# American Institute of Mathematical Sciences

June  2016, 36(6): 3107-3123. doi: 10.3934/dcds.2016.36.3107

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

 1 Ceremade UMR CNRS 7534 Universite Paris Dauphine, Place du Marechal DeLattre De Tassigny, 75775 PARIS CEDEX 16, France 2 Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland

Received  April 2015 Revised  October 2015 Published  December 2015

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 $\epsilon$.
Citation: Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
##### References:
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Zatorska, Approximate solutions to model of two-component reactive flow,, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 1079. doi: 10.3934/dcdss.2014.7.1079. Google Scholar [30] A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations,, , (2015). Google Scholar [31] A. F. Vasseur and C. Yu, Global weak solutions to compressible quantum Navier-Stokes equations with damping,, , (2015). Google Scholar [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. Google Scholar [33] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (2007). Google Scholar [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. doi: 10.1016/S0022-0396(03)00060-3. Google Scholar [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. doi: 10.1081/PDE-100002385. Google Scholar [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. doi: 10.1006/jdeq.2001.4140. Google Scholar [37] T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,, Comm. Math. Phys., 230 (2002), 329. doi: 10.1007/s00220-002-0703-6. Google Scholar [38] E. Zatorska, On the flow of chemically reacting gaseous mixture,, J. Differential Equations, 253 (2012), 3471. doi: 10.1016/j.jde.2012.08.043. Google Scholar

show all references

##### References:
 [1] D. Bresch and B. Desjardins, Some diffusive capillary models of Korteweg type,, C. R. Math. Acad. Sci. Paris, 332 (2004), 881. Google Scholar [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. doi: 10.1007/s00220-003-0859-8. Google Scholar [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. doi: 10.1016/j.matpur.2006.11.001. Google Scholar [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. doi: 10.1016/j.matpur.2015.05.004. Google Scholar [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. doi: 10.1016/j.matpur.2015.05.003. Google Scholar [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. doi: 10.1016/j.crma.2004.03.025. Google Scholar [7] J.-F. Coulombel, From gas dynamics to pressureless gas dynamics,, Proc. Amer. Math. Soc., 134 (2006), 683. doi: 10.1090/S0002-9939-05-08087-1. Google Scholar [8] R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations,, Ann. Sci. École Norm. Sup. (4), 35 (2002), 27. doi: 10.1016/S0012-9593(01)01085-0. Google Scholar [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. doi: 10.1098/rspa.1999.0403. Google Scholar [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. doi: 10.1016/S0021-7824(99)00032-X. Google Scholar [11] M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations,, Nonlinear Anal., 128 (2015), 106. doi: 10.1016/j.na.2015.07.006. Google Scholar [12] T. Goudon and S. Junca, Vanishing pressure in gas dynamics equations,, Z. angew. Math. Phys., 51 (2000), 143. doi: 10.1007/PL00001502. Google Scholar [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, (2012), 25. Google Scholar [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, (2013). Google Scholar [15] B. Haspot, Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D,, HAL Id: hal-01082319, (2014). Google Scholar [16] B. Haspot, New formulation of the compressible Navier-Stokes equations and parabolicity of the density,, HAL Id: hal-01081580, (2014). Google Scholar [17] B. Haspot, New entropy for Korteweg's system, existence of global weak solution and new blow-up criterion,, HAL . Id: hal-00778811, (2013). Google Scholar [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. doi: 10.1137/0151043. Google Scholar [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. doi: 10.4310/MAA.2005.v12.n3.a2. Google Scholar [20] Q. Jiu and Z. Xin, The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients,, Kinet. Relat. Models, 1 (2008), 313. doi: 10.3934/krm.2008.1.313. Google Scholar [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. doi: 10.1002/cpa.3160340405. Google Scholar [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, (1968). Google Scholar [23] M. Lewicka and P. B. Mucha, On temporal asymptotics for the $p$th power viscous reactive gas,, Nonlinear Anal., 57 (2004), 951. doi: 10.1016/j.na.2003.12.001. Google Scholar [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. doi: 10.1007/s00220-008-0495-4. Google Scholar [25] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl. (9), 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6. Google Scholar [26] A. Mellet and A. F. Vasseur, On the barotropic compressible Navier-Stokes equations,, Comm. Partial Differential Equations, 32 (2007), 431. doi: 10.1080/03605300600857079. Google Scholar [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 (): 1344. doi: 10.1137/060658199. Google Scholar [28] P. B. Mucha, Compressible Navier-Stokes system in 1-D,, Math. Methods Appl. Sci., 24 (2001), 607. doi: 10.1002/mma.232. Google Scholar [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. doi: 10.3934/dcdss.2014.7.1079. Google Scholar [30] A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations,, , (2015). Google Scholar [31] A. F. Vasseur and C. Yu, Global weak solutions to compressible quantum Navier-Stokes equations with damping,, , (2015). Google Scholar [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. Google Scholar [33] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs, (2007). Google Scholar [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. doi: 10.1016/S0022-0396(03)00060-3. Google Scholar [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. doi: 10.1081/PDE-100002385. Google Scholar [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. doi: 10.1006/jdeq.2001.4140. Google Scholar [37] T. Yang and C. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,, Comm. Math. Phys., 230 (2002), 329. doi: 10.1007/s00220-002-0703-6. Google Scholar [38] E. Zatorska, On the flow of chemically reacting gaseous mixture,, J. Differential Equations, 253 (2012), 3471. doi: 10.1016/j.jde.2012.08.043. Google Scholar
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