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Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum
1. | School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
2. | Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631 |
3. | School of Mathematics, South China University of Technology, Guangzhou 510641, China |
References:
[1] |
C. Baranger, L. Boudin, P. E. Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 mathematics and applications to biology and medicine, ESAIM Proc, 14 (2005), 41-47. |
[2] |
J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system. Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series on Applied Mathematics, 8 (2014), 301-308. |
[3] |
J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Ph.D thesis, University of Maryland, USA, 2014. |
[4] |
J. Ballew and K. Trivisa, Weakly dissipative solutions and weak strong uniqueness for the Navier Stokes Smoluchowski system, Nonlinear Analysis Series A: Theory, Methods Applications, 91 ( 2013), 1-19.
doi: 10.1016/j.na.2013.06.002. |
[5] |
J. Ballew and K. Trivisa, Viscous and inviscid models in fluid-particle interaction, Communications in Information Systems, 13 (2013), 45-78.
doi: 10.4310/CIS.2013.v13.n1.a2. |
[6] |
S. Berres, R. Bürger, K. H. Karlsen and E. M. Tory, Strongly degenerate parabolic hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.
doi: 10.1137/S0036139902408163. |
[7] |
H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equ., 190 (2003), 504-523.
doi: 10.1016/S0022-0396(03)00015-9. |
[8] |
H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier Stokes equations for the isentropic compressible fluids, Math. Methods Appl. Sci., 28 (2005), 1-28.
doi: 10.1002/mma.545. |
[9] |
J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.
doi: 10.1080/03605300500394389. |
[10] |
J. A. Carrillo, T. Karper and K. Trivisa, On the dynamics of a fluid-particle interaction model: The bubbling regime, Nonlinear Anal., 74 (2011), 2778-2801.
doi: 10.1016/j.na.2010.12.031. |
[11] |
Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.
doi: 10.1007/s00229-006-0637-y. |
[12] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.
doi: 10.1016/j.matpur.2003.11.004. |
[13] |
S. J. Ding, J. R. Huang and F. G. Xia, A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension, J. Diff. Equ., 255 (2013), 3848-3879.
doi: 10.1016/j.jde.2013.07.039. |
[14] |
S. J. Ding, H. Y. Wen, L. Yao and C. J. Zhu, Global spherically symmetric classical solution to compressible Navier Stokes equations with large initial data and vacuum, SIAM J. Math. Anal., 44 (2012), 1257-1278.
doi: 10.1137/110836663. |
[15] |
D. Y. Fang, R. Z. Zi and T. Zhang, Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime, J. Math. Phys., 53 (2012), 033706, 21pp.
doi: 10.1063/1.3693979. |
[16] |
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford: Oxford University Press, 2004. |
[17] |
E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[18] |
E. Feireisl and H. Petzeltová, Large-time behavior of solutions to the Navier-Stokes equations of compressible flow, Arch. Rational Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[19] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Second edition. Springer Monographs in Mathematics. Springer, New York, 2011.
doi: 10.1007/978-0-387-09620-9. |
[20] |
J. R. Huang and S. J. Ding, Compressible hydrodynamic flow of nematic liquid crystals with vacuum, J. Diff. Equ., 258 (2015), 1653-1684.
doi: 10.1016/j.jde.2014.11.008. |
[21] |
T. Huang, C. Y. Wang and H. Y. Wen, Blow Up Criterion for Compressible Nematic Liquid Crystal Flows in Dimension Three, Arch. Rational Mech. Anal., 204 (2012), 285-311.
doi: 10.1007/s00205-011-0476-1. |
[22] |
T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Diff. Equ., 252 (2012), 2222-2265.
doi: 10.1016/j.jde.2011.07.036. |
[23] |
X. D. Huang and J. Li, Global Well-Posedness of Classical Solutions to the Cauchy problem of Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data, preprint,, , ().
|
[24] |
X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure. Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[25] |
X. F. Hou, H. Y. Peng and C. J. Zhu, Global classical solution to 3D isentropic compressible Navier-Stokes equations with large initial data and vacuum, preprint,, , ().
|
[26] |
X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699.
doi: 10.1137/120898814. |
[27] |
J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows,, preprint, ().
|
[28] |
P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford University Press, Oxford, 1998. |
[29] |
S. Ma, Classical solutions for the compressible liquid crystal flows with nonnegative initial densities, J. Math. Anal. Appl., 397 (2013), 595-618.
doi: 10.1016/j.jmaa.2012.08.010. |
[30] |
Y. K. Song, H. J. Yuan, Y. Chen and Z. D. Guo, Strong solutions for a 1D fluid-particle interaction non-newtonian model: The bubbling regime, J. Math. Phys., 54 (2013), 091501, 12pp.
doi: 10.1063/1.4820446. |
[31] |
I. Vinkovic, C. Aguirre, S. Simoëns and M. Gorokhovski, Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow, International Journal of Multiphase Flow, 32 (2006), 344-364.
doi: 10.1016/j.ijmultiphaseflow.2005.10.005. |
[32] |
F. A. Williams, Combustion Theory, 2nd ed, Benjamin Cummings Publ, 1985. |
[33] |
F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555.
doi: 10.1063/1.1724379. |
[34] |
H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Advances in Mathematics, 248 (2013), 534-572.
