American Institute of Mathematical Sciences

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May  2022, 21(5): 1537-1565. doi: 10.3934/cpaa.2022028

Global Well-posedness and Optimal Decay Rate of the Quasi-static Incompressible Navier–Stokes–Fourier–Maxwell–Poisson System

Yuan Xu , , Fujun Zhou and  Weihua Gong

School of Mathematics, South China University of Technology, Guangzhou, 510640, China

* Corresponding author

Received  September 2021 Revised  December 2021 Published  May 2022 Early access  February 2022

Fund Project: The authors are supported by NSFC under the grant number 11371147, Fundamental Research Founds for the Central Universities under the grant number 2019MS112, and Foundation for Basic and Applied Basic Research of Guangdong under the grant number 2020A1515011562

This work aims to establish global classical solution and optimal $ L^p $ ($ p\ge 2 $) time decay rate of the quasi-static incompressible Navier–Stokes–Fourier–Maxwell–Poisson system with small initial data in $ \mathbb{R}^3 $. The optimal $ L^2 $ time decay rate for higher order spatial derivatives is also given. To deal with the difficulty induced by the degeneration of the coupled Maxwell equation, we adopt the vector-valued form of the electric field $ E $ to obtain the time decay rate.

Keywords: Quasi-static incompressible Navier–Stokes–Fourier–Maxwell–Poisson system, global solutions, optimal time decay rate.
Mathematics Subject Classification: Primary: 76D05, 76W05.
Citation: Yuan Xu, Fujun Zhou, Weihua Gong. Global Well-posedness and Optimal Decay Rate of the Quasi-static Incompressible Navier–Stokes–Fourier–Maxwell–Poisson System. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1537-1565. doi: 10.3934/cpaa.2022028
References:
[1]

D. Arsénio and L. Saint-Raymond, From the Vlasov–Maxwell–Boltzmann system to incompressible viscous electro–magneto–hydrodynamics, in Monographs in Mathematics, European Mathematical Society, Zürich, (2019). doi: 10.4171/193.

[2]

L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Commun. Pure Appl. Math, 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.

[3]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier–Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789. 

[4]

R. Duan, H. Liu, S. Ukai and T. Yang, Optimal $L^p$–$L^q$ convergence rates for the compressible Navier–Stokes equations with potential force, J. Differ. Equ., 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.

[5]

R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal convergence rates for the compressible Navier–Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758.  doi: 10.1142/S021820250700208X.

[6]

R. Duan, Green's function and large time behavior of the Navier–Stokes–Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078.

[7]

Y. Feng, Y. Peng and S. Wang, Asymptotic behavior of global smooth solutions for full compressible Navier–Stokes–Maxwell equations, Nonlinear Anal. RWA., 19 (2014), 105-116.  doi: 10.1016/j.nonrwa.2014.03.004.

[8]

Y. Fujigaki and T. Miyakawa, Asymptotic profiles of nonstationary incompressible Navier–Stokes flows in the whole space, SIAM J. Math. Anal., 33 (2001), 523-544.  doi: 10.1137/S0036141000367072.

[9]

H. Fujita and T. Kato, On the Navier–Stokes initial value problem. Ⅰ., Arch. Ration. Mech. Anal., 16 (1964), 269-315. 

[10]

P. Germain, S. Ibrahim and N. Masmoudi, Well-posedness of the Navier–Stokes–Maxwell equations, Proc. Roy. Soc. Edinb. Sect. A, 144 (2014), 71-86.  doi: 10.1017/S0308210512001242.

[11]

W. Gong, F. Zhou, W. Wu and Q. Hu, Optimal decay rate of the two-fluid incompressible Navier–Stokes–Fourier–Poisson system with Ohm's law, Nonlinear Analy. RWA., 63 (2022), 103392.  doi: 10.1016/j.nonrwa.2021.103392.

[12]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differ. Equ., 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[13]

D. Hoff and K. Zumbru, Multi-dimensional diffusion waves for the Navier–Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. 

[14]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.

[15]

S. Ibrahim and S. Keraani, Global small solutions for the Navier–Stokes–Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295.  doi: 10.1137/100819813.

[16]

N. Jiang and Y. Luo, Global classical solutions to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm's law, Commun. Math. Sci., 16 (2018), 561-578.  doi: 10.4310/CMS.2018.v16.n2.a12.

[17]

N. Jiang, Y. Luo and S. Tang, Convergence from two-fluid incompressible Navier–Stokes–Maxwell system with Ohm's law to solenoidal Ohm's law: classical solutions, J. Differ. Equ., 269 (2020), 349-376.  doi: 10.1016/j.jde.2019.12.006.

[18]

R. Kajikiya and T. Miyakawa, On $L^2$ decay of weak solutions of the Navier–Stokes equations in $ \mathbb{R}^n$, Math. Z., 192 (1986), 135-148. 

