Drifting solutions with elliptic symmetry for the ... · Due to the significance of the Navier-Stokes (NS) equations in various physical fields such as fluid, plasmas, astrophysics,

Drifting solutions with elliptic symmetry for the compressible Navier-Stokes equationswith density-dependent viscosityHongli An and Manwai Yuen

Citation: Journal of Mathematical Physics 55, 053506 (2014); doi: 10.1063/1.4872235 View online: http://dx.doi.org/10.1063/1.4872235 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/55/5?ver=pdfcov Published by the AIP Publishing

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

JOURNAL OF MATHEMATICAL PHYSICS 55, 053506 (2014)

Drifting solutions with elliptic symmetry for thecompressible Navier-Stokes equations withdensity-dependent viscosity

Hongli An1,a) and Manwai Yuen2,b)

1College of Science, Nanjing Agricultural University, Nanjing 210095,People’s Republic of China2Department of Mathematics and Information Technology, The Hong Kong Institute ofEducation, 10 Po Ling Road, Tai Po, New Territories, Hong Kong

(Received 8 August 2013; accepted 8 April 2014; published online 9 May 2014)

In this paper, we investigate the analytical solutions of the compressible Navier-Stokesequations with dependent-density viscosity. By using the characteristic method, wesuccessfully obtain a class of drifting solutions with elliptic symmetry for the Navier-Stokes model wherein the velocity components are governed by a generalized Emdendynamical system. In particular, when the viscosity variables are taken the same asYuen [M. W. Yuen, “Analytical solutions to the Navier-Stokes equations,” J. Math.Phys. 49, 113102 (2008)], our solutions constitute a generalization of that obtainedby Yuen. Interestingly, numerical simulations show that the analytical solutions canbe used to explain the drifting phenomena of the propagation wave like Tsunamis inoceans. C© 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4872235]

I. INTRODUCTION

In this paper, we consider the following compressible Navier-Stokes equations with density-dependent viscosity coefficients

ρt+div(ρU) =0, (1)

(ρU)t + div(ρU ⊗ U) − div(h(ρ)D(U)) − ∇(g(ρ)divU) + ∇ P(ρ) = 0, (2)

where t ∈ (0, + ∞) is the time and x ∈ RN(N ≥ 2) is the spacial coordinate, while ρ(x, t) denotesthe fluid density, U = U(x, t) = (u1, u2, · · · , uN ) stands for the fluid velocity and P(ρ) = κργ forthe pressure, respectively. And

D(U) = ∇U +t ∇U2

is the strain tensor, h(ρ) and g(ρ) are the Lame viscosity coefficients satisfying

h(ρ) > 0, h(ρ) + Ng(ρ) ≥ 0.

Due to the significance of the Navier-Stokes (NS) equations in various physical fields such asfluid, plasmas, astrophysics, oceanography, and atmospheric dynamics, the NS equations have beenstudied extensively and intensively, which is manifested by a large number of related papers. Forexample, the mathematical derivations were derived in the simulation of flow surface in shallowregion.1–3 The existence and uniqueness of the local strong solution were analyzed by Choe andKim.4 While, the existence of global weak solutions was discussed by Lions5 and other authors.6–10

There are also some interesting work done on analytical solutions of the NS equations. For instance,

a)Author to whom correspondence should be addressed. Electronic mail: [email protected])E-mail: [email protected]

0022-2488/2014/55(5)/053506/10/$30.00 C©2014 AIP Publishing LLC55, 053506-1

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-2 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

Yuen derived a class of self-similar solutions with radial symmetry for the NS equations withh(ρ) = κ1ρ

θ and g(ρ) = 0 in Ref. 11. Subsequently, Yuen constructed some self-similar solutionswith elliptic symmetry for the NS equations with h(ρ) = μ and g(ρ) = 0 in Ref. 12. It is noticed thatthese two works were based on the separation method. Recently, by using the characteristic method,An and Yuen obtained a new class of perturbational solutions with elliptic symmetry for the NSequations in Ref. 13.

