Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system
Rousset, Frédéric1 ; Sun, Changzhen1
1 Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay (UMR 8628), 91405 Orsay Cedex, France
Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1255-1294.
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We prove a stability result of constant equilibria for the three dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter ε while the incompressible part of the initial velocity is assumed to be small compared to ε. We then get a unique global smooth solution. We also prove a uniform in ε time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at ε fixed and the dispersive techniques (dispersive estimates and normal forms) that are useful for the inviscid irrotational system.

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.11.004
Mots-clés : Navier-Stokes-Poisson system, Small viscosity, Global existence
Rousset, Frédéric 1 ; Sun, Changzhen 1

1 Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay (UMR 8628), 91405 Orsay Cedex, France 
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     title = {Stability of equilibria uniformly in the inviscid limit for the {Navier-Stokes-Poisson} system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Rousset, Frédéric; Sun, Changzhen. Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1255-1294. doi : 10.1016/j.anihpc.2020.11.004. http://www.numdam.org/articles/10.1016/j.anihpc.2020.11.004/

[1] Bahouri, H.; Chemin, J.-Y.; Danchin, R. Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer Science & Business Media, 2011 | MR | Zbl

[2] Cai, Y.; Lei, Z.; Lin, F.; Masmoudi, N. Vanishing viscosity limit for incompressible viscoelasticity in two dimensions, Commun. Pure Appl. Math., Volume 72 (2019) no. 10, pp. 2063-2120 | DOI | MR | Zbl

[3] Chikami, N.; Danchin, R. On the global existence and time decay estimates in critical spaces for the Navier-Stokes-Poisson system, Math. Nachr., Volume 290 (2017) no. 13, pp. 1939-1970 | DOI | MR | Zbl

[4] Germain, P. Space-time resonances, Journ. Equ. Dériv. Partielles (2010), pp. 1-10

[5] Germain, P.; Masmoudi, N.; Pausader, B. Nonneutral global solutions for the electron Euler-Poisson system in three dimensions, SIAM J. Math. Anal., Volume 45 (2013) no. 1, pp. 267-278 | DOI | MR | Zbl

[6] Grafakos, L. Classical Fourier Analysis, Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008 | MR | Zbl

[7] Grafakos, L. Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009 | MR | Zbl

[8] Guo, Y. Smooth irrotational flows in the large to the Euler-Poisson system in R3+1 , Commun. Math. Phys., Volume 195 (1998) no. 2, pp. 249-265 | DOI | MR | Zbl

[9] Guo, Y.; Han, L.; Zhang, J. Absence of shocks for one dimensional Euler-Poisson system, Arch. Ration. Mech. Anal., Volume 223 (2017) no. 3, pp. 1057-1121 | DOI | MR | Zbl

[10] Guo, Y.; Pausader, B. Global smooth ion dynamics in the Euler-Poisson system, Commun. Math. Phys., Volume 303 (2011) no. 1, pp. 89-125 | DOI | MR | Zbl

[11] Guo, Y.; Wang, Y. Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Differ. Equ., Volume 37 (2012) no. 12, pp. 2165-2208 | DOI | MR | Zbl

[12] Guo, Z.; Peng, L.; Wang, B. Decay estimates for a class of wave equations, J. Funct. Anal., Volume 254 (2008) no. 6, pp. 1642-1660 | DOI | MR | Zbl

[13] Hao, C.; Li, H.-L. Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differ. Equ., Volume 246 (2009) no. 12, pp. 4791-4812 | DOI | MR | Zbl

[14] Hoff, D.; Zumbrun, K. Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., Volume 44 (1995) no. 2, pp. 603-676 | DOI | MR | Zbl

[15] Ionescu, A.D.; Pausader, B. The Euler-Poisson system in 2D: global stability of the constant equilibrium solution, Int. Math. Res. Not., Volume 4 (2013), pp. 761-826 | DOI | MR | Zbl

[16] Kato, T. Liapunov functions and monotonicity in the Navier-Stokes equation, Tokyo, 1989 (Lecture Notes in Math.), Volume vol. 1450, Springer, Berlin (1990), pp. 53-63 | DOI | MR | Zbl

[17] Kato, T.; Ponce, G. Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., Volume 41 (1988) no. 7, pp. 891-907 | DOI | MR | Zbl

[18] Li, D.; Wu, Y. The Cauchy problem for the two dimensional Euler-Poisson system, J. Eur. Math. Soc., Volume 16 (2014) no. 10, pp. 2211-2266 | DOI | MR | Zbl

[19] Li, H.-L.; Matsumura, A.; Zhang, G. Optimal decay rate of the compressible Navier-Stokes-Poisson system in R3 , Arch. Ration. Mech. Anal., Volume 196 (2010) no. 2, pp. 681-713 | DOI | MR | Zbl

[20] Matsumura, A.; Nishida, T. The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., Volume 20 (1980) no. 1, pp. 67-104 | MR | Zbl

[21] Nakanishi, K.; Schlag, W. Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2011 | DOI | MR | Zbl

[22] Shatah, J. Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., Volume 38 (1985) no. 5, pp. 685-696 | DOI | MR | Zbl

[23] Stein, E.M. Harmonic Analysis (PMS-43), Volume 43 – Real-Variable Methods, Orthogonality, and Oscillatory Integrals. (PMS-43) , vol. 43, Princeton University Press, 2016 | MR | Zbl

[24] Wang, Y. Decay of the Navier-Stokes-Poisson equations, J. Differ. Equ., Volume 253 (2012) no. 1, pp. 273-297 | DOI | MR | Zbl

[25] Zheng, X. Global well-posedness for the compressible Navier-Stokes-Poisson system in the Lp framework, Nonlinear Anal., Volume 75 (2012) no. 10, pp. 4156-4175 | DOI | MR | Zbl

[26] Zworski, M. Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012 | MR | Zbl

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