This article is the continued version of the analytical solutions for the pressureless Navier–Stokes equations with density-dependent viscosity [Yuen, M. W., “Analytical solutions to the Navier-Stokes equations,” J. Math. Phys.49, 113102 (2008)]. We are able to extend the similar solution structure to the case with pressure under some restriction to the constants γ and θ.

1.
Chen
,
G. Q.
and
Wang
,
D. H.
,
Handbook of Mathematical Fluid Dynamics
(
North-Holland
,
Amsterdam
,
2002
), Vol.
I
, pp.
421
543
.
2.
Deng
,
Y. B.
,
Xiang
,
J. L.
, and
Yang
,
T.
, “
Blowup phenomena of solutions to Euler-Poisson equations
,”
J. Math. Anal. Appl.
286
,
295
(
2003
).
3.
Landau
,
L. D.
and
Lifshitz
,
E. M.
,
Fluid Mechanics
(
Pergamon Press
,
London
,
1959
) [translated from the Russian by
Sykes
,
J. B.
and
Reid
,
W. H.
,
Course of Theoretical Physics
Vol. 6].
4.
Lions
,
P. L.
,
Mathematical Topics in Fluid Mechanics
(
Clarendon
,
Oxford
,
1998
), Vols.
1
and 2.
5.
Li
,
T. H.
, “
Some special solutions of the multidimensional Euler equations in RN
,”
Commun. Pure Appl. Anal.
4
,
757
(
2005
).
6.
Makino
,
T.
, “
Blowing up solutions of the Euler-Poission equation for the evolution of the gaseous stars
,”
Trans. Theory Stat. Phys.
21
,
615
(
1992
).
7.
Mellet
,
A.
and
Vasseur
,
A.
, “
On the barotropic compressible Navier-Stokes equations
,”
Commun. Partial Differ. Equ.
32
,
431
(
2007
).
8.
Nishida
,
T.
, “
Equations of fluid dynamics-free surface problems
,”
Commun. Pure Appl. Math.
39
,
221
(
1986
).
9.
Truesdell
,
C.
and
Noll
,
W.
,
The Nonlinear Flied Theories of Mechanics
, 2nd ed. (
Springer
,
New York
,
1992
).
10.
Yang
,
T.
,
Yao
,
Z. A.
, and
Zhu
,
C. J.
, “
Compressible Navier-Stokes equations with density-dependent viscosity and vacuum
,”
Commun. Partial Differ. Equ.
26
,
965
(
2001
).
11.
Yuen
,
M. W.
, “
Blowup solutions for a class of fluid dynamical equations in RN
,”
J. Math. Anal. Appl.
329
,
1064
(
2007
).
12.
Yuen
,
M. W.
, “
Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars
,”
J. Math. Anal. Appl.
341
,
445
(
2008
).
13.
Yuen
,
M. W.
, “
Analytical solutions to the Navier-Stokes equations
,”
J. Math. Phys.
49
,
113102
(
2008
).
You do not currently have access to this content.