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

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July 2021, 41(7): 3489-3530. 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 July 2021 Early access 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 and Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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