Self-Similar Solutions of the Compressible Flow in One-Space Dimension

Abstract

For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Guo and Jiang's work (2006) where the one-dimensional compressible fluids with constant viscosity are considered.

1. Introduction

Self-similar solutions have attracted much attention in mathematical physics because understanding them is fundamental and important for investigating the well-posedness, regularity, and asymptotic behavior of differential equations in physics. Since the pioneering work of Leray [1], self-similar solutions of the Navier-Stokes equations for incompressible fluids have been widely studied in different settings (e.g., [2, page 207]; [3, page 120]; [4–10]; [11, Chapter 23]; [12–20]). On the contrary, studies on the self-similar solutions of the compressible Navier-Stokes equations have been limited partially due to the complicated nonlinearities in the equations (see [21–24]).

In one-space dimension, the isentropic compressible fluid flow is governed by the Navier-Stokes equations: 𝜌_𝑡+𝜌𝑢_𝑥=0,𝜌𝑢_𝑡+𝜌𝑢²_𝑥+𝑃𝜌_𝑥=𝜇𝜌𝑢_𝑥𝑥,(𝑥,𝑡)∈ℝ×ℝ₊, (1) where 𝜌=𝜌(𝑥,𝑡) and 𝑢=𝑢(𝑥,𝑡) are the density and velocity of the fluid, 𝜇(𝜌) and 𝑃(𝜌) denote the density-dependent viscosity and pressure, respectively, and the subscripts mean partial derivations. Guo and Jiang [21] considered (1) with constant viscosity, 𝜇(𝜌)≡𝜇>0, and linear density-dependent pressure, 𝑃(𝜌)=𝑎𝜌, where 𝑎>0 is a constant, and proved that there exist neither forward nor backward self-similar solutions with finite total energy. Their investigation generalized the results for 3D incompressible fluids in Nečas et al.’s work [6] to the 1D compressible case with 𝑃(𝜌)=𝜌^𝛾, where 𝛾=1. The problem with 𝛾>1, however, is open. From a physical point of view, one can derive the compressible Navier-Stokes equations from the Boltzmann equations by exploiting the Chapman-Enskog expansion up to the second order and then find that the viscosity depends on the temperature. If considering an isentropic process, this dependence can be translated into that on the density, such as 𝜇(𝜌)=𝜌^𝜃, where 𝜃>0 is a constant (see [25]). Okada et al. [26] pointed out that, because of the hard sphere interaction, the relation between indices 𝜃 and 𝛾 is 𝜃=(𝛾−1)/2. In the first part of this paper, we are concerned with (1) where 𝜇𝜌=𝜌^𝜃,𝑃𝜌=𝜌^𝛾,𝛾−12≤𝜃<𝛾+12,𝛾≥1. (2)

When considering an ideal compressible gas flow, particularly in the thermodynamic analysis with exergy loss and entropy generation, both the viscosity and pressure rely on the entropy, so it is necessary to extend the nonisentropic fluid dynamics to include the transport of entropy (see [13, 27–35]). We consider the following coupled system of the Navier-Stokes equations with an entropy transport equation in a pure form: 𝜌𝑠_𝑡+𝜌𝑢𝑠_𝑥𝜌=0,_𝑡+𝜌𝑢_𝑥=0,𝜌𝑢_𝑡+𝜌𝑢²_𝑥+𝑃𝜌,𝑠_𝑥=𝜇𝜌,𝑠𝑢_𝑥𝑥,(𝑥,𝑡)∈ℝ×ℝ₊, (3)(4)(5) where 𝑠=𝑠(𝑥,𝑡) is the entropy of the fluid and 𝜇(𝜌,𝑠) and 𝑃(𝜌,𝑠) denote the density-entropy-dependent viscosity and pressure, respectively. In this system, we assume that 𝜇𝜌,𝑠=𝜌^𝜃𝑒^𝑠,𝑃𝜌,𝑠=𝜌^𝛾𝑒^𝑠,0≤𝜃<𝛾+12,𝛾≥1. (6)

