-acoustic shock in a collisional plasma. (English) Zbl 1437.35571 J. Differ. Equations 269, No. 4, 3721-3768 (2020). MSC: 35Q35 35C07 35B35 35B40 76L05 76X05 76Q05 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Differ. Equations 269, No. 4, 3721--3768 (2020; Zbl 1437.35571) Full Text: DOI arXiv
Duan, Renjun; Yu, Hongjun The Vlasov-Poisson-Landau system near a local Maxwellian. (English) Zbl 1433.35403 Adv. Math. 362, Article ID 106956, 83 p. (2020). MSC: 35Q83 35A01 35B35 35B40 35Q31 76P05 35Q20 PDFBibTeX XMLCite \textit{R. Duan} and \textit{H. Yu}, Adv. Math. 362, Article ID 106956, 83 p. (2020; Zbl 1433.35403) Full Text: DOI
Duan, Renjun; Huang, Feimin; Wang, Yong; Zhang, Zhu Effects of soft interaction and non-isothermal boundary upon long-time dynamics of rarefied gas. (English) Zbl 1428.35265 Arch. Ration. Mech. Anal. 234, No. 2, 925-1006 (2019). MSC: 35Q20 82C40 76N15 76P05 35B40 35B35 35A01 PDFBibTeX XMLCite \textit{R. Duan} et al., Arch. Ration. Mech. Anal. 234, No. 2, 925--1006 (2019; Zbl 1428.35265) Full Text: DOI arXiv
Duan, Renjun; Wang, Yong; Zhang, Zhu The Boltzmann equation with time-periodic boundary temperature. (English) Zbl 1417.35086 Acta Math. Appl. Sin., Engl. Ser. 35, No. 1, 174-208 (2019). MSC: 35Q20 35B20 35B35 35B45 35B10 PDFBibTeX XMLCite \textit{R. Duan} et al., Acta Math. Appl. Sin., Engl. Ser. 35, No. 1, 174--208 (2019; Zbl 1417.35086) Full Text: DOI arXiv
Duan, Renjun; Wang, Yong The Boltzmann equation with large-amplitude initial data in bounded domains. (English) Zbl 1406.35216 Adv. Math. 343, 36-109 (2019). MSC: 35Q20 76P05 35A01 35B45 PDFBibTeX XMLCite \textit{R. Duan} and \textit{Y. Wang}, Adv. Math. 343, 36--109 (2019; Zbl 1406.35216) Full Text: DOI arXiv
Duan, Renjun; Sakamoto, Shota Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces. (English) Zbl 1405.35127 Kinet. Relat. Models 11, No. 6, 1301-1331 (2018). MSC: 35Q20 76P05 82B40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Sakamoto}, Kinet. Relat. Models 11, No. 6, 1301--1331 (2018; Zbl 1405.35127) Full Text: DOI arXiv
Ueda, Yoshihiro; Duan, Renjun; Kawashima, Shuichi New structural conditions on decay property with regularity-loss for symmetric hyperbolic systems with non-symmetric relaxation. (English) Zbl 1407.35029 J. Hyperbolic Differ. Equ. 15, No. 1, 149-174 (2018). MSC: 35B40 35B35 35L40 PDFBibTeX XMLCite \textit{Y. Ueda} et al., J. Hyperbolic Differ. Equ. 15, No. 1, 149--174 (2018; Zbl 1407.35029) Full Text: DOI
Duan, Renjun; Zhang, Zhu Solutions to a moving boundary problem on the Boltzmann equation. arXiv:1812.04797 Preprint, arXiv:1812.04797 [math.AP] (2018). BibTeX Cite \textit{R. Duan} and \textit{Z. Zhang}, ``Solutions to a moving boundary problem on the Boltzmann equation'', Preprint, arXiv:1812.04797 [math.AP] (2018) Full Text: arXiv OA License
Duan, Renjun; Wang, Yong; Yang, Tong Global existence for the ellipsoidal BGK model with initial large oscillations. (Chinese. English summary) Zbl 1499.35465 Sci. Sin., Math. 47, No. 10, 1143-1154 (2017). MSC: 35Q20 35Q82 82C40 PDFBibTeX XMLCite \textit{R. Duan} et al., Sci. Sin., Math. 47, No. 10, 1143--1154 (2017; Zbl 1499.35465) Full Text: DOI arXiv
