Abstract
We use a general energy method to prove the optimal time decay rates of the solutions to the compressible Navier–Stokes–Korteweg system in the whole space. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained.
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Bresch, D., Desjardins, B., Lin, C.-K.: On some compressible fluid models: Korteweg, lubrication and shallow water system. Commun. Partial Differ. Equ. 28(3–4), 843–868
Dunn J.E., Serrin J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88(2), 95–133 (1985)
Deckelnick K.: Decay estimates for the compressible Navier-Stokes equations in unbounded domains. Math. Z. 209, 115–130 (1992)
Deckelnick K.: L 2-decay for the compressible Navier-Stokes equations in unbounded domains. Commun. Partial Differ. Equ. 18, 1445–1476 (1993)
Danchin R., Desjardins B.: Existence of solutions for compressible fluid models of korteweg type. Ann. Inst. H. Poincare Anal. Non Lineaire 18, 97–133 (2001)
Duan R.J., Ukai S., Yang T., Zhao H.-J.: Optimal L p-L q convergence rate for the compressible Navier-Stokes equations with potential force. J. Differ. Equ. 238, 220–223 (2007)
Duan R.J., Ukai S., Yang T., Zhao H.-J.: Optimal convergence rate for compressible Navier-Stokes equations with potential force. Math. Models Methods Appl. Sci. 17, 737–758 (2007)
Guo, Y., Wang, Y. J.: Decay of dissipative equations and negative Sobolev spaces, Preprint, 2011, [arXiv:1111.5660]
Haspot, B.: Existence of global weak solution for compressible fluid models of korteweg type. J. Math. Fluid Mech. (2009), doi:10.1007/s00021-009-0013-2
Hoff D.: Global solutions of the Navier-Stokes equations for mulitidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)
Hoff D.: Strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Ratio. Mech. Anal. 132, 1–14 (1995)
Hattori H., Li D.: Solutions for two dimensional system for materials of Korteweg type. SIAM J. Math. Anal. 25, 85–98 (1994)
Hattori H., Li D.: Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198(1), 84–97 (1996)
Ju N.: Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space. Commun. Math. Phys. 251, 365–376 (2004)
Korteweg, D.J.: Sur la forme que prennent les équations du mouvement des fluides si I’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans I’hypothè se d’une variation continue de la densiteé, Archives Néerlandaises de Sciences Exactes et Naturelles, 1991, pp. 1–24
Kotschote M.: Strong solutions for a compressible fluid m odel of Korteweg type. Ann. Inst. H. Poincare Anal. Non Lineaire 25(4), 679–696 (2008)
Kobayashi T.: Some estimates of solutions for the equations of compressible viscous fluid in an exterior domain in \({\mathbb{R}^3}\) . J. Differ. Equ. 184, 587–619 (2002)
Kobayashi T., Shibata Y.: Remark on the rate of decay of solutions to linearized compressible Navier-Stokes equations. Pacific J. Math. 207(1), 199–234 (2002)
Kobayashi T., Shibata Y.: Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in \({\mathbb{R}^3}\) . Commun. Math. Phys. 200, 621–659 (1999)
Kagei Y., Kobayashi T.: On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in \({\mathbb{R}^3}\) . Arch. Ration. Mech. Anal. 165, 89–159 (2002)
Li H.-L., Matsumura A., Zhang G.: Optimal decay rate of the compressible Navier-Stokes-Poisson system in \({\mathbb{R}^3}\) . Arch. Ration. Mech. Anal. 196, 681–713 (2010)
Liu T.-P., Wang W.-K.: The pointwise estiamtes of diffusion waves for Navier-Stokes equations in odd multi-dimensions. Commun. Math. Phys. 196, 145–173 (1998)
Li Y.P.: Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force. J. Math. Anal. Appl. 388, 1218–1232 (2011)
Matsumura A., Nishida T.: The initial value problems for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto univ. 20, 67–104 (1980)
Matsumura A., Nishida T.: Initial boundary value problem for equations of motion of compressible viscous and heat conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)
Matsumura A., Nishida T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A 55, 337–342 (1979)
Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959)
Ponce G.: Global existence of small solution to a class of nonlinear evolution equations. Nonlinear Anal. 9, 339–418 (1985)
Stein E.M.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton (1970)
Shibata Y., Tanaka K.: Rate of convergence of non-stationary flow to the steady fiow of compressible viscous fluid. Comput. Math. Appl. 53, 605–623 (2007)
Tan, Z., Wang, H. Q.: Optimal decay rates of the compressible Magnetohydrodynamic equations, Submitted, 2011
Tan Z., Wang H.Q., Xu J.K.: Global existence and optimal L 2 decay rate for the strong solutions to the compressible fluid models of Korteweg type. J. Math. Anal. Appl. 390, 181–187 (2012)
Ukai S., Yang T., Zhao H.-J.: Convergence rate for the compressible Navier-Stokes equations with external force. J. Hyperbolic differ. Equ. 3, 561–574 (2006)
Wang, Y.J.: Decay of the Vlasov-Poisson-Boltzmann system, Preprint, 2011, [arXiv:1111.6335]
Wang, Y.J.: Decay of the Navier-Stokes-Poisson equations, Preprint, 2011, [arXiv:1112.4902]
Wang Y.J., Tan Z.: Optimal decay rates for the compressible fluid models of Korteweg type. J. Math. Anal. Appl. 379, 256–271 (2011)
Wang Y.J., Tan Z.: Global existence and optimal decay rate for the strong solutions in H 2 to the compressible Navier-Stokes equations. Appl. Math. Lett. 24, 1778–1784 (2011)
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Supported by National Natural Science Foundation of China- NSAF (Grant No. 10976026).
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Tan, Z., Zhang, R. Optimal decay rates of the compressible fluid models of Korteweg type. Z. Angew. Math. Phys. 65, 279–300 (2014). https://doi.org/10.1007/s00033-013-0331-3
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DOI: https://doi.org/10.1007/s00033-013-0331-3