Navier-Stokes equations enjoy a scaling property: if (𝜌,𝑢) solves (1)-(2), then 𝜌^(𝜆),𝑢^(𝜆)=𝜆^𝑎𝜌𝜆^𝑐𝑥,𝜆^𝑑𝑡,𝜆^𝑏𝑢𝜆^𝑐𝑥,𝜆^𝑑𝑡 (7) does so for any 𝜆>0, by setting 𝑎=1/(𝛾−𝜃), 𝑏=(𝛾−1)/2(𝛾−𝜃), 𝑐=(𝛾+1−2𝜃)/2(𝛾−𝜃), and 𝑑=1. Note that, from (2), 𝑎≥𝑐>0 and 𝑏≥0. Solution (𝜌,𝑢) is called forward self-similar if 𝜌,𝑢=𝜌^(𝜆),𝑢^(𝜆),forevery𝜆>0. (8) In that case, 𝜌(𝑥,𝑡) and 𝑢(𝑥,𝑡) are decided by their values at the instant of 𝑡=1/𝜆: 𝜌()=1𝑥,𝑡𝑡^𝑎𝑄𝑥𝑡^𝑐()=1,𝑢𝑥,𝑡𝑡^𝑏𝑈𝑥𝑡^𝑐,𝑡>0, (9) where 𝑄(𝑦)=𝜌(𝑦,1) and 𝑈(𝑦)=𝑢(𝑦,1) are defined on ℝ. In the same manner, the backward self-similar solutions are of the form: 𝜌()=1𝑥,𝑡(𝑇−𝑡)^𝑎𝑄𝑥(𝑇−𝑡)^𝑐,𝑢()=1𝑥,𝑡(𝑇−𝑡)^𝑏𝑈𝑥(𝑇−𝑡)^𝑐,0<𝑡<𝑇, (10) where 𝑄(𝑦)=𝜌(𝑦,𝑇−1) and 𝑈(𝑦)=𝑢(𝑦,𝑇−1) for 𝑇>1. Substitution of (9) or (10) into (1) gives (𝑄𝑈)^′−𝑐𝑦𝑄^′+()𝑄𝑐−𝑎𝑄=0,_𝜃𝑈^′′−𝑄𝑈^2′+𝑐𝑦𝑄𝑈^′−𝑄^𝛾′+(𝑎+𝑏−𝑐)𝑄𝑈=0, (11)(12) for forward self-similar solutions, or (𝑄𝑈)^′+𝑐𝑦𝑄^′−()𝑄𝑐−𝑎𝑄=0,_𝜃𝑈^′′−𝑄𝑈^2′−𝑐𝑦𝑄𝑈^′−𝑄^𝛾′−(𝑎+𝑏−𝑐)𝑄𝑈=0, (13)(14) for backward self-similar solutions, respectively. In comparison with those in Guo and Jiang [21], forward (backward) self-similar equations above process necessary modifications and additional difficulties. For instance, (11) and (13) have solutions with an additional integral term, and thus the modified blow-up analysis needs an 𝐿^∞ estimate on the density and a new large-scale argument on the energy. In addition, conditions on 𝜃 and 𝛾 proposed in (2) are directly related to the energy estimate.