Duan, Renjun; Liu, Shuangqian The Vlasov-Poisson-Boltzmann system for a disparate mass binary mixture. (English) Zbl 1393.35245 J. Stat. Phys. 169, No. 3, 614-684 (2017). Reviewer: Dmitry Pelinovsky (Hamilton) MSC: 35Q83 35Q20 76P05 35B35 35B40 82D10 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, J. Stat. Phys. 169, No. 3, 614--684 (2017; Zbl 1393.35245) Full Text: DOI arXiv
Duan, Renjun; Li, Xie; Xiang, Zhaoyin Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system. (English) Zbl 1378.35160 J. Differ. Equations 263, No. 10, 6284-6316 (2017). MSC: 35K55 35Q92 35Q30 92C17 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Differ. Equations 263, No. 10, 6284--6316 (2017; Zbl 1378.35160) Full Text: DOI
Ueda, Yoshihiro; Duan, Renjun; Kawashima, Shuichi Decay structure of two hyperbolic relaxation models with regularity loss. (English) Zbl 1397.35035 Kyoto J. Math. 57, No. 2, 235-292 (2017). Reviewer: Michael Reissig (Freiberg) MSC: 35B40 35B35 35L45 PDFBibTeX XMLCite \textit{Y. Ueda} et al., Kyoto J. Math. 57, No. 2, 235--292 (2017; Zbl 1397.35035) Full Text: DOI arXiv
Duan, Renjun; Huang, Feimin; Wang, Yong; Yang, Tong Global well-posedness of the Boltzmann equation with large amplitude initial data. (English) Zbl 1367.35097 Arch. Ration. Mech. Anal. 225, No. 1, 375-424 (2017). MSC: 35Q20 35A01 35A02 35B40 35B45 PDFBibTeX XMLCite \textit{R. Duan} et al., Arch. Ration. Mech. Anal. 225, No. 1, 375--424 (2017; Zbl 1367.35097) Full Text: DOI arXiv
Duan, Renjun; Yu, Hongjun The relativistic Boltzmann equation for soft potentials. (English) Zbl 1367.35098 Adv. Math. 312, 315-373 (2017). MSC: 35Q20 35Q75 35B40 35B65 82C40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{H. Yu}, Adv. Math. 312, 315--373 (2017; Zbl 1367.35098) Full Text: DOI
Duan, Renjun; Lei, Yuanjie; Yang, Tong; Zhao, Huijiang The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials. (English) Zbl 1414.82027 Commun. Math. Phys. 351, No. 1, 95-153 (2017). Reviewer: Dominik Strzałka (Rzeszów) MSC: 82C40 35Q83 35Q61 35Q20 35L60 35A01 PDFBibTeX XMLCite \textit{R. Duan} et al., Commun. Math. Phys. 351, No. 1, 95--153 (2017; Zbl 1414.82027) Full Text: DOI arXiv
Duan, RenJun; Liu, ShuangQian; Yin, HaiYan; Zhu, ChangJiang Stability of the rarefaction wave for a two-fluid plasma model with diffusion. (English) Zbl 1362.35038 Sci. China, Math. 59, No. 1, 67-84 (2016). Reviewer: Simonetta Abenda (Bologna) MSC: 35B35 35B40 35Q35 82D10 76X05 35Q31 PDFBibTeX XMLCite \textit{R. Duan} et al., Sci. China, Math. 59, No. 1, 67--84 (2016; Zbl 1362.35038) Full Text: DOI
Duan, Renjun Large-time behavior for fluid and kinetic plasmas with collisions. (English) Zbl 1341.35095 Bull. Braz. Math. Soc. (N.S.) 47, No. 1, 307-321 (2016). MSC: 35Q20 76P05 35B35 35B40 35Q61 35Q83 76X05 PDFBibTeX XMLCite \textit{R. Duan}, Bull. Braz. Math. Soc. (N.S.) 47, No. 1, 307--321 (2016; Zbl 1341.35095) Full Text: DOI
Duan, Renjun; Liu, Shuangqian; Xu, Jiang Global well-posedness in spatially critical Besov space for the Boltzmann equation. (English) Zbl 1334.35199 Arch. Ration. Mech. Anal. 220, No. 2, 711-745 (2016). MSC: 35Q20 42B25 35B65 35D35 PDFBibTeX XMLCite \textit{R. Duan} et al., Arch. Ration. Mech. Anal. 220, No. 2, 711--745 (2016; Zbl 1334.35199) Full Text: DOI arXiv Link
