One-dimensional Compressible Navier-Stokes Equations with ...

The problem Former results Main problem and difficulties Main results: µ depends only on v Main results: µ depends on θ Outline of the proof of Theorem 1 Outline of the proof of Theorem 2 Outline of the proof of Theorem 3 Some remarks

One-dimensional CompressibleNavier-Stokes Equations with Degenerate

Density and Temperature DependentTransport Coefficients

Huijiang Zhao

School of Mathematics and Statistics, Wuhan University

2012 International Conference on Nonlinear Analysis:Evolutionary P.D.E. and Kinetic Theory

Institute of Mathematics, Academia Sinica, TaipeiOctober 29-November 2, 2012


Outline

1 The problem

2 Former results

3 Main problem and difficulties

4 Main results: µ depends only on v

5 Main results: µ depends on θ

6 Outline of the proof of Theorem 1



9 Some remarks


1. The problem1d compressible Navier-Stokes equation in the Lagrangian coordinate:

vt − ux = 0,

ut + p(v , θ)x =(µ(v,θ)ux

v

)x,

et + p(v , θ)ux =µ(v,θ)u2

xv +

(κ(v,θ)θx

v

)x

(1)

Here

• v = 1ρ

: specific volume;

• θ : absolute temperature;

• e : internal energy;

• p : pressure;

• s : entropy;

• u : velocity;

• µ(v , θ) > 0 for v > 0, θ > 0: viscosity coefficient;

• κ(v , θ) > 0 for v > 0, θ > 0: heat conductivity coefficient.


• The thermodynamic variables v ,θ, p, s, and e are related by Gibbsequation

de = θds − pdv . (2)

Throughout this talk, we are concentrated on the ideal, polytropic gases:

p(v , θ) =Rθv

= Av−γ exp(γ − 1

Rs), e = Cvθ =

Rθγ − 1

. (3)

Here

• R : the specific gas constant;• Cv = R

γ−1 : the specific heat at constant volume;• γ > 1 : adiabatic exponent.

• Our main concern: Construction of global solutions with large dataaway from vacuum to the Cauchy problem of the one-dimensionalcompressible Navier-Stokes equation (1) with

• µ ≡ µ(v), κ ≡ κ(v , θ) are degenerate functions of v and/or θ;• µ ≡ µ(θ), κ ≡ κ(θ) are degenerate functions of θ.


Remarks on the model:

• Certain class of solid-like materials: 1 viscosity depends on density, heatconductivity depends on density and temperature;

• Experimental results for gases at very high temperatures: 2 viscosityand heat conductivity may depends on density and temperature

• For compressible Navier-Stokes equations derived from Boltzmannsystem by using the Chapman-Enskog expansion: 3 the transportcoefficients µ, ν, κ depend on temperature and ν = − 2

3µ for themonatomic gas. If the inter-molecular potential is proportional to r−α

with α > 1, where r represents the intermolecular distance, then µ, νand κ satisfy:

µ, −ν, κ ∝ θα+42α =

{θ, for the Maxwellian molecule (α = 4)√θ, for the elastic spheres (α→ +∞).

1Dafermos, C. M., SIAM J. Math. Anal. 13 (1982), 397-4082Zel’dovich, Y. B. and Raizer, Y. P., Physics of Shock Waves and High Temperature

Hydrodynamic Phenomena, Vol. II. Academic Press, New York, 19673Vincenti, W. G. and Kruger, C. H., Introduction to Physical Gas Dynamics.

Cambridge Math. Lib., Krieger, Malabar, FL, 1975.


2. Former results• µ is a positive constant or depends only on v , κ is non-degenerate

• Thermoviscoelasticity (some special constitutive relations)• Dafermos, C. M., SIAM J. Math. Anal. 13 (1982), 397-408: µ = µ(v),κ = κ(v , θ) but uniformly bounded from below and above

• Real-gas effects that occur in high-temperature regimes (somespecial constitutive relations)• Kawohl, B., Journal of Differential Equations 58 (1985), 76-103:µ = µ(v), κ = κ(v , θ) but both of them should be uniformly boundedaway from zero

• Navier-Stokes equations for ideal polytropic gas• Kazhikhov, A. V. and Shelukhin,V. V., J. Appl. Math. Mech. 41 (1977),

no. 2, 273-282: µ = const ., κ = const .: Non-vacuum smoothsolutions

µ is a positive constant, κ is a degenerate function of θ

• Jenssen, H. K. and Karper, T. K., SIAM J. Math. Anal. 42(2010),904-930: µ = const ., κ = θb, b ∈ [0, 3