Mellet and Vasseur [25] obtained the global existence of strong solutions for the Cauchy problem of (1) with positive initial density having (possibly different) positive limits at 𝑥=±∞. Precisely, fix constant positive density 𝜌₊>0 and 𝜌₋>0, and let 𝜌(𝑥) be a smooth monotone function satisfying 𝜌(𝑥)=𝜌_±when±𝑥≥1,𝜌(𝑥)>0,∀𝑥∈ℝ. (15) Assume that the initial data 𝜌(𝑥,𝑡)∣_𝑡=0=𝜌₀(𝑥) and 𝑢(𝑥,𝑡)∣_𝑡=0=𝑢₀(𝑥) satisfy 0<𝜅₀≤𝜌₀(𝑥)≤𝜅₀𝜌<∞,₀−𝜌∈𝐻¹(ℝ),𝑢₀∈𝐻¹(ℝ), (16) for some constants 𝜅₀ and 𝜅₀. Assume also that 𝜇(𝜌) and 𝑃(𝜌) verify (2). Mellet and Vasseur [25] proved that there exists a global strong solution (𝜌,𝑢) of (1) on ℝ×ℝ₊ such that for every 𝑇>0: 𝜌−𝜌∈𝐿^∞0,𝑇;𝐻¹(ℝ),𝑢∈𝐿^∞0,𝑇;𝐻¹(ℝ)∩𝐿²0,𝑇;𝐻²(ℝ). (17) Moreover, for every 𝑇>0, there exist uniform bounds away from zero with respect to all strong solutions having the same initial data. Precisely, there exist some constants 𝐶(𝑇), 𝜅(𝑇), and 𝜅(𝑇) depending only on 𝑇, 𝜌₀(𝑥), and 𝑢₀(𝑥) such that the following bounds hold uniformly for any strong solution (𝜌,𝑢): 𝜌−𝜌_{𝐿∞(0,𝑇;𝐻1(ℝ))}(𝑇),‖𝑢‖≤𝐶_{𝐿∞(0,𝑇;𝐻1(ℝ))}≤𝐶(𝑇),0<𝜅(𝑇)≤𝜌(𝑥,𝑡)≤𝜅(𝑇),∀(𝑥,𝑡)∈ℝ×[0,𝑇]. (18)(19) Define 𝑝𝜌∣𝜌=1𝜌𝛾−1^𝛾−1𝛾−1𝜌^𝛾−𝛾𝛾−1𝜌^𝛾−1𝜌−𝜌 (20) as the relative potential energy density of (1), and 𝐸(𝑡)=_ℝ12𝜌(𝑥,𝑡)𝑢²(𝑥,𝑡)𝑑𝑥 (21) as the kinetic energy. Note that, since 𝑝 is strictly convex, 𝑝(𝜌∣𝜌) is nonnegative for every 𝜌, and 𝑝(𝜌∣𝜌)=0 if and only if 𝜌=𝜌. Mellet and Vasseur [25] also showed that, if the initial total energy is finite, that is, the sum of the kinetic and potential energy at time 0 satisfies _ℝ12𝜌₀𝑢²₀+𝑝𝜌₀∣𝜌𝑑𝑥<+∞, (22) then the following global-energy estimate on (−∞,∞)×[0,𝑇] holds uniformly with respect to all strong solutions; that is, for every 𝑇>0, there exists a positive constant 𝐶(𝑇) depending only on 𝑇, 𝜌₀(𝑥), and 𝑢₀(𝑥) such that sup_[]0,𝑇ℝ12𝜌𝑢²+𝑝𝜌∣𝜌+𝑑𝑥_𝑇0ℝ𝜇𝜌𝑢_𝑥²𝑑𝑥𝑑𝑡≤𝐶(𝑇) (23) holds for any strong solution (𝜌,𝑢). Correspondingly, for 𝑅>0, 0<𝑡₁<𝑇, and some constant 𝐶(𝑅,𝑡₁,𝑇), we call sup_{𝑡1≤𝑡≤𝑇}^𝑅_−𝑅12𝜌𝑢²+𝑝𝜌∣𝜌+𝑑𝑥_{𝑇𝑡1𝑅−𝑅}𝜇𝜌𝑢_𝑥²𝑑𝑥𝑑𝑡≤𝐶𝑅,𝑡₁,𝑇 (24) the local-energy estimate on [−𝑅,𝑅]×[𝑡₁,𝑇]. Note that the global-energy estimate implies the local-energy estimate.

The main result for the self-similar solutions of the isentropic compressible Navier-Stokes equations is as follows.

Theorem 1. Assume that 𝜇(𝜌) and 𝑃(𝜌) in (1) verify (2). Then the following statements are true.(1)There is no self-similar strong solution satisfying the global-energy estimate (23). (2)If there is a forward (backward) self-similar strong solution satisfying the local-energy estimate (24), then its kinetic energy (21) blows up as 𝑡↓0⁺(𝑡↑𝑇⁻).

For the self-similar solutions of the coupled system of the nonisentropic compressible Navier-Stokes equations with an entropy transport equation, the main result is as follows.

Theorem 2. Assume that 𝜇(𝜌,𝑠) and 𝑃(𝜌,𝑠) in (3)–(5) verify (6). Then the following statements are true. (1)There is no self-similar strong solution satisfying the global-energy estimate (23). (2)If there is a forward (backward) self-similar strong solution satisfying the local-energy estimate (24), then its kinetic energy (21) blows up as 𝑡↓0⁺(𝑡↑𝑇⁻).

Theorem 1 is proved in Section 2 and Theorem 2 in Section 3.

2. Proof of Theorem 1

Any self-similar solution of (1) is either forward or backward, so we first prove Theorem 1 for forward and then for backward self-similar solutions.

2.1. Forward Self-Similar Solutions

Lemma 3. If (𝑄,𝑈) solves (11)-(12), then the corresponding strong solution (𝜌,𝑢) defined by (9) of (1) does not satisfy the global-energy estimate (23).

