October  2016, 36(10): 5287-5307. doi: 10.3934/dcds.2016032

Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations

1. 

Department of Mathematics, South China University of Technology, Guangzhou 510641

2. 

Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631

3. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093

Received  September 2015 Revised  March 2016 Published  July 2016

This paper is concerned with the Cauchy problem of the compressible Navier-Stokes-Smoluchowski equations in $\mathbb{R}^3$. Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces, the existence theory of global solutions in $H^3$ to the stationary profile is established. Moreover, when the initial perturbation is bounded in $L^p$-norm with $1\leq p< \frac{6}{5}$, we obtain the optimal convergence rates of the solution in $L^q$-norm with $2\leq q\leq 6$ and its first order derivative in $L^2$-norm.
Citation: 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 - A, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032
References:
[1]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system,, Hyperbolic Problems: Theory, 8 (2014), 301. Google Scholar

[3]

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, 14 (2005), 41. 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, 91 (2013), 1. doi: 10.1016/j.na.2013.06.002. Google Scholar

[5]

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. doi: 10.1137/S0036139902408163. Google Scholar

[6]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model,, Commun. Partial Differ. Equ., 31 (2006), 1349. doi: 10.1080/03605300500394389. Google Scholar

[7]

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. doi: 10.1016/j.na.2010.12.031. Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics,, American Mathematical Society, (2003). Google Scholar

[9]

S. J. Ding, B. Y. Huang and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum,, preprint, (2015). Google Scholar

[10]

R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces,, Math. Models Methods Appl. Sci., 17 (2007), 737. doi: 10.1142/S021820250700208X. Google Scholar

[11]

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). doi: 10.1063/1.3693979. Google Scholar

[12]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow,, Indiana Univ. Math. J., 44 (1995), 603. doi: 10.1512/iumj.1995.44.2003. Google Scholar

[13]

N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space,, Comm. Math. Phys., 251 (2004), 365. doi: 10.1007/s00220-004-1062-2. Google Scholar

[14]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics,, Kyoto University, (1983). Google Scholar

[15]

T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbbR^3$,, Comm. Math. Phys., 200 (1999), 621. doi: 10.1007/s002200050543. Google Scholar

[16]

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

[17]

A. Matsumura and T. Nishida, Initial boundary problems for the equations of motion of compressible viscous and heat-conducive fluids,, Commun. Math. Phys., 89 (1983), 445. doi: 10.1007/BF01214738. Google Scholar

[18]

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). doi: 10.1063/1.4820446. Google Scholar

[19]

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. doi: 10.1016/j.ijmultiphaseflow.2005.10.005. Google Scholar

[20]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations,, J. Differential Equations, 253 (2012), 273. doi: 10.1016/j.jde.2012.03.006. Google Scholar

[21]

F. A. Williams, Combustion Theory,, Benjamin Cummings Publ., (1985). Google Scholar

[22]

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

[23]

J. W. Zhang and J. N. Zhao, Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics,, Commun. Math. Sci., 8 (2010), 835. doi: 10.4310/CMS.2010.v8.n4.a2. Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar

[2]

J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system,, Hyperbolic Problems: Theory, 8 (2014), 301. Google Scholar

[3]

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, 14 (2005), 41. 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, 91 (2013), 1. doi: 10.1016/j.na.2013.06.002. Google Scholar

[5]

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. doi: 10.1137/S0036139902408163. Google Scholar

[6]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model,, Commun. Partial Differ. Equ., 31 (2006), 1349. doi: 10.1080/03605300500394389. Google Scholar

[7]

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. doi: 10.1016/j.na.2010.12.031. Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics,, American Mathematical Society, (2003). Google Scholar

[9]

S. J. Ding, B. Y. Huang and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum,, preprint, (2015). Google Scholar

[10]

R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces,, Math. Models Methods Appl. Sci., 17 (2007), 737. doi: 10.1142/S021820250700208X. Google Scholar

[11]

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). doi: 10.1063/1.3693979. Google Scholar

[12]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow,, Indiana Univ. Math. J., 44 (1995), 603. doi: 10.1512/iumj.1995.44.2003. Google Scholar

[13]

N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space,, Comm. Math. Phys., 251 (2004), 365. doi: 10.1007/s00220-004-1062-2. Google Scholar

[14]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics,, Kyoto University, (1983). Google Scholar

[15]

T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbbR^3$,, Comm. Math. Phys., 200 (1999), 621. doi: 10.1007/s002200050543. Google Scholar

[16]

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

[17]

A. Matsumura and T. Nishida, Initial boundary problems for the equations of motion of compressible viscous and heat-conducive fluids,, Commun. Math. Phys., 89 (1983), 445. doi: 10.1007/BF01214738. Google Scholar

[18]

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). doi: 10.1063/1.4820446. Google Scholar

[19]

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. doi: 10.1016/j.ijmultiphaseflow.2005.10.005. Google Scholar

[20]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations,, J. Differential Equations, 253 (2012), 273. doi: 10.1016/j.jde.2012.03.006. Google Scholar

[21]

F. A. Williams, Combustion Theory,, Benjamin Cummings Publ., (1985). Google Scholar

[22]

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

[23]

J. W. Zhang and J. N. Zhao, Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics,, Commun. Math. Sci., 8 (2010), 835. doi: 10.4310/CMS.2010.v8.n4.a2. Google Scholar

[1]

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

[2]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

[3]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[4]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107

[5]

Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047

[6]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75

[7]

Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991

[8]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041

[9]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

[10]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609

[11]

Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083

[12]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[13]

Jishan Fan, Fucai Li, Gen Nakamura. Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. Kinetic & Related Models, 2016, 9 (3) : 443-453. doi: 10.3934/krm.2016002

[14]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609

[15]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

[16]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009

[17]

Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783

[18]

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

[19]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[20]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]