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doi: 10.3934/dcds.2021005

On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow

Ling-Bing He 1, and  Li Xu 2,

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, China
2. 

School of Mathematical Sciences, Beihang University, 100191 Beijing, China

Received  April 2020 Revised  November 2020 Published  January 2021

The present work aims at the mathematical derivation of the equations for the isentropic flow from those for the non-isentropic flow for perfect gases in the whole space. Suppose that the following things hold for the entropy equation: (1). both conduction of heat and its generation by dissipation of mechanical energy are sufficiently weak(with the order of
$ \varepsilon $
); (2). initially the entropy
$ S^{N}_ \varepsilon $
 is around a constant
$ c_S $
, that is,
$ S^{N}_ \varepsilon |_{t = 0} = c_S+O( \varepsilon ) $
. Then the non-isentropic compressible Navier-Stokes equations admit a unique and global solution
$ ( \rho^{N}_ \varepsilon , \, \boldsymbol{u}^{N}_ \varepsilon , \, S^{N}_ \varepsilon ) $
 with the initial data
$ ( \rho_0, \, \boldsymbol{u}_0, \, c_S+ \varepsilon S_0) $
, which is a perturbation of the equilibrium
$ (1, \boldsymbol{0}, c_S) $
. Moreover,
$ ( \rho^{N}_ \varepsilon , u^{N}_ \varepsilon ) $
 can be approximated by
$ ( \rho^{I}, \, u^{I}) $
, the solution to the associated isentropic compressible Navier-Stokes equations equipped with the initial data
$ ( \rho_0, \, \boldsymbol{u}_0) $
, in the sense that
$ \begin{eqnarray*} ( \rho^{N}_ \varepsilon (t), \, \boldsymbol{u}^{N}_ \varepsilon (t)) = ( \rho^{I}(t), \, \boldsymbol{u}^{I}(t))+O(\epsilon), \end{eqnarray*} $
which holds globally in the so-called critical Besov spaces for the compressible Navier-Stokes equations.
Keywords: Compressible Navier-Stokes equations, non isentropic flow, asymptotical analysis, isentropic flow, uniform error estimates.
Mathematics Subject Classification: 35Q35, 35B40, 76N10.
Citation: Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2021005
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, , Grundlehren der Mathematischen Wissenschaften, Springer 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les $\mathop {\rm{q}}\limits^{\rm{'}} $uations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209–246. doi: 10.24033/asens.1404.  Google Scholar

[3]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271.  doi: 10.1007/s00205-010-0306-x.  Google Scholar

[4]

Q. Chen, C. Miao and Z. Zhang, Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26 (2010), 915-946.  doi: 10.4171/RMI/621.  Google Scholar

[5]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.  doi: 10.1002/cpa.20325.  Google Scholar

[6]

Q. Chen, C. Miao and Z. Zhang, On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces., Rev. Mat. Iberoam., 31 (2015), 1375-1402.  doi: 10.4171/RMI/872.  Google Scholar

[7]

N. Chikami and R. Danchin, On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces, J. Differential Equations, 258 (2015), 3435-3467.  doi: 10.1016/j.jde.2015.01.012.  Google Scholar

[8]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.  Google Scholar

[9]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.  doi: 10.1081/PDE-100106132.  Google Scholar

[10]

R. Danchin, Global existence in critical spaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39.  doi: 10.1007/s002050100155.  Google Scholar

[11]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397.  doi: 10.1080/03605300600910399.  Google Scholar

[12]

R. Danchin and L. He, The incompressible limit in $L^p$ type critical spaces, Math. Ann., 366 (2016), 1365-1402.  doi: 10.1007/s00208-016-1361-x.  Google Scholar

[13]

R. Danchin and P. B. Mucha, Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., 320 (2017), 904-925.  doi: 10.1016/j.aim.2017.09.025.  Google Scholar

[14]

R. Danchin and J. Xu, Optimal Time-decay Estimates for the Compressible Navier-Stokes Equations in the Critical $L^p$Framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.  Google Scholar

[15]

D. Fang, T. Zhang and R. Zi, Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data, SIAM J. Math. Anal., 50 (2018), 4983–5026. doi: 10.1137/17M1122062.  Google Scholar