2 ): Non-vacuum, weaksolutions

• Pan, R. H., private communications: µ = const ., κ = θb,non-vacuum smooth solution for b ≥ 0


3. Main problem and difficulties• Main problem: Construction of global non-vacuum smooth solutions for

• µ ≡ µ(v), κ ≡ κ(v , θ) are degenerate functions of their arguments;• µ ≡ µ(θ), κ ≡ κ(θ) are degenerate functions of θ

• Main difficulties:

• Unlike the small perturbation results, such dependence has stronginfluence on the solution behavior and thus leads to difficulties inanalysis not for the case of constant coefficients

• Key estimates: Lower and upper bounds for v and θ• For the case when µ depends only on v , the ideas in former results

are first to deduce the lower and upper bounds on v based on thespecial constitutive relations and the assumptions imposed on µand/or κ. Since the maximum principle can deduce a lower boundon θ, the remaining problem is to deduce an upper bound on θ. Forthe case of degenerate dependence of both µ and κ on v and/or θ,these arguments can not be applied any longer

• For the case when the viscosity depends on θ, no result is availableup to now


4. Main results: µ depends only on vConsider the Navier-Stokes equations (1) with initial data

(v , u, θ)(0, x) = (v0(x), u0(x), θ0(x))→ (1, 0, 1) as |x | → +∞. (4)

Ifµ = const ., κ = κ(v , θ), (5)

κ(v , θ) > 0 for v > 0, θ > 0, infv≥V̄ ,θ≥Θ̄

κ(v , θ) ≥ C(V̄ , Θ̄) > 0 (6)

then we haveTheorem 1 In addition to the assumptions (5), (6), we assume further that

•(v0(x)− 1, u0(x), θ0(x)− 1) ∈ H1(R); (7)

• There exists positive constants Θ0 > 0, V0 > 0 such that

Θ−10 ≤ θ0(x) ≤ Θ0, V−1

0 ≤ v0(x) ≤ V0. (8)

Then (1), (4) admits a unique global solution (v , u, θ)(t , x) satisfying

Θ−11 ≤ θ(t , x) ≤ Θ1, V−1

1 ≤ v(t , x) ≤ V1, ∀(t , x) ∈ [0,T ]× R (9)

for any given T > 0 and some constants Θ1, V1 depending on T .


Our second result is concerned with the case

µ(v) = v−a, κ(v , θ) = θb, (10)

which is stated as follows.Theorem 2. Suppose

• The assumptions (7) and (8) hold;

• 13 < a < 1

2 ;

• b satisfies one of the following conditions

(i) 1 ≤ b < 2a1−a ,

(ii) 0 < b < 1, 2−b2 + (a2−a+2)(1−b)

(1−2a)(3a−1)< 1, (1−b)(3+a−2a2)

(3a−1)(1−2a)< 1.

Then the Cauchy problem (1), (4) with µ(v) and κ(v , θ) given by (10) admitsa unique global solution (v(t , x), u(t , x), θ(t , x)) satisfying(

v(t , x)− 1, u(t , x), θ(t , x)− 1)∈ C0 (0,T ; H1(R)

),(

ux (t , x), θx (t , x))∈ L2 (0,T ; H2(R)

),

V−12 ≤ v(t , x) ≤ V2, Θ−1

2 ≤ θ(t , x) ≤ Θ2, ∀(t , x) ∈ [0,T ]× R.

(11)

Here T > 0 is any given positive constant and V2,Θ2 are some positive

constants which may depend on T .


5. Main results: µ depends on θFor the case when µ and κ are degenerate functions of θ satisfying

µ(θ) > 0, κ(θ) > 0 ∀θ > 0, (12)

we need to take (v , u, s) as the unknown function and let

s =R

γ − 1ln

RA

be the far field of the initial entropy s0(x), that is,

lim|x|→+∞

s0(x) = lim|x|→+∞

Rγ − 1

lnRθ0(x)v0(x)γ−1

A= s :=

Rγ − 1

lnRA.

Then we haveTheorem 3. Suppose

• N0 := ‖(v0(x)− 1, u0(x), s0(x)− s)‖H3(R) is bounded by some positiveconstant independent of γ − 1 and (8) holds for someγ − 1−independent positive constants V0, Θ0;

• γ − 1 is sufficiently small.


