In this paper, we consider the 3D compressible isentropic Navier-Stokes equations when the shear viscosity μ is a positive constant and the bulk viscosity is λ(ρ) = ρβ with β > 2, which is a situation that was first introduced by Vaigant and Kazhikhov in [1]. The global axisymmetric classical solution with arbitrarily large initial data in a periodic domain Ω = {(r, z)|r = √x2 + y2, (x, y, z) ∈ R3, r ∈ I ⊂ (0, +∞), −∞ < z < +∞} is obtained. Here the initial density keeps a non-vacuum state p > 0 when z → ±∞. Our results also show that the solution will not develop the vacuum state in any finite time, provided that the initial density is uniformly away from the vacuum.
Mei WANG
,
Zilai LI
,
Zhenhua GUO
. GLOBAL SOLUTIONS TO A 3D AXISYMMETRIC COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY[J]. Acta mathematica scientia, Series B, 2022
, 42(2)
: 521
-539
.
DOI: 10.1007/s10473-022-0207-8
[1] Vaigant V A, Kazhikhov A V. On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid (in Russian). 1995, 36:1283-1316; translation in Siberian Math J, 1995, 36:1283-1316
[2] Hoff D, Smoller J. Non-Formation of Vacuum States for Compressible Navier-Stokes Equations. Comm Math Phys, 2001, 216:255-276
[3] Kazhikhov A V, Shelukhin V V. Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. (Russian) J Appl Math Mech, 1977, 41:273-282; translated from Prikl Mat Meh, 1977, 41:282-291
[4] Itaya N. On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluid. Kodai Math Sem Rep, 1971, 23:60-120
[5] Nash J. Le problème de Cauchy pour les équations différentielles d'un fluide général. Bull Soc Math France, 1962, 90:487-497
[6] Matsumura A, Nishida T. The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc Japan Acad Ser A Math Sci, 1979, 80:337-342; Appl Anal, 2011,10:459-478
[7] Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscripta Math, 2006, 120:91-129
[8] Luo Z. Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum. Commun Math Sci, 2012, 10:527-554
[9] Chen Q L, Miao C X, Zhang Z F. Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity. Comm Pure Appl Math, 2010,63:1173-1224
[10] Huang X D, Li J, Xin Z P. Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Comm Pure Appl Math, 2010, 65:549-585
[11] Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford:Oxford University Press, 2004
[12] Jiang S, Zhang P. On Spherically Symmetric Solutions of the Compressible Isentropic Navier-Stokes Equations. Comm Math Phys, 2001, 215:559-581
[13] Lions P L. Mathematical topics in fluid mechanics. Vol. 2. Compressible models. New York:Oxford University Press, 1998
[14] Rozanova O. Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations. J Differential Equations, 2008, 245:1762-1774
[15] Xin Z P. Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm Pure Appl Math, 1998, 51:229-240
[16] Xin Z P, Yan W. On Blowup of Classical Solutions to the Compressible Navier-Stokes Equations. Comm Math Phys, 2013, 32:529-541
[17] Guo Z H, Jiu Q S, Xin Z P. Spherically Symmetric Isentropic Compressible Flows with Density-Dependent Viscosity Coefficients. SIAM J Math Anal, 2008, 39:1402-1427
[18] Guo Z H, Li H L, Xin Z P. Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations. Comm Math Phys, 2012, 309:371-412
[19] Chen P, Fang D Y, Zhang T. Free boundary problem for compressible flows with density-dependent viscosity coefficients. Commun Pure Appl Anal, 2011, 10:459-478
[20] Perepelitsa M. On the Global Existence of Weak Solutions for the Navier-Stokes Equations of Compressible Fluid Flows. SIAM J Math Anal, 2006, 38:1126-1153
[21] Jiu Q S, Wang Y, Xin Z P. Global Well-Posedness of 2D Compressible Navier-Stokes Equations with Large Data and Vacuum. J Math Fluid Mech, 2014, 16:483-521
[22] Huang X D, Li J. Existence and Blowup Behavior of Global Strong Solutions to the Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data. arXiv:1205.5342v2, 2012
[23] Jiu Q S, Wang Y, Xin Z P. Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces. J Differential Equations, 2013, 255:351-404
[24] Huang X D, Li J. Global Well-Posedness of Classical Solutions to the Cauchy problem of Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data. arXiv:1207.3746v3, 2013
[25] Ye Y L. Global classical solution to the Cauchy problem of the 3-D compressible Navier-Stokes equations with density-dependent viscosity. Acta Mathematica Scientia, 2016, 36B(5):1419-1432
[26] Hoff D, Jenssen H K. Symmetric nonbarotropic flows with large data and forces. Arch Ration Mech Anal, 2004, 173:297-343
[27] Guo Z H, Wang M, Wang Y. Global solution to 3D spherically symmetric compressibile Navier-Stokes equations with large data.Nonlinear Analysis:Real World Applications, 2018, 40:260-289
[28] Wang M, Li Z L, Guo Z H. Global Weak Solution to 3D Compressible Flows with Density-dependent Viscosity and free Boundary. Commun Pure Appl Anal. doi:10.3934/cpaa.2017001
[29] Jiu Q S, Wang Y, Xin Z P. Global classical solutions to the two-dimensional compressible Navier-Stokes equations in R2. arXiv:1209.0157V1, 2012
[30] DelPino M, Dolbeault J. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J Math Pures Appl, 2002, 81:847-875