[19]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[20]

H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.

[21]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[22]

H. Li, A. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier–Stokes–Poisson system in $ \mathbb{R}^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4.

[23]

F. Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Commun. Pure Appl. Math., 51 (1998), 241-257.  doi: 10.1002/(sici)1097-0312(199803)51:3<241::aid-cpa2>3.0.co;2-a.

[24]

Q. Liu and Y. Su, Large time behavior for the non-isentropic Navier–Stokes–Maxwell system, Math. Methods Appl. Sci., 40(3) (2017), 663-679.  doi: 10.1002/mma.3999.

[25]

N. Masmoudi, Global well posedness for the Maxwell–Navier–Stokes system in 2D, J. Math. Pures Appl., 93(9) (2010), 559-571.  doi: 10.1016/j.matpur.2009.08.007.

[26]

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

[27]

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

[28]

M. Oliver and E. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier–Stokes equations in $ \mathbb{R}^n$, J. Funct. Anal., 172 (2000), 1-18.  doi: 10.1006/jfan.1999.3550.

[29]

F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier–Stokes equations in $ \mathbb{R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 319-336.  doi: 10.1016/S0294-1449(16)30107-X.

[30]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 339-418.  doi: 10.1016/0362-546X(85)90001-X.

[31]

M. Schonbek, $L^2$ decay for weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. 

[32]

M. Schonbek, Large time behaviour of solutions to the Navier–Stokes equations, Commun. Partial Differ. Equ., 11 (1986), 733-763.  doi: 10.1080/03605308608820443.

[33]

W. Wang and X. Xu, Large time behavior of solution for the full compressible Navier–Stokes–Maxwell system, Commun. Pure Appl. Anal., 14 (2015), 2283-2313.  doi: 10.3934/cpaa.2015.14.2283.

[34]

Y. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271.  doi: 10.1016/j.jmaa.2011.01.006.

[35]

Y. Wang, Decay of the Navier–Stokes–Poisson equations, J. Differ. Equ., 253 (2012), 273-297.  doi: 10.1016/j.jde.2012.03.006.

[36]

Z. Wu and W. Wang, Pointwise estimates for bipolar compressible Navier–Stokes–Poisson system in dimension three, Arch. Ration. Mech. Anal., 226 (2017), 587-638.  doi: 10.1007/s00205-017-1140-1.

[37]

Z. Wu and W. Wang, Generalized Huygens' principle for bipolar non-isentropic compressible Navier–Stokes–Poisson system in dimension three, J. Differ. Equ., 269 (2020), 7906-7930.  doi: 10.1016/j.jde.2020.05.046.

[38]

G. Zhang, H. Li and C. Zhu, Optimal decay rate of the non-isentropic compressible Navier–Stokes–Poisson system in $ \mathbb{R}^3$, J. Differ. Equ., 250 (2011), 866-891.  doi: 10.1016/j.jde.2010.07.035.

show all references

References:
[1]

D. Arsénio and L. Saint-Raymond, From the Vlasov–Maxwell–Boltzmann system to incompressible viscous electro–magneto–hydrodynamics, in Monographs in Mathematics, European Mathematical Society, Zürich, (2019). doi: 10.4171/193.

[2]

L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Commun. Pure Appl. Math, 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.

[3]

P. Constantin and C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier–Stokes equations, Indiana Univ. Math. J., 42 (1993), 775-789. 

[4]

R. Duan, H. Liu, S. Ukai and T. Yang, Optimal $L^p$–$L^q$ convergence rates for the compressible Navier–Stokes equations with potential force, J. Differ. Equ., 238 (2007), 220-233.  doi: 10.1016/j.jde.2007.03.008.

[5]

R. Duan, S. Ukai, T. Yang and H. Zhao, Optimal convergence rates for the compressible Navier–Stokes equations with potential forces, Math. Models Methods Appl. Sci., 17 (2007), 737-758.  doi: 10.1142/S021820250700208X.

[6]

R. Duan, Green's function and large time behavior of the Navier–Stokes–Maxwell system, Anal. Appl., 10 (2012), 133-197.  doi: 10.1142/S0219530512500078.

[7]

Y. Feng, Y. Peng and S. Wang, Asymptotic behavior of global smooth solutions for full compressible Navier–Stokes–Maxwell equations, Nonlinear Anal. RWA., 19 (2014), 105-116.  doi: 10.1016/j.nonrwa.2014.03.004.

[8]

Y. Fujigaki and T. Miyakawa, Asymptotic profiles of nonstationary incompressible Navier–Stokes flows in the whole space, SIAM J. Math. Anal., 33 (2001), 523-544.  doi: 10.1137/S0036141000367072.