Then the Cauchy problem (1), (4) admits a unique global solution(v(t , x), u(t , x), θ(t , x)) satisfying (9) for some time-independent V1 and Θ1

and the following large time behavior

limt→∞

supx∈R

∣∣∣ (v(t , x)− 1, u(t , x), θ(t , x)− 1, s(t , x)− s)∣∣∣ = 0 (13)

holds.Moreover, there exists a function C(N0) satisfying C(0) = 0 such that∥∥∥∥∥(

v − 1, u,θ − 1√γ − 1

)∥∥∥∥∥2

H3(R)

+

∫ t

0

(‖vx‖2

H2(R) + ‖(ux , θx )‖2H3(R)

)dτ ≤ c(N0).

(14)Several remarks concerning Theorem 3 are listed below:

• Theorem 3 is somewhat a Nishida-Smoller type4 result;

• Even when µ and κ are functions of v and θ satisfyingµ(v , θ) > 0, κ(v , θ) > 0 for v > 0, θ > 0, similar result still holds.

4Nishida, T. and Smoller, J. A., Comm. Pure Appl. Math. 26 (1973), 183-200


6. Outline of the proof of Theorem 1

• By the argument developed by Kazhikhov and Shelukhin5 forµ = const ., κ = const , an explicit formula for v(t , x) can bederived and based on this, the lower bound for v(t , x) can beobtained;

• The maximum principle together with the equation for θ can yieldthe lower bound on θ(t , x);

• The lower bounders on v and θ can deduce an upper bound onv(t , x) provided that κ(v , θ) satisfies

infv≥V̄ ,θ≥Θ̄

κ(v , θ) ≥ C(V̄ , Θ̄) > 0;

• The upper bound on θ(t , x)

5Kazhikhov, A. V. and Shelukhin,V. V., J. Appl. Math. Mech. 41 (1977), no. 2,273-282


7. Outline of the proof of Theorem 2• Local solvability: well-established• Basic energy estimates: Note that

η(v ,u, θ) = Rφ(v) +u2

2+

Rφ(θ)

γ − 1, with φ(x) = x − ln x − 1,

is a convex entropy to (1) which satisfies

η(v ,u, θ)t +

{(Rθv− R

)u}

x−{

uux

v1+a +(θ − 1)θx

vθ1−b

}x

(15)

+

{u2

x

v1+aθ+

θ2x

vθ2−b

}= 0,

we have∫Rη(v ,u, θ)(t , x)dx +

∫ t

0

∫R

(u2

x

v1+aθ+

θ2x

vθ2−b

)(τ, x)dxdτ

=

∫Rη(v0,u0, θ0)(x)dx . (16)


• Lower bound estimates on θ(t , x):

1θ(t , x)

≤ O(1) + O(1)

∥∥∥∥1v

∥∥∥∥1−a

L∞T ,x

, x ∈ R, 0 ≤ t ≤ T . (17)

Cv

(1θ

)t

= − u2x

θ2v1+a +Rux

vθ− 2θ1+b

v

[(1θ

)x

]2

+

[(θb

v

)(1θ

)x

]x

= −

{2θ1+b

v

[(1θ

)x

]2

+1

v1+aθ2

(ux −

Rθva

2

)2}

(18)

+R2

4v1−a +

[(θb

v

)(1θ

)x

]x

.

• Estimate on∥∥ vx

v1+a

∥∥:∥∥∥ vx

v1+a

∥∥∥2+

∫ t

0

∫R

θv2x

v3+a dxdτ (19)

≤ O(1) ‖(v0 − 1, u0, θ0 − 1)‖2 + O(1)

(∥∥∥∥1v

∥∥∥∥a

L∞T ,x

+

∥∥∥∥1v

∥∥∥∥1−a

L∞T ,x

)‖θ1−b‖L∞T ,x

.