[16]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.  Google Scholar

[17]

E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system., Arch. Ration. Mech. Anal., 204 (2012), 683-706.  doi: 10.1007/s00205-011-0490-3.  Google Scholar

[18]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460.  doi: 10.1007/s00205-011-0430-2.  Google Scholar

[19]

B. Haspot, Global existence of strong solutions for viscous shallow water system with large initial data on the rotational part, J. Differential Equations, 262 (2017), 4931-4978.  doi: 10.1016/j.jde.2017.01.010.  Google Scholar

[20]

X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar

[21]

X. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal., 227 (2018), 995-1059.  doi: 10.1007/s00205-017-1188-y.  Google Scholar

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.  Google Scholar

[23]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.  Google Scholar

[24]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bulletin de la Soc. Math. de France, 90 (1962), 487-497.  doi: 10.24033/bsmf.1586.  Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, , Grundlehren der Mathematischen Wissenschaften, Springer 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

J.-M. Bony, Calcul symbolique et propagation des singularités pour les $\mathop {\rm{q}}\limits^{\rm{'}} $uations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209–246. doi: 10.24033/asens.1404.  Google Scholar

[3]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271.  doi: 10.1007/s00205-010-0306-x.  Google Scholar

[4]

Q. Chen, C. Miao and Z. Zhang, Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26 (2010), 915-946.  doi: 10.4171/RMI/621.  Google Scholar

[5]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.  doi: 10.1002/cpa.20325.  Google Scholar

[6]

Q. Chen, C. Miao and Z. Zhang, On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces., Rev. Mat. Iberoam., 31 (2015), 1375-1402.  doi: 10.4171/RMI/872.  Google Scholar

[7]

N. Chikami and R. Danchin, On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces, J. Differential Equations, 258 (2015), 3435-3467.  doi: 10.1016/j.jde.2015.01.012.  Google Scholar

[8]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.  Google Scholar

[9]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.  doi: 10.1081/PDE-100106132.  Google Scholar

[10]

R. Danchin, Global existence in critical spaces for flows of compressible viscous and heat-conductive gases, Arch. Ration. Mech. Anal., 160 (2001), 1-39.  doi: 10.1007/s002050100155.  Google Scholar

[11]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations, 32 (2007), 1373-1397.  doi: 10.1080/03605300600910399.  Google Scholar

[12]

R. Danchin and L. He, The incompressible limit in $L^p$ type critical spaces, Math. Ann., 366 (2016), 1365-1402.  doi: 10.1007/s00208-016-1361-x.  Google Scholar

[13]

R. Danchin and P. B. Mucha, Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., 320 (2017), 904-925.  doi: 10.1016/j.aim.2017.09.025.  Google Scholar

[14]

R. Danchin and J. Xu, Optimal Time-decay Estimates for the Compressible Navier-Stokes Equations in the Critical $L^p$Framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.  Google Scholar

[15]

D. Fang, T. Zhang and R. Zi, Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data, SIAM J. Math. Anal., 50 (2018), 4983–5026. doi: 10.1137/17M1122062.  Google Scholar

[16]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.  Google Scholar

[17]

E. Feireisl and A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system., Arch. Ration. Mech. Anal., 204 (2012), 683-706.  doi: 10.1007/s00205-011-0490-3.  Google Scholar

[18]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460.  doi: 10.1007/s00205-011-0430-2.  Google Scholar

[19]

B. Haspot, Global existence of strong solutions for viscous shallow water system with large initial data on the rotational part, J. Differential Equations, 262 (2017), 4931-4978.  doi: 10.1016/j.jde.2017.01.010.  Google Scholar

[20]

X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.  Google Scholar

[21]

X. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal., 227 (2018), 995-1059.  doi: 10.1007/s00205-017-1188-y.  Google Scholar

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.  doi: 10.3792/pjaa.55.337.  Google Scholar

[23]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.  Google Scholar

[24]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bulletin de la Soc. Math. de France, 90 (1962), 487-497.  doi: 10.24033/bsmf.1586.  Google Scholar

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