[9]

H. Fujita and T. Kato, On the Navier–Stokes initial value problem. Ⅰ., Arch. Ration. Mech. Anal., 16 (1964), 269-315. 

[10]

P. Germain, S. Ibrahim and N. Masmoudi, Well-posedness of the Navier–Stokes–Maxwell equations, Proc. Roy. Soc. Edinb. Sect. A, 144 (2014), 71-86.  doi: 10.1017/S0308210512001242.

[11]

W. Gong, F. Zhou, W. Wu and Q. Hu, Optimal decay rate of the two-fluid incompressible Navier–Stokes–Fourier–Poisson system with Ohm's law, Nonlinear Analy. RWA., 63 (2022), 103392.  doi: 10.1016/j.nonrwa.2021.103392.

[12]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differ. Equ., 37 (2012), 2165-2208.  doi: 10.1080/03605302.2012.696296.

[13]

D. Hoff and K. Zumbru, Multi-dimensional diffusion waves for the Navier–Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. 

[14]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.

[15]

S. Ibrahim and S. Keraani, Global small solutions for the Navier–Stokes–Maxwell system, SIAM J. Math. Anal., 43 (2011), 2275-2295.  doi: 10.1137/100819813.

[16]

N. Jiang and Y. Luo, Global classical solutions to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm's law, Commun. Math. Sci., 16 (2018), 561-578.  doi: 10.4310/CMS.2018.v16.n2.a12.

[17]

N. Jiang, Y. Luo and S. Tang, Convergence from two-fluid incompressible Navier–Stokes–Maxwell system with Ohm's law to solenoidal Ohm's law: classical solutions, J. Differ. Equ., 269 (2020), 349-376.  doi: 10.1016/j.jde.2019.12.006.

[18]

R. Kajikiya and T. Miyakawa, On $L^2$ decay of weak solutions of the Navier–Stokes equations in $ \mathbb{R}^n$, Math. Z., 192 (1986), 135-148. 

[19]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[20]

H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.

[21]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[22]

H. Li, A. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier–Stokes–Poisson system in $ \mathbb{R}^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4.

[23]

F. Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Commun. Pure Appl. Math., 51 (1998), 241-257.  doi: 10.1002/(sici)1097-0312(199803)51:3<241::aid-cpa2>3.0.co;2-a.

[24]

Q. Liu and Y. Su, Large time behavior for the non-isentropic Navier–Stokes–Maxwell system, Math. Methods Appl. Sci., 40(3) (2017), 663-679.  doi: 10.1002/mma.3999.

[25]

N. Masmoudi, Global well posedness for the Maxwell–Navier–Stokes system in 2D, J. Math. Pures Appl., 93(9) (2010), 559-571.  doi: 10.1016/j.matpur.2009.08.007.

[26]

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

[27]

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

[28]

M. Oliver and E. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier–Stokes equations in $ \mathbb{R}^n$, J. Funct. Anal., 172 (2000), 1-18.  doi: 10.1006/jfan.1999.3550.

[29]

F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier–Stokes equations in $ \mathbb{R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 319-336.  doi: 10.1016/S0294-1449(16)30107-X.

[30]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 339-418.  doi: 10.1016/0362-546X(85)90001-X.

[31]

M. Schonbek, $L^2$ decay for weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. 

[32]

M. Schonbek, Large time behaviour of solutions to the Navier–Stokes equations, Commun. Partial Differ. Equ., 11 (1986), 733-763.  doi: 10.1080/03605308608820443.

[33]

W. Wang and X. Xu, Large time behavior of solution for the full compressible Navier–Stokes–Maxwell system, Commun. Pure Appl. Anal., 14 (2015), 2283-2313.  doi: 10.3934/cpaa.2015.14.2283.

[34]

Y. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271.  doi: 10.1016/j.jmaa.2011.01.006.

[35]

Y. Wang, Decay of the Navier–Stokes–Poisson equations, J. Differ. Equ., 253 (2012), 273-297.  doi: 10.1016/j.jde.2012.03.006.

[36]

Z. Wu and W. Wang, Pointwise estimates for bipolar compressible Navier–Stokes–Poisson system in dimension three, Arch. Ration. Mech. Anal., 226 (2017), 587-638.  doi: 10.1007/s00205-017-1140-1.

[37]

Z. Wu and W. Wang, Generalized Huygens' principle for bipolar non-isentropic compressible Navier–Stokes–Poisson system in dimension three, J. Differ. Equ., 269 (2020), 7906-7930.  doi: 10.1016/j.jde.2020.05.046.

[38]

G. Zhang, H. Li and C. Zhu, Optimal decay rate of the non-isentropic compressible Navier–Stokes–Poisson system in $ \mathbb{R}^3$, J. Differ. Equ., 250 (2011), 866-891.  doi: 10.1016/j.jde.2010.07.035.

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