# American Institute of Mathematical Sciences

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

## 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

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 $t>0$.
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 [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

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
 [1] Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032 [2] Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 [3] Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 [4] Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263 [5] Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure & Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987 [6] Ben Duan, Zhen Luo. Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2543-2564. doi: 10.3934/cpaa.2013.12.2543 [7] Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163 [8] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [9] Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127 [10] Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785 [11] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348 [12] Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080 [13] Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure & Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459 [14] Tong Tang, Hongjun Gao. On the compressible Navier-Stokes-Korteweg equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2745-2766. doi: 10.3934/dcdsb.2016071 [15] Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure & Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161 [16] Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005 [17] Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099 [18] Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093 [19] Wenjun Wang, Weike Wang. Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 513-536. doi: 10.3934/dcds.2015.35.513 [20] Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

2020 Impact Factor: 2.425

## Metrics

• PDF downloads (66)
• HTML views (0)
• Cited by (11)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]