Duan, Renjun; Kawashima, Shuichi; Ueda, Yoshihiro Dissipative structure of the coupled kinetic-fluid models. (English) Zbl 1342.35167 Ei, Shin-Ichiro (ed.) et al., Nonlinear dynamics in partial differential equations. Proceedings of the 4th MSJ-SI international conference, Kyushu, Japan, September 12–21, 2011. Tokyo: Mathematical Society of Japan (ISBN 978-4-86497-022-8/hbk). Advanced Studies in Pure Mathematics 64, 327-335 (2015). MSC: 35L40 35B35 35B40 PDFBibTeX XMLCite \textit{R. Duan} et al., Adv. Stud. Pure Math. 64, 327--335 (2015; Zbl 1342.35167)
Ueda, Yoshihiro; Duan, Renjun; Kawashima, Shuichi Large time behavior of solutions to symmetric hyperbolic systems with non-symmetric relaxation. (English) Zbl 1342.35168 Ei, Shin-Ichiro (ed.) et al., Nonlinear dynamics in partial differential equations. Proceedings of the 4th MSJ-SI international conference, Kyushu, Japan, September 12–21, 2011. Tokyo: Mathematical Society of Japan (ISBN 978-4-86497-022-8/hbk). Advanced Studies in Pure Mathematics 64, 295-302 (2015). MSC: 35L40 35B35 35B40 PDFBibTeX XMLCite \textit{Y. Ueda} et al., Adv. Stud. Pure Math. 64, 295--302 (2015; Zbl 1342.35168)
Duan, Renjun; Liu, Shuangqian Time-periodic solutions of the Vlasov-Poisson-Fokker-Planck system. (English) Zbl 1340.35348 Acta Math. Sci., Ser. B, Engl. Ed. 35, No. 4, 876-886 (2015). MSC: 35Q84 82D10 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, Acta Math. Sci., Ser. B, Engl. Ed. 35, No. 4, 876--886 (2015; Zbl 1340.35348) Full Text: DOI
Duan, Renjun; Liu, Qingqin; Zhu, Changjiang Darcy’s law and diffusion for a two-fluid Euler-Maxwell system with dissipation. (English) Zbl 1330.35318 Math. Models Methods Appl. Sci. 25, No. 11, Article ID 2089, 2089-2151 (2015). MSC: 35Q35 35B40 35P20 76S05 35Q60 76K05 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 25, No. 11, Article ID 2089, 2089--2151 (2015; Zbl 1330.35318) Full Text: DOI arXiv
Duan, Renjun; Liu, Shuangqian Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system. (English) Zbl 1326.35221 SIAM J. Math. Anal. 47, No. 5, 3585-3647 (2015). Reviewer: Thomas Ernst (Uppsala) MSC: 35Q20 35Q83 35B35 35B40 82D10 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, SIAM J. Math. Anal. 47, No. 5, 3585--3647 (2015; Zbl 1326.35221) Full Text: DOI arXiv
Duan, Renjun; Liu, Shuangqian Stability of rarefaction waves of the Navier-Stokes-Poisson system. (English) Zbl 1309.35074 J. Differ. Equations 258, No. 7, 2495-2530 (2015). MSC: 35Q35 35B45 35B35 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, J. Differ. Equations 258, No. 7, 2495--2530 (2015; Zbl 1309.35074) Full Text: DOI arXiv
Duan, Renjun; Xiang, Zhaoyin A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. (English) Zbl 1323.35184 Int. Math. Res. Not. 2014, No. 7, 1833-1852 (2014). MSC: 35Q92 92C17 PDFBibTeX XMLCite \textit{R. Duan} and \textit{Z. Xiang}, Int. Math. Res. Not. 2014, No. 7, 1833--1852 (2014; Zbl 1323.35184) Full Text: DOI
Duan, Renjun Global smooth dynamics of a fully ionized plasma with long-range collisions. (English) Zbl 1305.82057 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 4, 751-778 (2014). Reviewer: Claudia-Veronika Meister (Darmstadt) MSC: 82D10 35Q84 35Q83 PDFBibTeX XMLCite \textit{R. Duan}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 4, 751--778 (2014; Zbl 1305.82057) Full Text: DOI arXiv
