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 & 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.  Google Scholar

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J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Ph.D thesis, University of Maryland, USA, 2014.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford: Oxford University Press, 2004.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,, , ().   Google Scholar

[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.  Google Scholar

[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,, , ().   Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[28]

P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford University Press, Oxford, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

F. A. Williams, Combustion Theory, 2nd ed, Benjamin Cummings Publ, 1985. Google Scholar

[33]

F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555. doi: 10.1063/1.1724379.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[3]

J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Ph.D thesis, University of Maryland, USA, 2014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford: Oxford University Press, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,, , ().   Google Scholar

[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.  Google Scholar

[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,, , ().   Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[28]

P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford University Press, Oxford, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

F. A. Williams, Combustion Theory, 2nd ed, Benjamin Cummings Publ, 1985. Google Scholar

[33]

F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555. doi: 10.1063/1.1724379.  Google Scholar

[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.  Google Scholar

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