• Estimate on ‖u‖:

‖u(t)‖2 +

∫ t

0

∫R

u2x

v1+a dxdτ

≤ O(1) ‖(v0 − 1, u0, θ0 − 1)‖2 + O(1)

∥∥∥∥1v

∥∥∥∥a

L∞T ,x

‖θ1−b‖L∞T ,x, (20)

• Estimate on v(t , x) in terms of ‖θ1−b‖L∞T ,x: To this end, set

Ψ(v) =

∫ v

1

√φ(z)

z1+a dz. (21)

Note that there exist positive constants A2,A3 such that

Ψ(v) ≥ A2

(v−a + v

12−a)− A3. (22)

Since

Ψ(v) =

∫ x

−∞Ψ(v(t , y))y dy

≤∫

R

∣∣∣∣∣√φ(v)

v1+a vx

∣∣∣∣∣ dx

≤ O(1)

(1 +

(∥∥∥∥1v

∥∥∥∥ a2

L∞T ,x

+

∥∥∥∥1v

∥∥∥∥ 1−a2

L∞T ,x

)‖θ1−b‖

12L∞T ,x

),


we have from (16), (19), and (22) that∥∥∥∥1v

∥∥∥∥a

L∞T ,x

+ ‖v‖12−aL∞T ,x≤ O(1)

(1 +

(∥∥∥∥1v

∥∥∥∥ a2

L∞T ,x

+

∥∥∥∥1v

∥∥∥∥ 1−a2

L∞T ,x

)‖θ1−b‖

12L∞T ,x

).

(23)Thus if 1

3 < a < 12 , we can deduce from (24) that

1v(t , x)

≤ O(1)

(1 + ‖θ1−b‖

13a−1L∞T ,x

)(24)

and

v(t , x) ≤ O(1)

(1 + ‖θ1−b‖

2a(3a−1)(1−2a)

L∞T ,x

)(25)

hold for any (t , x) ∈ [0,T ]× R.


• Estimates on ‖u‖ and∥∥ vx

v1+a

∥∥ revised: (19), (20), (24), and (25)imply

‖u(t)‖2 +

∫ t

0

∫R

u2x

v1+a dxdτ ≤ O(1)

(1 + ‖θ1−b‖

2a3a−1L∞T ,x

), (26)

∥∥∥ vx

v1+a

∥∥∥2+

∫ t

0

∫R

θv2x

v3+a dxdτ ≤ O(1)

(1 + ‖θ1−b‖

2a3a−1L∞T ,x

). (27)

• Estimates on ‖ux‖:

‖ux (t)‖2 +

∫ t

0

∫R

u2xx

v1+a dxdτ

≤ O(1) ‖(v0 − 1,u0, θ0 − 1)‖2

+O(1)∥∥θ2−b

∥∥L∞T ,x

(1 + ‖θ1−b‖

2a2(3a−1)(1−2a)

L∞T ,x

)+O(1)

(1 + ‖θ1−b‖

2(2a−2a2+1)(3a−1)(1−2a)

L∞T ,x

). (28)


• The upper bound on θ(t , x):

‖θ‖L∞T ,x≤ O(1)

{1 +

∫ t

0

(∥∥∥∥ u2x

v1+a

∥∥∥∥L∞x

+

∥∥∥∥u2x

v2

∥∥∥∥L∞x

+ ‖θ‖2L∞x

)dτ

}. (29)

(29) together with∫ t

0‖ux (s)‖2

L∞x ds

≤ O(1)∥∥∥θ2−b

∥∥∥ 12

L∞T ,x

(1 + ‖θ1−b‖

3a+a2(3a−1)(1−2a)

L∞T ,x

)+ ‖θ1−b‖

5a−2a2+1(3a−1)(1−2a)

L∞T ,x(30)

imply

‖θ‖L∞T ,x≤ O(1) + O(1)

∥∥∥θ2−b∥∥∥ 1

2

L∞T ,x

(1 +

∥∥∥θ1−b∥∥∥ a2−a+2

(3a−1)(1−2a)

L∞T ,x

)(31)

+O(1)∥∥∥θ1−b

∥∥∥ 3+a−2a2(3a−1)(1−2a)

L∞T ,x.

Thus if a and b satisfy certain conditions, we can deduce the desired upper

and lower bounds on v(t , x) and θ(t , x).