Proof. From (11), 𝑄𝑦𝑈𝑦=𝑐𝑦𝑄𝑦+(𝑎−𝑐)^𝑦₀𝑄(𝑧)𝑑𝑧+𝐶₀, (25) where 𝐶₀ is an arbitrary constant. From (19), 0<𝜅(1)≤𝜌𝑦,1=𝑄𝑦≤𝜅(1),∀𝑦∈ℝ. (26) Since 𝑎≥𝑐>0, (25) and (26) imply that, for 𝑦≥𝑌₀>0 where 𝑌₀ is large enough, 𝑄𝑦𝑈𝑦=𝑐2𝑦𝑦𝑦𝑄+𝑐2𝑄𝑦+(𝑎−𝑐)×^𝑦₀𝑄(𝑧)𝑑𝑧+𝐶₀≥𝑐2𝑦+𝑐𝑦𝑄2𝑦𝑄𝑦+𝐶₀≥𝑐2𝑦+𝑐𝑦𝑄2𝑌₀𝜅(1)+𝐶₀≥𝑐2𝑦𝑄𝑦. (27) Thus, from (9) and (21), for any 𝑡>0, 𝐸(𝑡)=_ℝ12⋅1𝑡^𝑎𝑄𝑥𝑡^𝑐⋅1𝑡^2𝑏𝑈²𝑥𝑡^𝑐=1𝑑𝑥2𝑡^{𝑎+2𝑏−𝑐}_ℝ1𝑄𝑦𝑄𝑦𝑈𝑦²≥1𝑑𝑦2𝑡^{𝑎+2𝑏−𝑐}_𝑌^+∞₀1𝑄𝑦𝑐2𝑦𝑄𝑦²≥𝑐𝑑𝑦₂𝜅(1)8𝑡^{𝑎+2𝑏−𝑐}_𝑌^+∞₀𝑦²𝑑𝑦=+∞. (28) This proves the lemma.

Lemma 4. If (𝑄,𝑈) solves (11)-(12) and the corresponding strong solution (𝜌,𝑢) defined by (9) of (1) satisfies the local-energy estimate (24), then as 𝑡↓0, the kinetic energy (21) must blow up.

Proof. Similar to the proof of Lemma 3, for any 𝑡>0 and 𝑅>𝑌₀𝑡^𝑐>0, 𝐸(𝑡)≥𝑐²𝜅(1)8𝑡^{𝑎+2𝑏−𝑐𝑅/𝑡𝑐}_𝑌0𝑦²=𝑐𝑑𝑦₂𝜅(1)24𝑡^{𝑎+2𝑏−𝑐}𝑅³𝑡^3𝑐−𝑌³₀=𝑐²𝜅(1)24𝑡^{𝑎+2𝑏+2𝑐}𝑅³−𝑌₀𝑡^𝑐3⟶+∞,as𝑡↓0. (29) This proves the lemma.

2.2. Backward Self-Similar Solutions

Lemma 5. If (𝑄,𝑈) solves (13)-(14), then the corresponding strong solution (𝜌,𝑢) defined by (10) of (1) does not satisfy the global-energy estimate (23).

Proof. Fix 𝑇>1. From (13), 𝑄𝑦𝑈𝑦=−𝑐𝑦𝑄𝑦−(𝑎−𝑐)^𝑦₀𝑄(𝑧)𝑑𝑧+𝐶₀, (30) where 𝐶₀ is an arbitrary constant. From (19), 0<𝜅(𝑇)≤𝜌𝑦,𝑇−1=𝑄𝑦≤𝜅(𝑇),∀𝑦∈ℝ. (31) Since 𝑎≥𝑐>0, (30) and (31) imply that, for 𝑦≥𝑌₀>0 where 𝑌₀ is large enough, 𝑄𝑦𝑈𝑦𝑐=−2𝑦𝑦𝑦𝑄−𝑐2𝑄𝑦−(𝑎−𝑐)×^𝑦₀𝑄(𝑧)𝑑𝑧+𝐶₀𝑐≤−2𝑦𝑄𝑦. (32) Thus, from (10) and (21), for any 𝑡<𝑇, 𝐸(𝑡)=_ℝ12⋅1(𝑇−𝑡)^𝑎𝑄𝑥(𝑇−𝑡)^𝑐⋅1(𝑇−𝑡)^2𝑏𝑈²×𝑥(𝑇−𝑡)^𝑐≥𝑐𝑑𝑥₂𝜅(𝑇)8(𝑇−𝑡)^{𝑎+2𝑏−𝑐𝑌+∞}₀𝑦²𝑑𝑦=+∞. (33) This proves the lemma.

