In this paper, we construct a class of analytical solutions to the one-dimensional compressible isothermal Navier–Stokes equations with density-dependent viscosity in the real line . Precisely, we take the pressure p(ρ) = a1ρ and the viscosity coefficient μ(ρ) = a2ρ with a1, a2 > 0. We show that the system has an exact solution with the initial data satisfying ρ0(x) = ex and u0(x) = x. The large-time asymptotic behavior of the density is exhibited according to various a1. The analytical solutions to the compressible isothermal Euler equations and the pressureless Euler equations are obtained as by-products.
REFERENCES
1.
F.
Cavalletti
, M.
Sedjro
, and M.
Westdickenberg
, “A simple proof of global existence for the 1D pressureless gas dynamics equations
,” SIAM J. Math. Anal.
47
(1
), 66
–79
(2015
).2.
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. Methods Appl. Sci.
33
, 103
–116
(2009
).3.
M.
Gugat
and S.
Ulbrich
, “The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up
,” J. Math. Anal. Appl.
454
, 439
–452
(2017
).4.
Z.
Guo
, S.
Jiang
, and F.
Xie
, “Global weak solutions and asymptotic behavior to 1D compressible Navier–Stokes equations with degenerate viscosity coefficient and discontinuities initial density
,” Asymptotic Anal.
60
, 101
–123
(2008
).5.
Z.
Guo
and Z.
Xin
, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries
,” J. Differ. Equations
253
, 1
–19
(2012
).6.
Z.
Guo
and C.
Zhu
, “Global weak solutions and asymptotic behavior to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum
,” J. Differ. Equations
248
, 2768
–2799
(2010
).7.
Z. H.
Guo
and C. J.
Zhu
, “Remarks on one-dimensional compressible Navier-Stokes equations with density-dependent viscosity and vacuum
,” Acta Math. Sin.
26
, 2015
–2030
(2010
).8.
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
).9.
S.
Jiang
, Z.
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
).10.
Q.
Jiu
, M.
Li
, and Y.
Ye
, “Global classical solution of the Cauchy problem to 1D compressible Navier–Stokes equations with large initial data
,” J. Differ. Equations
257
, 311
–350
(2014
).11.
Q.
Jiu
, Y.
Wang
, and Z.
Xin
, “Remarks on blow-up of smooth solutions to the compressible fluid with constant and degenerate viscosities
,” J. Differ. Equations
259
, 2981
–3003
(2015
).12.
H.-L.
Li
, J.
Li
, and Z.
Xin
, “Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
,” Commun. Math. Phys.
281
, 401
–444
(2008
).13.
H.-L.
Li
, Y.
Wang
, and Z.
Xin
, “Non-existence of classical solutions with finite energy to the Cauchy problem of the compressible Navier–Stokes equations
,” Arch. Ration. Mech. Anal.
232
, 557
–590
(2019
).14.
W. M.
Li
, X. J.
Liu
, and Q. S.
Jiu
, “The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients
,” Commun. Pure Appl. Anal.
12
(2
), 647
–661
(2013
).15.
T.-P.
Liu
, Z. P.
Xin
, Z.
Xin
, and T.
Yang
, “Vacuum states for compressible flow
,” Discrete Contin. Dyn. Syst.
4
, 1
–32
(1998
).16.
B.
Lü
, Y.
Wang
, and Y.
Wu
, “On global classical solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum
,” Math. Methods Appl. Sci.
43
(8
), 5127
–5150
(2020
).17.
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
).18.
Y.
Qin
, L.
Huang
, and Z.-a.
Yao
, “A remark on regularity of 1D compressible isentropic Navier–Stokes equations with density-dependent viscosity
,” J. Math. Anal. Appl.
351
, 497
–508
(2009
).19.
S.-W.
Vong
, T.
Yang
, and C.
Zhu
, “Compressible Navier–Stokes equations with degenerate viscosity coefficient and vacuum (II)
,” J. Differ. Equations
192
(2
), 475
–501
(2003
).20.
H.
Wen
and L.
Yao
, “Global existence of strong solutions of the Navier–Stokes equations for isentropic compressible fluids with density-dependent viscosity
,” J. Math. Anal. Appl.
349
, 503
–515
(2009
).21.
Z.
Xin
, “Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density
,” Commun. Pure Appl. Math.
51
, 229
–240
(1998
).22.
Z.
Xin
and W.
Yan
, “On blowup of classical solutions to the compressible Navier-Stokes equations
,” Commun. Math. Phys.
321
, 529
–541
(2013
).23.
T.
Yang
, Z. A.
Yao
, and C. J.
Zhu
, “Compressible Navier-Stokes equations with density-dependent viscosity and vacuum
,” Commun. Partial Differ. Equations
26
(5–6
), 965
–981
(2001
).24.
T.
Yang
and H.
Zhao
, “A vacuum problem for the one-dimensional compressible Navier–Stokes equations with density-dependent viscosity
,” J. Differ. Equations
184
(1
), 163
–184
(2002
).25.
T.
Yang
and C. J.
Zhu
, “Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum
,” Commun. Math. Phys.
230
(2
), 329
–363
(2002
).26.
L. H.
Yeung
and Y.
Manwai
, “Analytical solutions to the Navier–Stokes equations with density-dependent viscosity and with pressure
,” J. Math. Phys.
50
, 083101
(2009
).27.
M. W.
Yuen
, “Analytical solutions to the Navier-Stokes equations
,” J. Math. Phys.
49
(11
), 113102
(2008
).28.
M. W.
Yuen
, “Self-similar solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations in RN
,” Commun. Nonlinear Sci. Numer. Simul.
17
, 4524
–4528
(2012
).29.
M.
Yuen
, “Vortical and self-similar flows of 2D compressible Euler equations
,” Commun. Nonlinear Sci. Numer. Simul.
19
, 2172
–2180
(2014
).30.
M. W.
Yuen
, “Rotational and self-similar solutions for the compressible Euler equations in R3
,” Commun. Nonlinear Sci. Numer. Simul.
20
, 634
–640
(2015
).31.
T.
Zhang
and D.
Fang
, “Remark on compressible Navier–Stokes equations with density-dependent viscosity and discontinuous initial data
,” J. Math. Anal. Appl.
339
, 1413
–1424
(2008
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
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