8. Outline of the proof of Theorem 3• Motivated by the arguments used by Kawashima and Nishida6 and

Nishihara, Yang, and Zhao7 for constant transport coefficients;

• A key identity(µ(v , θ)vx

v

)t

= ut +

(Rθv

)x

+µθ(v , θ)

v(vxθt − uxθx ) (32)

• Energy type estimates based on the following a priori assumption‖θ(t , x)− 1‖H3(R) ≤ ε, ∀(t , x) ∈ [0,T ]× R,0 < M−1

1 ≤ v(t , x) ≤ M1, ∀(t , x) ∈ [0,T ]× R,‖(v(t , x)− 1, u(t , x))‖H3(R) ≤ N1, ∀(t , x) ∈ [0,T ]× R

(33)

• Main purpose is to deduce some a priori estimates independent ofM1 and N1

• To use the smallness of γ − 1 and ε to control the possible growthof the solutions caused by the nonlinearities of the Navier-Stokesequations

6Kawashima, S. and Nishida, T., J. Math. Kyoto Univ. 21 (1981), 825-8377Nishihara, K., Yang, T., and Zhao, H.-J., SIAM J. Math. Anal. 35 (2004), 1561-1597


• Main steps in performing the a priori estimates• The smallness of ε implies that 1

2 ≤ θ(t , x) ≤ 2, ∀(t , x) ∈ [0,T ]× R• Basic entropy estimate: Let φ(x) = x − ln x − 1∥∥∥∥∥

(√φ(v), u,

θ − 1√γ − 1

)∥∥∥∥∥2

+

∫ t

0

∫R

u2x + θ2

x

vdxdτ ≤ C (N0,V0,Θ0)

(34)• If

εM1N1 ≤ 1,(γ − 1)

(M4

1 + N41)≤ 1,

(γ − 1)M1N21 � 1,

(35)

then we have∥∥∥vx

v

∥∥∥2+

∫ t

0

∥∥∥ vx

v3/2

∥∥∥2dτ (36)

≤ C (N0,V0,Θ0) + C (V0,Θ0) (γ − 1)

∫ t

0

∥∥∥∥ θxx

v1/2

∥∥∥∥2

dτ

• Under the assumption (35), we have∥∥∥∥∥ θx√γ − 1

∥∥∥∥∥2

+

∫ t

0

∥∥∥∥ θxx

v1/2

∥∥∥∥2

dτ ≤ C (N0,V0,Θ0)(

M21 N2

1 + N21

)(37)


• Main steps in performing the a priori estimates (continued)• (36), (37) together with the assumption (35) imply∥∥∥vx

v

∥∥∥2+

∫ t

0

∥∥∥ vx

v3/2

∥∥∥2dτ ≤ C (N0,V0,Θ0) (38)

• (34), (38) together with Kanel’s argument8 to deduce that thereexists positive constant V1 depending only on N0,V0,Θ0 such that

0 < V−11 ≤ v(t , x) ≤ V1, ∀(t , x) ∈ [0,T ]× R (39)

• Under the assumption (35), we have the following first order energytype estimates∥∥∥∥∥(

vx , ux ,θx√γ − 1

)∥∥∥∥∥2

+

∫ t

0‖(vx , uxx , θxx )‖2 dτ ≤ C (N0,V0,Θ0)

(40)

8Kanel’, Y., Differencial’nya Uravnenija 4 (1968), 374-380.


• Main steps in performing the a priori estimates (continued)• Second order energy type estimates on θ and u:

‖uxx‖2 +

∫ t

0‖uxxx‖2dτ

≤ C (N0,V0,Θ0)

(1 +

∫ t

0‖vxx‖2dτ

)+C (N0,V0,Θ0)

∫ t

0

∫R|(vx , ux , θx )|2|(uxx , θxx )|2dxdτ

+C (N0,V0,Θ0)

∫ t

0‖(vx , θx )‖2‖(vxx , θxx )‖2dτ (41)

∥∥∥∥∥ θxx√γ − 1

∥∥∥∥∥2

+

∫ t

0‖θxxx‖2dτ

≤ C (N0,V0,Θ0)

(1 +

∫ t

0‖θx‖2‖vxx‖2dτ

)(42)

+C (N0,V0,Θ0)

∫ t

0

∫R

[|(vx , θx )|2|θxx |2 + u2

x u2xx + θ2

x v2xx

]dxdτ


• Main steps in performing the a priori estimates (continued)• Key points to deduce the second order energy type estimate on v :

• We need to use the identity(µ(θ)vx

v

)xt

= uxt +

(Rθv

)xx

+

(µ′(θ)

v(vxθt − uxθx )

)x

(43)

• Since θt = (γ − 1)

(−Rθux

v +µ(θ)u2

xv +

(κ(θ)θx

v

)x

), we need to deal

with the term (γ − 1)∫ t

0

∫Rµ(θ)µ′(θ)κ(θ)

v3 θxx v2xx dxdτ . It is easy to see

that to bound such a term, we need to close the energy typeestimates in H3(R)