Lemma 6. If (𝑄,𝑈) solves (13)-(14) and the corresponding strong solution (𝜌,𝑢) defined by (10) of (1) satisfies the local-energy estimate (24), then as 𝑡↑𝑇, the kinetic energy (21) must blow up.

Proof. Recalling the proofs of Lemmas 4 and 5, for 𝑅>𝑌₀(𝑇−𝑡)^𝑐>0, 𝐸(𝑡)≥𝑐²𝜅(𝑇)8(𝑇−𝑡)^{𝑎+2𝑏−𝑐𝑅/(𝑇−𝑡)𝑐𝑌}₀𝑦²=𝑐𝑑𝑦₂𝜅(𝑇)24(𝑇−𝑡)^{𝑎+2𝑏+2𝑐}×𝑅³−𝑌³₀(𝑇−𝑡)^3𝑐⟶+∞,as𝑡↑𝑇. (34) This proves the lemma.

Now, Theorem 1 follows from the four lemmas above.

3. Proof of Theorem 2

If (𝜌,𝑢,𝑠) solves (3)–(6), then 𝜌^(𝜆),𝑢^(𝜆)=𝜆^𝑎𝜌𝜆^𝑐𝑥,𝜆^𝑑𝑡,𝜆^𝑏𝑢𝜆^𝑐𝑥,𝜆^𝑑𝑡,𝜆^𝑙𝑠𝜆^𝑐𝑥,𝜆^𝑑𝑡 (35) does so for any 𝜆>0, by setting 𝑎=1/(𝛾−𝜃), 𝑏=(𝛾−1)/2(𝛾−𝜃), 𝑐=(𝛾+1−2𝜃)/2(𝛾−𝜃), 𝑑=1, and 𝑙=0. Note that, from (6), 𝑎>0, 𝑏≥0, and 𝑐>0. The forward self-similar solutions have the following form: 𝜌()=1𝑥,𝑡𝑡^𝑎𝑄𝑥𝑡^𝑐,𝑢()=1𝑥,𝑡𝑡^𝑏𝑈𝑥𝑡^𝑐,𝑠()𝑥𝑥,𝑡=𝑆𝑡^𝑐,𝑡>0, (36) where 𝑄(𝑦)=𝜌(𝑦,1), 𝑈(𝑦)=𝑢(𝑦,1), and 𝑆(𝑦)=𝑠(𝑦,1). The backward self-similar solutions are 𝜌()=1𝑥,𝑡(𝑇−𝑡)^𝑎𝑄𝑥(𝑇−𝑡)^𝑐,𝑢()=1𝑥,𝑡(𝑇−𝑡)^𝑏𝑄𝑥(𝑇−𝑡)^𝑐,𝑠()𝑥𝑥,𝑡=𝑆(𝑇−𝑡)^𝑐,0<𝑡<𝑇, (37) where 𝑄(𝑦)=𝜌(𝑦,𝑇−1), 𝑈(𝑦)=𝑢(𝑦,𝑇−1), and 𝑆(𝑦)=𝑠(𝑦,𝑇−1) for 𝑇>1.

Lions [29] investigated the coupled system of the Navier-Stokes equations with an entropy transport equation in a pure form and obtained the existence of weak solutions satisfying (19) and (23).

Proof of Theorem 2. Suppose that (𝜌,𝑢,𝑠) is a forward self-similar solution. Inserting (36) into (3), one gets 𝑈(𝑦)=𝑐𝑦, and thus 𝑢(𝑥,𝑡)=𝑐𝑥/𝑡. Therefore, for any 𝑡>0, (36), (19), and (21) yield 𝐸(𝑡)=_ℝ12⋅1𝑡^𝑎𝑄𝑥𝑡^𝑐⋅𝑐𝑥𝑡²≥𝑐²𝜅(1)2𝑡^𝑎+2_ℝ𝑥²𝑑𝑥=+∞. (38) This means that the global-energy estimate (23) does not hold and that the kinetic energy (21) blows up as 𝑡↓0.
The case of backward self-similar solutions can be proved similarly, so Theorem 2 is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by Zhejiang Provincial Natural Science Foundation of China (no. LQ13G030018) and National Natural Science Foundation of China (nos. 11001049 and 11226184).