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

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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

Yingshan Chen ^1, , Shijin Ding ^2, and Wenjun Wang ^3,

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

Abstract
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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.

Keywords: the Cauchy problem, global existence, uniqueness, Compressible Navier-Stokes-Smoluchowski equations, optimal convergence rate..

Mathematics Subject Classification: Primary: 35Q30, 76N10, 46E3.

Citation: Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032

References:

[1]	R. Adams, Sobolev Spaces, Academic Press, Now York, 1975.
[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]	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.
[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]	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.
[6]	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.
[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), 2778-2801. doi: 10.1016/j.na.2010.12.031.
[8]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 2003.
[9]	S. J. Ding, B. Y. Huang and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum, preprint, 2015, http://202.116.32.252:8080/maths/uploadfile/2015/1212/20151212030124241.pdf.
[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), 737-758. doi: 10.1142/S021820250700208X.
[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), 033706, 21pp. doi: 10.1063/1.3693979.
[12]	D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003.
[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), 365-376. doi: 10.1007/s00220-004-1062-2.
[14]	S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics, Kyoto University, 1983.
[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), 621-659. doi: 10.1007/s002200050543.
[16]	P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford University Press, Oxford, 1998.
[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), 445-464. doi: 10.1007/BF01214738.
[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), 091501, 12pp. doi: 10.1063/1.4820446.
[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), 344-364. doi: 10.1016/j.ijmultiphaseflow.2005.10.005.
[20]	Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.
[21]	F. A. Williams, Combustion Theory, Benjamin Cummings Publ., 1985.
[22]	F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555. doi: 10.1063/1.1724379.
[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), 835-850. doi: 10.4310/CMS.2010.v8.n4.a2.

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References:

[1]	R. Adams, Sobolev Spaces, Academic Press, Now York, 1975.
[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]	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.
[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]	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.
[6]	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.
[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), 2778-2801. doi: 10.1016/j.na.2010.12.031.
[8]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, American Mathematical Society, 2003.
[9]	S. J. Ding, B. Y. Huang and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum, preprint, 2015, http://202.116.32.252:8080/maths/uploadfile/2015/1212/20151212030124241.pdf.
[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), 737-758. doi: 10.1142/S021820250700208X.
[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), 033706, 21pp. doi: 10.1063/1.3693979.
[12]	D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003.
[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), 365-376. doi: 10.1007/s00220-004-1062-2.
[14]	S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics, Kyoto University, 1983.
[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), 621-659. doi: 10.1007/s002200050543.
[16]	P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models, Oxford University Press, Oxford, 1998.
[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), 445-464. doi: 10.1007/BF01214738.
[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), 091501, 12pp. doi: 10.1063/1.4820446.
[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), 344-364. doi: 10.1016/j.ijmultiphaseflow.2005.10.005.
[20]	Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential Equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.
[21]	F. A. Williams, Combustion Theory, Benjamin Cummings Publ., 1985.
[22]	F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555. doi: 10.1063/1.1724379.
[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), 835-850. doi: 10.4310/CMS.2010.v8.n4.a2.

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