• Fortunately the term∫ t

0

∫R v2

x v2xx dxdτ does not appear

• Second order energy type estimate on v :

‖vxx‖2 +

∫ t

0‖vxx‖2dτ

≤ C (N0,V0,Θ0)

(1 +

∫ t

0‖(vx , θx )‖2‖(vxx , θxx )‖2dτ

)+(γ − 1)

∫ t

0‖θxxx‖2dτ (44)

+C (N0,V0,Θ0)

∫ t

0

∫R

[|(ux , θx )|2|(uxx , vxx , θxx )|2 + v2

x θ2xx

]dxdτ


• Main steps in performing the a priori estimates (continued)• A suitable linear combination of (41), (42), (44) together with the

fact that the term∫ t

0

∫R v2

x v2xx dxdτ does not appear yield the

following second order energy type estimates on (v , u, θ)∥∥∥∥∥(

vxx , uxx ,θxx√γ − 1

)∥∥∥∥∥2

+

∫ t

0‖(vxx , uxxx , θxxx )‖2dτ

≤ C (N0,V0,Θ0) (45)

• Third order energy type estimates

‖vxxx‖2 +

∫ t

0‖vxxx‖2dτ (46)

≤ C (N0,V0,Θ0)

(1 + (γ − 1)2

∫ t

0‖θxxxx‖2dτ

)

‖uxxx‖2 +

∫ t

0‖uxxxx‖2dτ (47)

≤ C (N0,V0,Θ0)

(1 +

∫ t

0‖vxxx‖2dτ

)


• Main steps in performing the a priori estimates (continued)• Third order energy type estimates (continued)∥∥∥∥∥ θxxx√

γ − 1

∥∥∥∥∥2

+

∫ t

0‖θxxxx‖2dτ (48)

≤ C (N0,V0,Θ0)

(1 +

∫ t

0

∫Rθ2

x V 2xxx dxdτ

)• A suitable linear combination of (46), (47), (48) yields∥∥∥∥∥

(vxxx , uxxx ,

θxxx√γ − 1

)∥∥∥∥∥2

+

∫ t

0‖(vxxx , uxxxx , θxxxx )‖2dτ (49)

≤ C (N0,V0,Θ0)


• In summary, if the local solution (v ,u, θ) has been extended tothe time step t = T and satisfies the a priori assumption (33),then if ε and γ − 1 are sufficiently small such that (35) holds, thenthere exists positive constant C > 0 depending only onN0,V0,Θ0 such that∥∥∥∥(v − 1,u,

θ − 1√γ − 1

)∥∥∥∥2

H3(R)

(50)

+

∫ t

0

(‖vx‖2

H2(R) + ‖(ux , θx )‖2H3(R)

)dτ

≤ C

• The a priori estimate (50) together with the continuationargument can lead to Theorem 3.


9. Some remarks• The arguments used here can also be used to treat the one-dimensional

Navier-Stokes-Poisson system9 and the one-dimensionalMagnetohydrodynamics system10

• One-dimensional initial-boundary value problem11

• u(0, t) = u(1, t) = 0, θx (0, t) = θx (1, t) = 0: Results similar toTheorems 1 and 3 holds;

• σ(0, t) = σ(1, t) = −Q(t) < 0, θx (0, t) = θx (1, t) = 0 withσ(v , θ, ux ) := −p(v , θ) + µ(v)ux

v . Here the outer pressure Q(t) ∈ C1

is a given function: µ(v) = v−a, κ(θ) = θb with 0 ≤ a < 12 , b ≥ 1

2 ;• σ(0, t) = σ(1, t) = 0, θx (0, t) = θx (1, t) = 0:µ(v) = v−a (0 ≤ a < 1

5 ), κ(θ) = θb (b ≥ 2).

• Even for the case when µ and κ are positive constants, the globalexistence results available up to now for the Cauchy problem focus onthe ideal, polytropic gas. How to deduce the corresponding result forgeneral gas remains open!

9Tan, Z., Yang, T., Zhao, H.-J., Zou, Q.-Y., preprint 201210Wang, T., Xiong, L.-J., and Zhao, H.-J., work in progress11Chen, Q., Zhao, H.-J., and Zou, Q.-Y., preprint 2012


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