Duan, Renjun; Xiang, Zhaoyin On the Cauchy problem for the two-component Euler-Poincaré equations. (English) Zbl 1303.35064 J. Funct. Anal. 267, No. 8, 2698-2730 (2014). MSC: 35Q31 35Q35 35G25 35B44 42B25 35D30 PDFBibTeX XMLCite \textit{R. Duan} and \textit{Z. Xiang}, J. Funct. Anal. 267, No. 8, 2698--2730 (2014; Zbl 1303.35064) Full Text: DOI
Duan, Renjun; Liu, Shuangqian Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. (English) Zbl 1292.35295 Kinet. Relat. Models 6, No. 4, 687-700 (2013). MSC: 35Q84 35Q31 35B35 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, Kinet. Relat. Models 6, No. 4, 687--700 (2013; Zbl 1292.35295) Full Text: DOI
Duan, Renjun; Liu, Shuangqian The Vlasov-Poisson-Boltzmann system without angular cutoff. (English) Zbl 1285.35115 Commun. Math. Phys. 324, No. 1, 1-45 (2013). Reviewer: Oleg A. Sinkevich (Moskva) MSC: 35Q60 35Q83 35Q20 82D10 PDFBibTeX XMLCite \textit{R. Duan} and \textit{S. Liu}, Commun. Math. Phys. 324, No. 1, 1--45 (2013; Zbl 1285.35115) Full Text: DOI arXiv
Duan, Renjun; Yang, Tong; Zhao, Huijiang The Vlasov-Poisson-Boltzmann system for soft potentials. (English) Zbl 1437.76056 Math. Models Methods Appl. Sci. 23, No. 6, 979-1028 (2013). MSC: 76X05 76P05 35Q20 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 23, No. 6, 979--1028 (2013; Zbl 1437.76056) Full Text: DOI arXiv
Duan, Renjun; Yang, Xiongfeng Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. (English) Zbl 1264.76132 Commun. Pure Appl. Anal. 12, No. 2, 985-1014 (2013). MSC: 76X05 35M33 35B40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{X. Yang}, Commun. Pure Appl. Anal. 12, No. 2, 985--1014 (2013; Zbl 1264.76132) Full Text: DOI
Duan, Renjun; Liu, Shuangqian; Yang, Tong; Zhao, Huijiang Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials. (English) Zbl 1288.35476 Kinet. Relat. Models 6, No. 1, 159-204 (2013). Reviewer: Vladimir Čadež (Beograd) MSC: 35Q83 35Q61 35Q20 35B35 82D10 76P05 PDFBibTeX XMLCite \textit{R. Duan} et al., Kinet. Relat. Models 6, No. 1, 159--204 (2013; Zbl 1288.35476) Full Text: DOI
Duan, Renjun; Strain, Robert M. On the full dissipative property of the Vlasov-Poisson-Boltzmann system. (English) Zbl 1293.35339 Li, Tatsien (ed.) et al., Hyperbolic problems. Theory, numerics and applications. Vol. 2. Proceedings of the 13th international conference on hyperbolic problems, HYP 2010, Beijing, China, June 15–19, 2010. Hackensack, NJ: World Scientific; Beijing: Higher Education Press (ISBN 978-981-4417-08-2/v.2; 978-981-4417-06-8/set). Series in Contemporary Applied Mathematics CAM 18, 398-405 (2012). MSC: 35Q83 35Q20 PDFBibTeX XMLCite \textit{R. Duan} and \textit{R. M. Strain}, Ser. Contemp. Appl. Math. CAM 18, 398--405 (2012; Zbl 1293.35339)
Ueda, Yoshihiro; Duan, Renjun; Kawashima, Shuichi Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application. (English) Zbl 1273.35051 Arch. Ration. Mech. Anal. 205, No. 1, 239-266 (2012). Reviewer: Argiris I. Delis (Chania) MSC: 35B40 74H40 35L65 35L45 PDFBibTeX XMLCite \textit{Y. Ueda} et al., Arch. Ration. Mech. Anal. 205, No. 1, 239--266 (2012; Zbl 1273.35051) Full Text: DOI arXiv
