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

Boris  Haspot; Ewelina  Zatorska

doi:10.3934/dcds.2016.36.3107

Previous Article
On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$
DCDS Home
This Issue
Next Article
Global existence and optimal decay rates of solutions for compressible Hall-MHD equations

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

Boris Haspot ^1, and Ewelina Zatorska ^2,

1.	Ceremade UMR CNRS 7534 Universite Paris Dauphine, Place du Marechal 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

Abstract
Related Papers

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 $\epsilon$.

Keywords: large Mach number., Compressible Navier-Stokes equations, porous medium equation.

Mathematics Subject Classification: Primary: 35B25, 35Q30; Secondary: 35K5.

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:

[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

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

[1]	Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[2]	Tao Wang, Huijiang Zhao, Qingyang Zou. One-dimensional compressible Navier-Stokes equations with large density oscillation. Kinetic & Related Models, 2013, 6 (3) : 649-670. doi: 10.3934/krm.2013.6.649
[3]	Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[4]	Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[5]	Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[6]	Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[7]	Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609
[8]	Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719
[9]	Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[10]	Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
[11]	C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[12]	Fucai Li, Yanmin Mu. Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1669-1705. doi: 10.3934/dcds.2018069
[13]	Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[14]	Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[15]	Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[16]	Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[17]	Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409
[18]	Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[19]	Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373
[20]	Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors