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 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

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:
[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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

[18]

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

[19]

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

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]