Duan, Renjun; Li, Wei-Xi Hypocoercivity for the linear Boltzmann equation with confining forces. (English) Zbl 1248.35139 J. Stat. Phys. 148, No. 2, 306-324 (2012). MSC: 35Q20 35B30 35B40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{W.-X. Li}, J. Stat. Phys. 148, No. 2, 306--324 (2012; Zbl 1248.35139) Full Text: DOI arXiv
Duan, Renjun; Ruan, Lizhi; Zhu, Changjiang Optimal decay rates to conservation laws with diffusion-type terms of regularity-gain and regularity-loss. (English) Zbl 1241.35133 Math. Models Methods Appl. Sci. 22, No. 7, 1250012, 39 p. (2012). MSC: 35L65 35B40 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 22, No. 7, 1250012, 39 p. (2012; Zbl 1241.35133) Full Text: DOI arXiv
Duan, Renjun; Yang, Tong; Zhao, Huijiang The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case. (English) Zbl 1247.35174 J. Differ. Equations 252, No. 12, 6356-6386 (2012). Reviewer: Stephen Wollman (New York) MSC: 35Q83 35Q20 82C40 82D10 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Differ. Equations 252, No. 12, 6356--6386 (2012; Zbl 1247.35174) Full Text: DOI
Duan, Renjun Green’s function and large time behavior of the Navier-Stokes-Maxwell system. (English) Zbl 1241.35145 Anal. Appl., Singap. 10, No. 2, 133-197 (2012). MSC: 35Q30 35Q61 35M30 35B35 PDFBibTeX XMLCite \textit{R. Duan}, Anal. Appl., Singap. 10, No. 2, 133--197 (2012; Zbl 1241.35145) Full Text: DOI arXiv
Duan, Renjun; Liu, Qingqing; Zhu, Changjiang The Cauchy problem on the compressible two-fluids Euler-Maxwell equations. (English) Zbl 1236.35116 SIAM J. Math. Anal. 44, No. 1, 102-133 (2012). MSC: 35Q31 35Q60 35A01 35B40 PDFBibTeX XMLCite \textit{R. Duan} et al., SIAM J. Math. Anal. 44, No. 1, 102--133 (2012; Zbl 1236.35116) Full Text: DOI
Duan, Renjun Dissipative property of the Vlasov-Maxwell-Boltzmann system with a uniform ionic background. (English) Zbl 1233.35152 SIAM J. Math. Anal. 43, No. 6, 2732-2757 (2011). MSC: 35Q20 35Q83 76X05 76P05 35B35 35B40 PDFBibTeX XMLCite \textit{R. Duan}, SIAM J. Math. Anal. 43, No. 6, 2732--2757 (2011; Zbl 1233.35152) Full Text: DOI arXiv
Duan, Renjun; Strain, Robert M. Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space. (English) Zbl 1244.35010 Commun. Pure Appl. Math. 64, No. 11, 1497-1546 (2011). MSC: 35B40 35Q20 35Q83 35Q61 PDFBibTeX XMLCite \textit{R. Duan} and \textit{R. M. Strain}, Commun. Pure Appl. Math. 64, No. 11, 1497--1546 (2011; Zbl 1244.35010) Full Text: DOI arXiv
Duan, Renjun Hypocoercivity of linear degenerately dissipative kinetic equations. (English) Zbl 1237.35125 Nonlinearity 24, No. 8, 2165-2189 (2011). Reviewer: Guanggan Chen (Chengdu) MSC: 35Q20 35Q84 35B40 PDFBibTeX XMLCite \textit{R. Duan}, Nonlinearity 24, No. 8, 2165--2189 (2011; Zbl 1237.35125) Full Text: DOI arXiv
Duan, Renjun; Strain, Robert M. Optimal time decay of the Vlasov-Poisson-Boltzmann system in \({\mathbb R^3}\). (English) Zbl 1232.35169 Arch. Ration. Mech. Anal. 199, No. 1, 291-328 (2011). Reviewer: Vincent Lescarret (Gif-sur-Yvette) MSC: 35Q83 35Q20 76X05 76P05 35B40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{R. M. Strain}, Arch. Ration. Mech. Anal. 199, No. 1, 291--328 (2011; Zbl 1232.35169) Full Text: DOI arXiv
