In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = ρα(α > 0) and pressure P(ρ) = ργ (γ > 1). We will establish the global existence and asymptotic behavior of weak solutions for any α > 0 and γ > 1 under the assumption that the density function keeps a constant state at far fields. In particular, in the case that $0<\alpha <\frac{1}{2}$, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).
REFERENCES
1.
D.
Bresch
and B.
Desjardins
, “Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model
,” Commun. Math. Phys.
238
(1–2
), 211
–223
(2003
).2.
D.
Bresch
and B.
Desjardins
, “Quelques modeles diffusifs capillaires de type Korteweg
,” C. R. Acad. Sci. Paris Section Mécanique
332
(11
), 881
–886
(2004
).3.
D.
Bresch
and B.
Desjardins
, “On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models
,” J. Math. Pures Appl.
86
, 362
–368
(2006
).4.
D.
Bresch
, B.
Desjardins
, and C.-K.
Lin
, “On some compressible fluid models: Korteweg, lubrication, and shallow water systems
,” Commun. Partial Differ. Equ.
28
(3–4
), 843
–868
(2003
).5.
D.
Bresch
, B.
Desjardins
, and D.
Gerard-Varet
, “On compressible Navier-Stokes equations with density dependent viscosities in bounded domains
,” J. Math. Pures Appl.
87
(2
), 227
–235
(2007
).6.
R.
Danchin
, “Global existence in critical spaces for compressible Navier-Stokes equations
,” Invent. Math.
141
, 579
–614
(2000
).7.
C. S.
Dou
and Q. S.
Jiu
, “A remark on free boundary problem of 1-D compressible Navier-Stokes equations with density-dependent viscosity
,” Math. Meth. Appl. Sci.
33
, 103
–116
(2010
).8.
E.
Feireisl
, A.
Novotný
, and H.
Petzeltová
, “On the existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids
,” J. Math. Fluid Mech.
3
, 358
–392
(2001
).9.
J. F.
Gerbeau
and B.
Perthame
, “Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation
,” Discrete Contin. Dyn. Syst., Ser. B
1
(1
), 89
–102
(2001
).10.
Z.
Guo
, Q.
Jiu
, and Z.
Xin
, “Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients
,” SIAM J. Math. Anal.
39
(5
), 1402
–1427
(2008
).11.
D.
Hoff
, “Global existence of 1D compressible isentropic Navier-Stokes equations with large initial data
,” Trans. Am. Math. Soc.
303
(1
), 169
–181
(1987
).12.
D.
Hoff
, “Strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data
,” Arch. Ration. Mech. Anal.
132
, 1
–14
(1995
).13.
D.
Hoff
, “The zero-Mach limit of compressible flows
,” Commun. Math. Phys.
192
, 543
–554
(1998
).14.
D.
Hoff
and D.
Serre
, “The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow
,” SIAM J. Appl. Math.
51
, 887
–898
(1991
).15.
D.
Hoff
and J.
Smoller
, “Non-formation of vacuum states for compressible Navier-Stokes equations
,” Commun. Math. Phys.
216
(2
), 255
–276
(2001
).16.
I.
Straškraba
and A.
Zlotnik
, “Global properties of solutions to 1D viscous compressible barotropic fluid equations with density dependent viscosity
,” Z. Angew. Math. Phys.
54
, 593
–607
(2003
).17.
S.
Jiang
, “Global smooth solutions of the equations of a viscous, heat-conducting one-dimensional gas with density-dependent viscosity
,” Math. Nachr.
190
, 169
–183
(1998
).18.
S.
Jiang
, Z. P.
Xin
, and P.
Zhang
, “Global weak solutions to 1D compressible isentropy Navier-Stokes with density-dependent viscosity
,” Methods Appl. Anal.
12
(3
), 239
–252
(2005
).19.
Q.
Jiu
and Z. P.
Xin
, “The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients
,” Kinet. Relat. Models
1
(2
), 313
–330
(2008
).20.
Q.
Jiu
, Y.
Wang
, and Z. P.
Xin
, “Stability of rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity
,” Commun. Partial Differ. Equ.
36
, 602
–634
(2011
).21.
Q.
Jiu
, Y.
Wang
, and Z. P.
Xin
, “Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity
,” preprint, arXiv:1109.0871.22.
J. I.
Kanel
, “A model system of equations for the one-dimensional motion of a gas
,” Differencial'nye Uravnenija
4
, 721
–734
(1968
) (in Russian).23.
A. V.
Kazhikhov
and V. V.
Shelukhin
, “Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas
,” J. Appl. Math. Mech.
41
, 273
–282
(1977
)A. V.
Kazhikhov
and V. V.
Shelukhin
, [Prikl. Mat. Mekh.
41
, 282
–291
(1977
) (in Russian)].24.
H. L.
Li
, J.
Li
, and Z. P.
Xin
, “Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
,” Commun. Math. Phys.
281
(2
), 401
–444
(2008
).25.
P. L.
Lions
, Mathematical Topics in Fluid Dynamics 2, Compressible Models
(Oxford Science Publication
, Oxford
, 1998
).26.
T. P.
Liu
, Z. P.
Xin
, and T.
Yang
, “Vacuum states of compressible flow
,” Discrete Contin. Dyn. Syst.
4
(1
), 1
–32
(1998
).27.
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
(1
), 67
–104
(1980
).28.
A.
Mellet
and A.
Vasseur
, “On the isentropic compressible Navier-Stokes equation
,” Commun. Partial Differ. Equ.
32
, 431
–452
(2007
).29.
A.
Mellet
and A.
Vasseur
, “Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations
,” SIAM J. Math. Anal.
39
(4
), 1344
–1365
(2008
).30.
J.
Nash
, “Le probleme de Cauchy pour Les equations differentielles d'un fluids general
,” Bull. Soc. Math. France
90
, 487
–497
(1962
).31.
M.
Okada
, S.
Matusu-Necasova
, and T.
Makino
, “Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity
,” Ann. Univ. Ferrara Sez. VII (N.S.)
48
, 1
–20
(2002
).32.
D.
Serre
, “Sur léquation monodimensionnelle dún fluide visqueux, compressible et conducteur de chaleur
,” C. R. Acad. Sci., Paris, Sér. I Math.
303
, 703
–706
(1986
).33.
V. A.
Vaigant
and A. V.
Kazhikhov
, “On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid
,” Siberian Math. J.
36
(6
), 1108
–1141
(1995
).34.
S. W.
Vong
, T.
Yang
, and C. J.
Zhu
, “Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum II
,” J. Differ. Equations
192
(2
), 475
–501
(2003
).35.
J.
Wei
, L.
He
, and Z.
Guo
, “A remark on the Cauchy problem of 1D compressible Navier-Stokes equations with density-dependent viscosity coefficients
,” Acta Math. Appl. Sin. English Ser.
(2011
).36.
Z. P.
Xin
, “Blow-up of smooth solution to the compressible Navier-Stokes equations with compact density
,” Commun. Pure Appl. Math.
51
, 229
–240
(1998
).37.
T.
Yang
, Z. A.
Yao
, and C. J.
Zhu
, “Compressible Navier-Stokes equations with density-dependent viscosity and vacuum
,” Commun. Partial Differ. Equ.
26
(5–6
), 965
–981
(2001
).38.
T.
Yang
and H. J.
Zhao
, “A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity
,” J. Differ. Equations
184
(1
), 163
–184
(2002
).39.
T.
Yang
and C. J.
Zhu
, “Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum
,” Commun. Math. Phys.
230
(2
), 329
–363
(2002
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.