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Article Contents

# From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence

• We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to $\epsilon^{-1/2}$ for $\epsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations --$\rho_\epsilon$ converge to the unique solution to the porous medium equation [14,13]. For viscosity coefficient $\mu(\rho_\epsilon)=\rho_\epsilon^\alpha$ with $\alpha>1$, we obtain a rate of convergence of $\rho_\epsilon$ in $L^\infty(0,T; H^{-1}(\mathbb{R}))$; for $1<\alpha\leq\frac{3}{2}$ the solution $\rho_\epsilon$ converges in $L^\infty(0,T;L^2(\mathbb{R}))$. For compactly supported initial data, we prove that most of the mass corresponding to solution $\rho_\epsilon$ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of $\epsilon$.
Mathematics Subject Classification: Primary: 35B25, 35Q30; Secondary: 35K55.

 Citation:

•  [1] D. Bresch and B. Desjardins, Some diffusive capillary models of Korteweg type, C. R. Math. Acad. Sci. Paris, Section Mécanique, 332 (2004), 881-886. [2] D. Bresch and B. Desjardins, Existence of global weak solution for 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.doi: 10.1007/s00220-003-0859-8. [3] D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl. (9), 87 (2007), 57-90.doi: 10.1016/j.matpur.2006.11.001. [4] D. Bresch, B. Desjardins and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part II existence of global $\kappa$-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities, J. Math. Pures Appl. (9), 104 (2015), 801-836.doi: 10.1016/j.matpur.2015.05.004. [5] D. Bresch, V. Giovangigli and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part I well posedness for zero Mach number systems, J. Math. Pures Appl. (9), 104 (2015), 762-800.doi: 10.1016/j.matpur.2015.05.003. [6] J. A. Carrillo, M. P. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, C. R. Math. Acad. Sci. Paris, 338 (2004), 815-818.doi: 10.1016/j.crma.2004.03.025. [7] J.-F. Coulombel, From gas dynamics to pressureless gas dynamics, Proc. Amer. Math. Soc., 134 (2006), 683-688.doi: 10.1090/S0002-9939-05-08087-1. [8] R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4), 35 (2002), 27-75.doi: 10.1016/S0012-9593(01)01085-0. [9] B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271-2279.doi: 10.1098/rspa.1999.0403. [10] B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9), 78 (1999), 461-471.doi: 10.1016/S0021-7824(99)00032-X. [11] M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal., 128 (2015), 106-121.doi: 10.1016/j.na.2015.07.006. [12] T. Goudon and S. Junca, Vanishing pressure in gas dynamics equations, Z. angew. Math. Phys., 51 (2000), 143-148.doi: 10.1007/PL00001502. [13] B. Haspot, Porous media equations, fast diffusions equations and the existence of global weak solution for the quasi-solutions of compressible Navier-Stokes equations, in Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the 14th International Conference on Hyperbolic Problems held in Padova, June 25-29, 2012, 2014, 667-674. [14] B. Haspot, From the highly compressible Navier-Stokes equations to fast diffusion and porous media equations, existence of global weak solution for the quasi-solutions, to appear in Journal of Mathematical Fluid Mechanics, HAL Id: hal-00770248, arXiv:1304.4502, 2013. [15] B. Haspot, Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D, HAL Id: hal-01082319, arXiv:1411.5503, 11 2014. [16] B. Haspot, New formulation of the compressible Navier-Stokes equations and parabolicity of the density, HAL Id: hal-01081580, arXiv:1411.5501, 11 2014. [17] B. Haspot, New entropy for Korteweg's system, existence of global weak solution and new blow-up criterion, HAL . Id: hal-00778811, 2013. [18] D. Hoff and D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow, SIAM J. Appl. Math., 51 (1991), 887-898.doi: 10.1137/0151043. [19] S. Jiang, Z. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251.doi: 10.4310/MAA.2005.v12.n3.a2. [20] Q. Jiu and Z. Xin, The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, Kinet. Relat. Models, 1 (2008), 313-330.doi: 10.3934/krm.2008.1.313. [21] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.doi: 10.1002/cpa.3160340405. [22] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968. [23] M. Lewicka and P. B. Mucha, On temporal asymptotics for the $p$th power viscous reactive gas, Nonlinear Anal., 57 (2004), 951-969.doi: 10.1016/j.na.2003.12.001. [24] H.-L. Li, J. Li and Z. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444.doi: 10.1007/s00220-008-0495-4. [25] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 77 (1998), 585-627.doi: 10.1016/S0021-7824(98)80139-6. [26] A. Mellet and A. F. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452.doi: 10.1080/03605300600857079. [27] A. Mellet and A. F. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39 (2007/08), 1344-1365. doi: 10.1137/060658199. [28] P. B. Mucha, Compressible Navier-Stokes system in 1-D, Math. Methods Appl. Sci., 24 (2001), 607-622.doi: 10.1002/mma.232. [29] P. B. Mucha, M. Pokorný and E. Zatorska, Approximate solutions to model of two-component reactive flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 1079-1099.doi: 10.3934/dcdss.2014.7.1079. [30] A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, arXiv:1501.06803, 03 2015. [31] A. F. Vasseur and C. Yu, Global weak solutions to compressible quantum Navier-Stokes equations with damping, arXiv:1503.06894, 03 2015. [32] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and Its Applications, 2006.doi: 10.1093/acprof:oso/9780199202973.001.0001. [33] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, 2007. [34] S.-W. Vong, T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. II, J. Differential Equations, 192 (2003), 475-501.doi: 10.1016/S0022-0396(03)00060-3. [35] T. Yang, Z.-a. Yao and C. Zhu, Compressible Navier-Stokes equations with density-dependent viscosity and vacuum, Comm. Partial Differential Equations, 26 (2001), 965-981.doi: 10.1081/PDE-100002385. [36] T. Yang and H. Zhao, A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 184 (2002), 163-184.doi: 10.1006/jdeq.2001.4140. [37] T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363.doi: 10.1007/s00220-002-0703-6. [38] E. Zatorska, On the flow of chemically reacting gaseous mixture, J. Differential Equations, 253 (2012), 3471-3500.doi: 10.1016/j.jde.2012.08.043.