Duan, Renjun Global smooth flows for the compressible Euler-Maxwell system: the relaxation case. (English) Zbl 1292.76080 J. Hyperbolic Differ. Equ. 8, No. 2, 375-413 (2011). MSC: 76X05 76N10 35Q35 35Q60 PDFBibTeX XMLCite \textit{R. Duan}, J. Hyperbolic Differ. Equ. 8, No. 2, 375--413 (2011; Zbl 1292.76080) Full Text: DOI arXiv
Carrillo, José A.; Duan, Renjun; Moussa, Ayman Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. (English) Zbl 1215.35127 Kinet. Relat. Models 4, No. 1, 227-258 (2011). Reviewer: Alain Brillard (Riedisheim) MSC: 35Q35 35Q31 35Q84 76T20 35B40 PDFBibTeX XMLCite \textit{J. A. Carrillo} et al., Kinet. Relat. Models 4, No. 1, 227--258 (2011; Zbl 1215.35127) Full Text: DOI
Duan, Renjun; Yang, Tong; Zhao, Huijiang Global solutions to the Vlasov-Poisson-Landau System. arXiv:1112.3261 Preprint, arXiv:1112.3261 [math.AP] (2011). BibTeX Cite \textit{R. Duan} et al., ``Global solutions to the Vlasov-Poisson-Landau System'', Preprint, arXiv:1112.3261 [math.AP] (2011) Full Text: arXiv OA License
Duan, Renjun; Yang, Tong Stability of the one-species Vlasov-Poisson-Boltzmann system. (English) Zbl 1323.82036 SIAM J. Math. Anal. 41, No. 6, 2353-2387 (2010). MSC: 82C40 35Q83 76P05 82D05 PDFBibTeX XMLCite \textit{R. Duan} and \textit{T. Yang}, SIAM J. Math. Anal. 41, No. 6, 2353--2387 (2010; Zbl 1323.82036) Full Text: DOI
Duan, Renjun; Fornasier, Massimo; Toscani, Giuseppe A kinetic flocking model with diffusion. (English) Zbl 1213.35395 Commun. Math. Phys. 300, No. 1, 95-145 (2010). Reviewer: Giovanni Mascali (Arcavacata di Rende) MSC: 35Q84 82B05 35Q82 35A01 35B35 35B40 PDFBibTeX XMLCite \textit{R. Duan} et al., Commun. Math. Phys. 300, No. 1, 95--145 (2010; Zbl 1213.35395) Full Text: DOI
Duan, Renjun; Lorz, Alexander; Markowich, Peter Global solutions to the coupled chemotaxis-fluid equations. (English) Zbl 1275.35005 Commun. Partial Differ. Equations 35, No. 9, 1635-1673 (2010). MSC: 35A01 35Q92 35Q35 92C17 35B45 PDFBibTeX XMLCite \textit{R. Duan} et al., Commun. Partial Differ. Equations 35, No. 9, 1635--1673 (2010; Zbl 1275.35005) Full Text: DOI
Duan, Renjun; Fellner, Klemens; Zhu, Changjiang Energy method for multi-dimensional balance laws with non-local dissipation. (English) Zbl 1197.35050 J. Math. Pures Appl. (9) 93, No. 6, 572-598 (2010). MSC: 35B40 35B45 35L65 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Math. Pures Appl. (9) 93, No. 6, 572--598 (2010; Zbl 1197.35050) Full Text: DOI
Duan, Renjun; Ukai, Seiji; Yang, Tong A combination of energy method and spectral analysis for study of equations of gas motion. (English) Zbl 1180.35103 Front. Math. China 4, No. 2, 253-282 (2009). MSC: 35B40 35Q30 76P05 35B35 35Q20 PDFBibTeX XMLCite \textit{R. Duan} et al., Front. Math. China 4, No. 2, 253--282 (2009; Zbl 1180.35103) Full Text: DOI
Duan, Renjun Stability of the Boltzmann equation with potential forces on torus. (English) Zbl 1173.35638 Physica D 238, No. 17, 1808-1820 (2009). MSC: 35Q35 76P05 82B40 35B35 PDFBibTeX XMLCite \textit{R. Duan}, Physica D 238, No. 17, 1808--1820 (2009; Zbl 1173.35638) Full Text: DOI
Duan, Renjun; Ma, Hongfang Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity. (English) Zbl 1166.35029 Indiana Univ. Math. J. 57, No. 5, 2299-2319 (2008). MSC: 35Q30 65M12 93D20 34B45 PDFBibTeX XMLCite \textit{R. Duan} and \textit{H. Ma}, Indiana Univ. Math. J. 57, No. 5, 2299--2319 (2008; Zbl 1166.35029) Full Text: DOI
