# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2015, 35 (1) : 513-536. doi: 10.3934/dcds.2015.35.513
##### References:
 [1] R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [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. doi: 10.1081/PDE-120020499. Google Scholar [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. doi: 10.1016/j.matpur.2013.06.005. Google Scholar [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. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar [5] K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains,, Math. Z., 209 (1992), 115. doi: 10.1007/BF02570825. Google Scholar [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. doi: 10.1016/j.jde.2007.03.008. Google Scholar [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. doi: 10.1142/S021820250700208X. Google Scholar [8] J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working,, Arch. Rational Mech. Anal., 88 (1985), 95. doi: 10.1007/BF00250907. Google Scholar [9] Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar [10] B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type,, J. Math. Fluid Mech., 13 (2011), 223. doi: 10.1007/s00021-009-0013-2. Google Scholar [11] H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type,, SIAM J. Math. Anal., 25 (1994), 85. doi: 10.1137/S003614109223413X. Google Scholar [12] H. Hattori and D. Li, Global solutions of a high dimensionl system for Korteweg materials,, J. Math. Anal. Appl., 198 (1996), 84. doi: 10.1006/jmaa.1996.0069. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00220-004-1062-2. Google Scholar [15] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics,, Ph.D thesis, (1983). Google Scholar [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. doi: 10.2140/pjm.2002.207.199. Google Scholar [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. doi: 10.1007/s002200050543. Google Scholar [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. Google Scholar [19] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 679. doi: 10.1016/j.anihpc.2007.03.005. Google Scholar [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. doi: 10.1016/j.jmaa.2011.11.006. Google Scholar [21] D. Q. Li and Y. M. Chen, Nonlinear Evolution Equation,, Science Press, (1989). Google Scholar [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. doi: 10.1007/s002200050418. Google Scholar [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. Google Scholar [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. doi: 10.3792/pjaa.55.337. Google Scholar [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. doi: 10.1007/BF01214738. Google Scholar [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. doi: 10.1016/S0252-9602(09)60108-9. Google Scholar [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. doi: 10.2969/jmsj/1191419003. Google Scholar [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. doi: 10.1016/j.camwa.2006.02.030. Google Scholar [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar [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. doi: 10.1007/s00033-013-0331-3. Google Scholar [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. doi: 10.1142/S0219891606000902. Google Scholar [32] 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 [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. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar [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. doi: 10.1016/j.aml.2011.04.028. Google Scholar

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##### References:
 [1] R. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [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. doi: 10.1081/PDE-120020499. Google Scholar [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. doi: 10.1016/j.matpur.2013.06.005. Google Scholar [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. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar [5] K. Deckelnick, Decay estimates for the compressible Navier-Stokes equations in unbounded domains,, Math. Z., 209 (1992), 115. doi: 10.1007/BF02570825. Google Scholar [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. doi: 10.1016/j.jde.2007.03.008. Google Scholar [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. doi: 10.1142/S021820250700208X. Google Scholar [8] J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working,, Arch. Rational Mech. Anal., 88 (1985), 95. doi: 10.1007/BF00250907. Google Scholar [9] Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar [10] B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type,, J. Math. Fluid Mech., 13 (2011), 223. doi: 10.1007/s00021-009-0013-2. Google Scholar [11] H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type,, SIAM J. Math. Anal., 25 (1994), 85. doi: 10.1137/S003614109223413X. Google Scholar [12] H. Hattori and D. Li, Global solutions of a high dimensionl system for Korteweg materials,, J. Math. Anal. Appl., 198 (1996), 84. doi: 10.1006/jmaa.1996.0069. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00220-004-1062-2. Google Scholar [15] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations in Magnetohydrodynamics,, Ph.D thesis, (1983). Google Scholar [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. doi: 10.2140/pjm.2002.207.199. Google Scholar [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. doi: 10.1007/s002200050543. Google Scholar [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. Google Scholar [19] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 679. doi: 10.1016/j.anihpc.2007.03.005. Google Scholar [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. doi: 10.1016/j.jmaa.2011.11.006. Google Scholar [21] D. Q. Li and Y. M. Chen, Nonlinear Evolution Equation,, Science Press, (1989). Google Scholar [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. doi: 10.1007/s002200050418. Google Scholar [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. Google Scholar [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. doi: 10.3792/pjaa.55.337. Google Scholar [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. doi: 10.1007/BF01214738. Google Scholar [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. doi: 10.1016/S0252-9602(09)60108-9. Google Scholar [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. doi: 10.2969/jmsj/1191419003. Google Scholar [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. doi: 10.1016/j.camwa.2006.02.030. Google Scholar [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar [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. doi: 10.1007/s00033-013-0331-3. Google Scholar [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. doi: 10.1142/S0219891606000902. Google Scholar [32] 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 [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. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar [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. doi: 10.1016/j.aml.2011.04.028. Google Scholar
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