January  2015, 35(1): 513-536. doi: 10.3934/dcds.2015.35.513

Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces

1. 

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

2. 

Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai

Received  January 2014 Revised  June 2014 Published  August 2014

In this paper, the compressible Navier-Stokes-Korteweg equations with a potential external force is considered in $\mathbb{R}^3$. Under the smallness assumption on both the external force and the initial perturbation of the stationary solution in some Sobolev spaces, we establish the existence theory of global solutions to the stationary profile. What's more, when the initial perturbation is bounded in $L^p$-norm with $1\leq p<2$, the optimal time decay rates of the solution in $L^q$-norm with $2\leq q\leq 6$ and its first order derivative in $L^2$-norm are shown. On the other hand, when the $\dot{H}^{-s}$ norm $(s\in(0,\frac{3}{2}])$ of the perturbation is finite, we obtain the optimal time decay rates of the solution and its first order derivative in $L^2$-norm.
Citation: Wenjun Wang, Weike Wang. Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 513-536. doi: 10.3934/dcds.2015.35.513
References:
[1]

R. Adams, Sobolev Spaces, Academic Press, Now York, 1975.

[2]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.

[3]

Z. Z. Chen and H. J. Zhao, Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system, J. Math. Pures Appl., 101 (2014), 330-371. doi: 10.1016/j.matpur.2013.06.005.

[4]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri Poincaré Anal. Nonlinear, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1.

[5]

K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains, Math. Z., 209 (1992), 115-130. doi: 10.1007/BF02570825.

[6]

R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008.

[7]

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.

[8]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907.

[9]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.

[10]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249. doi: 10.1007/s00021-009-0013-2.

[11]

H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.

[12]

H. Hattori and D. Li, Global solutions of a high dimensionl system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.

[13]

H. Hattori and D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Differ. Equ., 9 (1996), 323-342.

[14]

N. Ju, Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space, Commun. Math. Phys., 251 (2004), 365-376. doi: 10.1007/s00220-004-1062-2.

[15]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1983.

[16]

T. Kobayashi and Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations, Pacific J. Math., 207 (2002), 199-234. doi: 10.2140/pjm.2002.207.199.

[17]

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 $\mathbbR^3$, Commun. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.

[18]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Archives Néerlandaises de Sciences Exactes et Naturelles II, 6 (1901), 1-24.

[19]

M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005.

[20]

Y. P. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232. doi: 10.1016/j.jmaa.2011.11.006.

[21]

D. Q. Li and Y. M. Chen, Nonlinear Evolution Equation, Science Press, 1989.

[22]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418.

[23]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[24]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342. doi: 10.3792/pjaa.55.337.

[25]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.

[26]

J. Z. Qian and H. Yin, Convergence rates for the compressible Navier-Stokes equations with general forces, Acta Math. Sci. Ser. B, 29 (2009), 1351-1365. doi: 10.1016/S0252-9602(09)60108-9.

[27]

Y. Shibata and K. Tanaka, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance, J. Math. Soc. Jpn., 55 (2003), 797-826. doi: 10.2969/jmsj/1191419003.

[28]

Y. Shibata and K. Tanaka, Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid, Comput. Math. Appl., 53 (2007), 605-623. doi: 10.1016/j.camwa.2006.02.030.

[29]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[30]

Z. Tan and R. F. Zhang, Optimal decay rates of the compressible fluid models of Korteweg type, Z. Angew. Math. Phys., 65 (2014), 279-300. doi: 10.1007/s00033-013-0331-3.

[31]

S. Ukai, T. Yang and H. J. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differential Equations, 3 (2006), 561-574. doi: 10.1142/S0219891606000902.

[32]

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.

[33]

Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006.

[34]

Y. J. Wang and Z. Tan, Global existence and optimal decay rate for the strong solutions in $H^2$ to the compressible Navier-Stokes equations, Appl. Math. Lett., 24 (2011), 1778-1784. doi: 10.1016/j.aml.2011.04.028.

show all references

References:
[1]

R. Adams, Sobolev Spaces, Academic Press, Now York, 1975.

[2]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.

[3]

Z. Z. Chen and H. J. Zhao, Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system, J. Math. Pures Appl., 101 (2014), 330-371. doi: 10.1016/j.matpur.2013.06.005.

[4]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri Poincaré Anal. Nonlinear, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1.

[5]

K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains, Math. Z., 209 (1992), 115-130. doi: 10.1007/BF02570825.

[6]

R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008.

[7]

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.

[8]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907.

[9]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.

[10]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249. doi: 10.1007/s00021-009-0013-2.

[11]

H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.

[12]

H. Hattori and D. Li, Global solutions of a high dimensionl system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.

[13]

H. Hattori and D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Differ. Equ., 9 (1996), 323-342.

[14]

N. Ju, Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space, Commun. Math. Phys., 251 (2004), 365-376. doi: 10.1007/s00220-004-1062-2.

[15]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1983.

[16]

T. Kobayashi and Y. Shibata, Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations, Pacific J. Math., 207 (2002), 199-234. doi: 10.2140/pjm.2002.207.199.

[17]

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 $\mathbbR^3$, Commun. Math. Phys., 200 (1999), 621-659. doi: 10.1007/s002200050543.

[18]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Archives Néerlandaises de Sciences Exactes et Naturelles II, 6 (1901), 1-24.

[19]

M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005.

[20]

Y. P. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232. doi: 10.1016/j.jmaa.2011.11.006.

[21]

D. Q. Li and Y. M. Chen, Nonlinear Evolution Equation, Science Press, 1989.

[22]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418.

[23]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[24]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342. doi: 10.3792/pjaa.55.337.

[25]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464. doi: 10.1007/BF01214738.

[26]

J. Z. Qian and H. Yin, Convergence rates for the compressible Navier-Stokes equations with general forces, Acta Math. Sci. Ser. B, 29 (2009), 1351-1365. doi: 10.1016/S0252-9602(09)60108-9.

[27]

Y. Shibata and K. Tanaka, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance, J. Math. Soc. Jpn., 55 (2003), 797-826. doi: 10.2969/jmsj/1191419003.

[28]

Y. Shibata and K. Tanaka, Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid, Comput. Math. Appl., 53 (2007), 605-623. doi: 10.1016/j.camwa.2006.02.030.

[29]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[30]

Z. Tan and R. F. Zhang, Optimal decay rates of the compressible fluid models of Korteweg type, Z. Angew. Math. Phys., 65 (2014), 279-300. doi: 10.1007/s00033-013-0331-3.

[31]

S. Ukai, T. Yang and H. J. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differential Equations, 3 (2006), 561-574. doi: 10.1142/S0219891606000902.

[32]

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.

[33]

Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006.

[34]

Y. J. Wang and Z. Tan, Global existence and optimal decay rate for the strong solutions in $H^2$ to the compressible Navier-Stokes equations, Appl. Math. Lett., 24 (2011), 1778-1784. doi: 10.1016/j.aml.2011.04.028.

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