American Institute of Mathematical Sciences

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December  2016, 9(6): 1717-1752. doi: 10.3934/dcdss.2016072

Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum

Bingyuan Huang 1, , Shijin Ding 2, and  Huanyao Wen 3,

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

Received  July 2015 Revised  September 2016 Published  November 2016

This paper is concerned with the Cauchy problem for compressible Navier-Stokes-Smoluchowski equations with vacuum in $\mathbb{R}^3$. We prove both existence and uniqueness of the local strong solution, and then obtain a local classical solution by deriving the smoothing effect of the strong solution for $t>0$.
Keywords: vacuum., Classical solution, compressible Navier-Stokes-Smoluchowski equations.
Mathematics Subject Classification: Primary: 35Q30, 76N10; Secondary: 46E3.
Citation: Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072
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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