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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 194704, 5 pages

http://dx.doi.org/10.1155/2013/194704

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

^{1}School of Economics & Management, Zhejiang Sci-Tech University, Hangzhou 310018, China^{2}Shanghai Baosteel Industry Technological Service Co., Ltd., Tongji Road 3521, Baoshan Area, Shanghai 201900, China^{3}Department of Mathematics, Hangzhou Normal University, Hangzhou 310016, China

Received 29 July 2013; Accepted 17 September 2013

Academic Editor: Hui-Shen Shen

Copyright © 2013 Tailong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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: where and are the density and velocity of the fluid, and denote the density-dependent viscosity and pressure, respectively, and the subscripts mean partial derivations. Guo and Jiang [21] considered (1) with constant viscosity, , and linear density-dependent pressure, , where 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 , where . The problem with , 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 , where is a constant (see [25]). Okada et al. [26] pointed out that, because of the hard sphere interaction, the relation between indices and is . In the first part of this paper, we are concerned with (1) where

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: where is the entropy of the fluid and and denote the density-entropy-dependent viscosity and pressure, respectively. In this system, we assume that

Navier-Stokes equations enjoy a scaling property: if solves (1)-(2), then does so for any , by setting , , , and . Note that, from (2), and . Solution is called forward self-similar if In that case, and are decided by their values at the instant of : where and are defined on . In the same manner, the backward self-similar solutions are of the form: where and for . Substitution of (9) or (10) into (1) gives for forward self-similar solutions, or 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 estimate on the density and a new large-scale argument on the energy. In addition, conditions on and 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 . Precisely, fix constant positive density and , and let be a smooth monotone function satisfying Assume that the initial data and satisfy for some constants and . Assume also that and verify (2). Mellet and Vasseur [25] proved that there exists a global strong solution of (1) on such that for every : Moreover, for every , there exist uniform bounds away from zero with respect to all strong solutions having the same initial data. Precisely, there exist some constants , , and depending only on , , and such that the following bounds hold uniformly for any strong solution : Define as the relative potential energy density of (1), and as the kinetic energy. Note that, since is strictly convex, is nonnegative for every , and if and only if . 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 satisfies then the following global-energy estimate on holds uniformly with respect to all strong solutions; that is, for every , there exists a positive constant depending only on , , and such that holds for any strong solution . Correspondingly, for , , and some constant , we call the local-energy estimate on . 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 and 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 . *

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 and 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 . *

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 solves (11)-(12), then the corresponding strong solution defined by (9) of (1) does not satisfy the global-energy estimate (23).*

*Proof. *From (11),
where is an arbitrary constant. From (19),
Since , (25) and (26) imply that, for where is large enough,
Thus, from (9) and (21), for any ,
This proves the lemma.

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

*Proof. *Similar to the proof of Lemma 3, for any and ,
This proves the lemma.

##### 2.2. Backward Self-Similar Solutions

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

*Proof. *Fix . From (13),
where is an arbitrary constant. From (19),
Since , (30) and (31) imply that, for where is large enough,
Thus, from (10) and (21), for any ,
This proves the lemma.

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

*Proof. *Recalling the proofs of Lemmas 4 and 5, for ,
This proves the lemma.

Now, Theorem 1 follows from the four lemmas above.

#### 3. Proof of Theorem 2

If solves (3)–(6), then does so for any , by setting , , , , and . Note that, from (6), , , and . The forward self-similar solutions have the following form: where , , and . The backward self-similar solutions are where , , and for .

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 is a forward self-similar solution. Inserting (36) into (3), one gets , and thus . Therefore, for any , (36), (19), and (21) yield
This means that the global-energy estimate (23) does not hold and that the kinetic energy (21) blows up as .

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).