doi: 10.1016/j.aim.2013.07.018. |
show all references
References:
[1] |
C. Baranger, L. Boudin, P. E. Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 mathematics and applications to biology and medicine, ESAIM Proc, 14 (2005), 41-47. |
[2] |
J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system. Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series on Applied Mathematics, 8 (2014), 301-308. |
[3] |
J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Ph.D thesis, University of Maryland, USA, 2014. |
[4] |
J. Ballew and K. Trivisa, Weakly dissipative solutions and weak strong uniqueness for the Navier Stokes Smoluchowski system, Nonlinear Analysis Series A: Theory, Methods Applications, 91 ( 2013), 1-19.
doi: 10.1016/j.na.2013.06.002. |
[5] |
J. Ballew and K. Trivisa, Viscous and inviscid models in fluid-particle interaction, Communications in Information Systems, 13 (2013), 45-78.
doi: 10.4310/CIS.2013.v13.n1.a2. |
[6] |
S. Berres, R. Bürger, K. H. Karlsen and E. M. Tory, Strongly degenerate parabolic hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.
doi: 10.1137/S0036139902408163. |
[7] |
H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equ., 190 (2003), 504-523.
doi: 10.1016/S0022-0396(03)00015-9. |
[8] |
H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier Stokes equations for the isentropic compressible fluids, Math. Methods Appl. Sci., 28 (2005), 1-28.
doi: 10.1002/mma.545. |
[9] |
J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.
doi: 10.1080/03605300500394389. |
[10] |
J. A. Carrillo, T. Karper and K. Trivisa, On the dynamics of a fluid-particle interaction model: The bubbling regime, Nonlinear Anal., 74 (2011), 2778-2801.
doi: 10.1016/j.na.2010.12.031. |
[11] |
Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.
doi: 10.1007/s00229-006-0637-y. |
[12] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.
doi: 10.1016/j.matpur.2003.11.004. |
[13] |
S. J. Ding, J. R. Huang and F. G. Xia, A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension, J. Diff. Equ., 255 (2013), 3848-3879.
doi: 10.1016/j.jde.2013.07.039. |
[14] |
S. J. Ding, H. Y. Wen, L. Yao and C. J. Zhu, Global spherically symmetric classical solution to compressible Navier Stokes equations with large initial data and vacuum, SIAM J. Math. Anal., 44 (2012), 1257-1278.
doi: 10.1137/110836663. |
[15] |
D. Y. Fang, R. Z. Zi and T. Zhang, Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime, J. Math. Phys., 53 (2012), 033706, 21pp.
doi: 10.1063/1.3693979. |
[16] |
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford: Oxford University Press, 2004. |
[17] |
E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids, J. Math. Fluid Mech., 3 (2001), 358-392.
doi: 10.1007/PL00000976. |
[18] |
E. Feireisl and H. Petzeltová, Large-time behavior of solutions to the Navier-Stokes equations of compressible flow, Arch. Rational Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[19] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Second edition. Springer Monographs in Mathematics. Springer, New York, 2011.
doi: 10.1007/978-0-387-09620-9. |
[20] |
J. R. Huang and S. J. Ding, Compressible hydrodynamic flow of nematic liquid crystals with vacuum, J. Diff. Equ., 258 (2015), 1653-1684.
doi: 10.1016/j.jde.2014.11.008. |
[21] |
T. Huang, C. Y. Wang and H. Y. Wen, Blow Up Criterion for Compressible Nematic Liquid Crystal Flows in Dimension Three, Arch. Rational Mech. Anal., 204 (2012), 285-311.
doi: 10.1007/s00205-011-0476-1. |
[22] |
T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow, J. Diff. Equ., 252 (2012), 2222-2265.
doi: 10.1016/j.jde.2011.07.036. |
[23] |
X. D. Huang and J. Li, Global Well-Posedness of Classical Solutions to the Cauchy problem of Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data, preprint,, , ().
|
[24] |
X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure. Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[25] |
X. F. Hou, H. Y. Peng and C. J. Zhu, Global classical solution to 3D isentropic compressible Navier-Stokes equations with large initial data and vacuum, preprint,, , ().
|
[26] |
X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699.
doi: 10.1137/120898814. |
[27] |
J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows,, preprint, ().
|
[28] |
P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford University Press, Oxford, 1998. |
[29] |
S. Ma, Classical solutions for the compressible liquid crystal flows with nonnegative initial densities, J. Math. Anal. Appl., 397 (2013), 595-618.
doi: 10.1016/j.jmaa.2012.08.010. |
[30] |
Y. K. Song, H. J. Yuan, Y. Chen and Z. D. Guo, Strong solutions for a 1D fluid-particle interaction non-newtonian model: The bubbling regime, J. Math. Phys., 54 (2013), 091501, 12pp.
doi: 10.1063/1.4820446. |
[31] |
I. Vinkovic, C. Aguirre, S. Simoëns and M. Gorokhovski, Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow, International Journal of Multiphase Flow, 32 (2006), 344-364.
doi: 10.1016/j.ijmultiphaseflow.2005.10.005. |
[32] |
F. A. Williams, Combustion Theory, 2nd ed, Benjamin Cummings Publ, 1985. |
[33] |
F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555.
doi: 10.1063/1.1724379. |
[34] |
H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Advances in Mathematics, 248 (2013), 534-572.
doi: 10.1016/j.aim.2013.07.018. |
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