Duan, Renjun; Li, Meng-Rong; Yang, Tong Propagation of singularities in the solutions to the Boltzmann equation near equilibrium. (English) Zbl 1147.76054 Math. Models Methods Appl. Sci. 18, No. 7, 1093-1114 (2008). MSC: 76P05 45K05 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 18, No. 7, 1093--1114 (2008; Zbl 1147.76054) Full Text: DOI
Duan, Renjun On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in \(L_{\xi}^{2}(H_{x}^{N})\). (English) Zbl 1150.82019 J. Differ. Equations 244, No. 12, 3204-3234 (2008). Reviewer: Oleg A. Sinkevich (Moskva) MSC: 82C40 82D05 76P05 PDFBibTeX XMLCite \textit{R. Duan}, J. Differ. Equations 244, No. 12, 3204--3234 (2008; Zbl 1150.82019) Full Text: DOI
Duan, Renjun; Ukai, Seiji; Yang, Tong; Zhao, Huijiang Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications. (English) Zbl 1175.82047 Commun. Math. Phys. 277, No. 1, 189-236 (2008). Reviewer: Bernard Ducomet (Bruyères le Châtel) MSC: 82C40 35Q35 35B30 76D03 76D05 PDFBibTeX XMLCite \textit{R. Duan} et al., Commun. Math. Phys. 277, No. 1, 189--236 (2008; Zbl 1175.82047) Full Text: DOI
Duan, Renjun The Boltzmann equation near equilibrium states in \(\mathbb R^N\). (English) Zbl 1157.76042 Methods Appl. Anal. 14, No. 3, 227-249 (2007). MSC: 76P05 76-02 PDFBibTeX XMLCite \textit{R. Duan}, Methods Appl. Anal. 14, No. 3, 227--249 (2007; Zbl 1157.76042) Full Text: DOI
Duan, Renjun; Ukai, Seiji; Yang, Tong; Zhao, Huijiang Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. (English) Zbl 1122.35093 Math. Models Methods Appl. Sci. 17, No. 5, 737-758 (2007). Reviewer: Michael Jung (Dresden) MSC: 35Q30 65M12 93D20 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 17, No. 5, 737--758 (2007; Zbl 1122.35093) Full Text: DOI
Duan, Renjun; Liu, Hongxia; Ukai, Seiji; Yang, Tong Optimal \(L^{p}\) - \(L^{q}\) convergence rates for the compressible Navier-Stokes equations with potential force. (English) Zbl 1121.35096 J. Differ. Equations 238, No. 1, 220-233 (2007). Reviewer: Vadim Komkov (Florida) MSC: 35Q30 76N10 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Differ. Equations 238, No. 1, 220--233 (2007; Zbl 1121.35096) Full Text: DOI
Duan, Renjun; Lin, Saipan; Zhu, Changjiang Optimal \(L^{p}(1\leq p \leq \infty)\) rates of decay to linear diffusion waves for nonlinear evolution equations with ellipticity and dissipation. (English) Zbl 1113.35088 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 11, 2335-2344 (2007). MSC: 35K45 35B40 35K55 PDFBibTeX XMLCite \textit{R. Duan} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 11, 2335--2344 (2007; Zbl 1113.35088) Full Text: DOI
Duan, Renjun; Yang, Tong; Zhu, Changjiang Navier-Stokes equations with degenerate viscosity, vacuum and gravitational force. (English) Zbl 1128.35082 Math. Methods Appl. Sci. 30, No. 3, 347-374 (2007). Reviewer: Klaus Deckelnick (Magdeburg) MSC: 35Q30 76N10 35K60 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Methods Appl. Sci. 30, No. 3, 347--374 (2007; Zbl 1128.35082) Full Text: DOI