#### References

- J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,”
*Acta Mathematica*, vol. 63, no. 1, pp. 193–248, 1934. View at Publisher · View at Google Scholar · View at MathSciNet - G. K. Batchelor,
*An Introduction to Fluid Dynamics*, Cambridge University Press, Cambridge, UK, 1999. View at MathSciNet - L. I. Sedov,
*Similarity and Dimensional Methods in Mechanics*, CRC Press, Boca Raton, Fla, USA, 10th edition, 1993. View at MathSciNet - O. A. Barraza, “Self-similar solutions in weak ${L}^{p}$-spaces of the Navier-Stokes equations,”
*Revista Matemática Iberoamericana*, vol. 12, no. 2, pp. 411–439, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - M. Cannone and F. Planchon, “Self-similar solutions for Navier-Stokes equations in ${\mathbb{R}}^{3}$,”
*Communications in Partial Differential Equations*, vol. 21, no. 1-2, pp. 179–193, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - J. Nečas, M. Ružička, and V. Šverák, “On Leray's self-similar solutions of the Navier-Stokes equations,”
*Acta Mathematica*, vol. 176, no. 2, pp. 283–294, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - F. Planchon, “Asymptotic behavior of global solutions to the Navier-Stokes equations in ${\mathbb{R}}^{3}$,”
*Revista Matemática Iberoamericana*, vol. 14, no. 1, pp. 71–93, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - T.-P. Tsai, “On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates,”
*Archive for Rational Mechanics and Analysis*, vol. 143, no. 1, pp. 29–51, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - T.-P. Tasi, “Forward discrete self-similar solutions of the Navier-Stokes equations,” submitted, http://arxiv.org/abs/1210.2783.
- J. R. Miller, M. O'Leary, and M. Schonbek, “Nonexistence of singular pseudo-self-similar solutions of the Navier-Stokes system,”
*Mathematische Annalen*, vol. 319, no. 4, pp. 809–815, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - P. G. Lemarié-Rieusset,
*Recent Developments in the Navier-Stokes Problem*, vol. 431 of*Chapman & Hall/CRC Research Notes in Mathematics*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - D. Chae, “Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations,”
*Mathematische Annalen*, vol. 338, no. 2, pp. 435–449, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - D. Chae, “Notes on the incompressible Euler and related equations on ${R}^{N}$,”
*Chinese Annals of Mathematics B*, vol. 30, no. 5, pp. 513–526, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - D. Chae, “Nonexistence of self-similar singularities in the ideal magnetohydrodynamics,”
*Archive for Rational Mechanics and Analysis*, vol. 194, no. 3, pp. 1011–1027, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - L. Brandolese, “Fine properties of self-similar solutions of the Navier-Stokes equations,”
*Archive for Rational Mechanics and Analysis*, vol. 192, no. 3, pp. 375–401, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - X.-Y. Jiao, “Some similarity reduction solutions to two-dimensional incompressible Navier-Stokes equation,”
*Communications in Theoretical Physics*, vol. 52, no. 3, pp. 389–394, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - I. F. Barna, “Self-similar solutions of three-dimensional Navier—stokes equation,”
*Communications in Theoretical Physics*, vol. 56, no. 4, pp. 745–750, 2011. View at Publisher · View at Google Scholar · View at Scopus - G. Karch and D. Pilarczyk, “Asymptotic stability of Landau solutions to Navier-Stokes system,”
*Archive for Rational Mechanics and Analysis*, vol. 202, no. 1, pp. 115–131, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - V. Šverák, “On Landau's solutions of the Navier-Stokes equations,”
*Journal of Mathematical Sciences*, vol. 179, no. 1, pp. 208–228, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - H. Jia and V. Šverák, “Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions,”
*Inventiones Mathematicae*, 2013. View at Publisher · View at Google Scholar - Z. Guo and S. Jiang, “Self-similar solutions to the isothermal compressible Navier-Stokes equations,”
*IMA Journal of Applied Mathematics*, vol. 71, no. 5, pp. 658–669, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Guo and Z. Xin, “Analytical solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients and free boundaries,”
*Journal of Differential Equations*, vol. 253, no. 1, pp. 1–19, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - M. Yuen, “Self-similar solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations in ${\mathbb{R}}^{N}$,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 12, pp. 4524–4528, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. An and M. Yuen, “Supplement to ‘Self-similar solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations in ${\mathbb{R}}^{N}$’,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 18, no. 6, pp. 1558–1561, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - A. Mellet and A. Vasseur, “Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations,”
*SIAM Journal on Mathematical Analysis*, vol. 39, no. 4, pp. 1344–1365, 2008. View at Publisher · View at Google Scholar · View at Scopus - M. Okada, S. Matušocircu-Nečasová, and T. Makino, “Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity,”
*Annali dell'Università di Ferrara. Nuova Serie. Sezione VII. Scienze Matematiche*, vol. 48, pp. 1–20, 2002. View at MathSciNet - H. Teng, C. M. Kinoshita, S. M. Masutani, and J. Zhou, “Entropy generation in multicomponent reacting flows,”
*Journal of Energy Resources Technology, Transactions of the ASME*, vol. 120, no. 3, pp. 226–232, 1998. View at Scopus - J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird,
*Molecular Theory of Gases and Liquids*, John Wiley & Sons, New York, NY, USA, 1954. - P.-L. Lions,
*Mathematical Topics in Fluid Mechanics*, vol. 2 of*Oxford Lecture Series in Mathematics and Its Applications*, Oxford University Press, New York, NY, USA, 1996. View at MathSciNet - K. Nishida, T. Takagi, and S. Kinoshita, “Analysis of entropy generation and exergy loss during combustion,”
*Proceedings of the Combustion Institute*, vol. 29, pp. 869–874, 2002. - Z. W. Li, S. K. Chou, C. Shu, and W. M. Yang, “Entropy generation during microcombustion,”
*Journal of Applied Physics*, vol. 97, no. 8, Article ID 084914, 2005. View at Publisher · View at Google Scholar · View at Scopus - H. Yapıcı, N. Kayataş, B. Albayrak, and G. Baştürk, “Numerical calculation of local entropy generation in a methaneair burner,”
*Energy Conversion and Management*, vol. 46, pp. 1885–1919, 2005. - N. Lior, W. Sarmiento-Darkin, and H. S. Al-Sharqawi, “The exergy fields in transport processes: their calculation and use,”
*Energy*, vol. 31, no. 5, pp. 553–578, 2006. View at Publisher · View at Google Scholar · View at Scopus - D. Stanciu, M. Marinescu, and A. Dobrovicescu, “The influence of swirl angle on the irreversibilities in turbulent diffusion flames,”
*International Journal of Thermodynamics*, vol. 10, no. 4, pp. 143–153, 2007. View at Scopus - M. Safari, M. R. H. Sheikhi, M. Janbozorgi, and H. Metghalchi, “Entropy transport equation in large eddy simulation for exergy analysis of turbulent combustion systems,”
*Entropy*, vol. 12, no. 3, pp. 434–444, 2010. View at Publisher · View at Google Scholar · View at Scopus