# American Institute of Mathematical Sciences

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

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##### 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
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