Duan, Renjun; Yang, Tong; Zhu, Changjiang Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system. (English) Zbl 1253.82079 J. Math. Anal. Appl. 327, No. 1, 425-434 (2007). MSC: 82D10 35Q83 35J60 76P05 76X05 82C40 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Math. Anal. Appl. 327, No. 1, 425--434 (2007; Zbl 1253.82079) Full Text: DOI
Duan, Renjun; Yang, Tong; Zhu, Changjiang Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. (English) Zbl 1185.35035 Discrete Contin. Dyn. Syst. 16, No. 1, 253-277 (2006). MSC: 35F25 35Q20 76P05 82C40 PDFBibTeX XMLCite \textit{R. Duan} et al., Discrete Contin. Dyn. Syst. 16, No. 1, 253--277 (2006; Zbl 1185.35035) Full Text: DOI
Duan, Renjun; Tang, Shaoqiang; Zhu, Changjiang Asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant. (English) Zbl 1107.35026 J. Math. Anal. Appl. 323, No. 2, 1152-1170 (2006). MSC: 35B40 34B45 35K50 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Math. Anal. Appl. 323, No. 2, 1152--1170 (2006; Zbl 1107.35026) Full Text: DOI
Duan, Renjun; Zhang, Mei; Zhu, Changjiang \(L^{1}\) stability for the Vlasov-Poisson-Boltzmann system around vacuum. (English) Zbl 1096.76050 Math. Models Methods Appl. Sci. 16, No. 9, 1505-1526 (2006). MSC: 76P05 76X05 45K05 PDFBibTeX XMLCite \textit{R. Duan} et al., Math. Models Methods Appl. Sci. 16, No. 9, 1505--1526 (2006; Zbl 1096.76050) Full Text: DOI
Duan, Renjun; Yang, Tong; Zhu, Changjiang \(L^{1}\) and BV-type stability of the Boltzmann equation with external forces. (English) Zbl 1101.76051 J. Differ. Equations 227, No. 1, 1-28 (2006). MSC: 76P05 45K05 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Differ. Equations 227, No. 1, 1--28 (2006; Zbl 1101.76051) Full Text: DOI
Duan, Renjun; Yang, Tong; Zhu, Changjiang Global existence to Boltzmann equation with external force in infinite vacuum. (English) Zbl 1110.82034 J. Math. Phys. 46, No. 5, 053307, 13 p. (2005). MSC: 82C40 82B40 PDFBibTeX XMLCite \textit{R. Duan} et al., J. Math. Phys. 46, No. 5, 053307, 13 p. (2005; Zbl 1110.82034) Full Text: DOI
Duan, Renjun; Zhu, Changjiang Asymptotics of dissipative nonlinear evolution equations with ellipticity: different end states. (English) Zbl 1063.35069 J. Math. Anal. Appl. 303, No. 1, 15-35 (2005). MSC: 35K45 35K50 35B45 35B40 PDFBibTeX XMLCite \textit{R. Duan} and \textit{C. Zhu}, J. Math. Anal. Appl. 303, No. 1, 15--35 (2005; Zbl 1063.35069) Full Text: DOI
Zhu, Changjiang; Duan, Renjun Existence and uniqueness of entropy solution to initial boundary value problem for the inviscid Burgers equation. (English) Zbl 1168.35393 J. Phys. A, Math. Gen. 36, No. 8, 2099-2107 (2003). MSC: 35L65 35D05 35K05 PDFBibTeX XMLCite \textit{C. Zhu} and \textit{R. Duan}, J. Phys. A, Math. Gen. 36, No. 8, 2099--2107 (2003; Zbl 1168.35393) Full Text: DOI
Duan, Renjun; Zhu, Changjiang A note on semiconcave function. (English) Zbl 1067.26010 Appl. Anal. 82, No. 9, 889-894 (2003). Reviewer: Pavol Quittner (Bratislava) MSC: 26B25 26D20 PDFBibTeX XMLCite \textit{R. Duan} and \textit{C. Zhu}, Appl. Anal. 82, No. 9, 889--894 (2003; Zbl 1067.26010) Full Text: DOI
Duan, Ren Jun; Xiao, Xing Juan Curves controlled by regular polygons and their properties. (Chinese. English summary) Zbl 1504.65044 Math. Theory Appl. 21, No. 3, 29-34 (2001). MSC: 65D17 PDFBibTeX XMLCite \textit{R. J. Duan} and \textit{X. J. Xiao}, Math. Theory Appl. 21, No. 3, 29--34 (2001; Zbl 1504.65044)