What needs to point out is that most works mentioned above only hold for NS equations withspecial viscosity coefficients h(ρ) and g(ρ). A natural idea is that the analytical solutions should alsoexist for the NS equations with general and reasonable viscosity coefficients h(ρ) and g(ρ). Sincethe choice of viscosity coefficients is key to obtain some physically important solutions. However,up to now, except the work of Guo and Xin,14 not much related work has been done. It is remarkablethat here we derive the drifting solutions with elliptic symmetry for the compressible NS equationswith density-dependent viscosity via a characteristic approach. Interestingly, numerical simulationresults show that such solutions can be used to explain the drifting phenomena of the propagationwave like Tsunamis in oceans.

II. DRIFTING SOLUTIONS OF THE NS EQUATIONS

Here, we consider the general viscosity coefficients h(ρ) and g(ρ), which take a form of

h(ρ) = κ1ρθ , g(ρ) = κ2ρ

θ . (3)

Then, the compressible Navier-Stokes system with density-dependent viscosity coefficients become

ρt+div(ρU) =0, (4)

ρ [Ut + U · ∇U] − div(κ1ρθ D(U)) − ∇(κ2ρ

θ divU) + ∇ P(ρ) = 0. (5)

For simplicity, we shall take D(U) = ∇U as what has been chosen by Guo and Xin in Ref. 14.In the following, we shall give a lemma that proves important to the constructions of drifting

solutions of NS equations with dependent-density viscosity.

Lemma: For the continuity equation of the NS system, namely,

ρt + div (ρU) = 0, (6)

there exist solutions ⎧⎪⎪⎨⎪⎪⎩

ρ = f(

x1−d1a1

,x2−d2

a2,··· , xN −dN

aN

)N�i=1

ai

ui = aiai

(xi − di ) + di , for i = 1, 2, · · · , N

(7)

where di = di(t), ai = ai(t) > 0 and an arbitrary C1 function f ≥ 0.

Proof. Inspired by the works of Yuen15 and Rogers and An,16 we perturb the velocity as thisform:

ρ = ρ(t, x), ui = ai

ai(xi − di ) + di . (8)

Substitution of this ansatz into the continuity equation (6), yields

ρt + div (ρU) = ρt + ∇ρ · U + ρ∇ · U

= ∂

∂tρ +

N∑i=1

∂

∂xiρ

[ai

ai(xi − di ) + di

]+

N∑i=1

ρai

ai= 0. (9)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-3 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

According to the classical characteristic approach,17 we have

dt

1= dx

aiai

(xi − di ) + di= dρ

−N∑

i=1

ρai

ai

(10)

whence, the solution is

F

(N�i=1

aiρ,x1 − d1

a1,

x2 − d2

a2, · · · ,

xN − dN

aN

)= 0 (11)

with an arbitrary C1 function F such that ρ ≥ 0.For convenience, we rewrite (11) into an explicit form

ρ =f(

x1−d1a1

, x2−d2a2

, · · · , xN −dNaN

)N�i=1

ai

. (12)

Therefore, the proof is completed. �

Remark 1: It is necessary to point out that the negative symbol in the perturbational non-constantfunctions di for the velocity in (7) is critical to guarantee the use of the characteristic method.

On the application of the above lemma, we construct a class of drifting solutions with ellipticsymmetry for the Navier-Stokes equations (4)(5). The main result is described as follows:

Theorem 1: For the compressible Navier-Stokes equations with dependent-density viscositycoefficients, there exists a class of drifting solutions:⎧⎪⎨

⎪⎩ρ = f (s)

N�

k=1ak

ui = aiai

(xi − di ) + di , for i = 1, 2, · · · , N

(13)

where

f (s) =

⎧⎪⎪⎨⎪⎪⎩

αe− ξ

2θs for θ = 1

max

((− ξ (θ−1)

2θs + α

) 1θ−1

, 0

)for θ �= 1

(14)

with

di = di0 + tdi1, s =N∑

k=1

(xk − dk

ak

)2

. (15)

In the above ξ , di0, di1, and α ≥ 0 are arbitrary constants. While the auxiliary functions ai = ai(t)are governed by the generalized Emden dynamical system:⎧⎪⎪⎨

⎪⎪⎩ai (t) = −ξ

[k1

∑Nk=1

ak (t)ak (t) +k2

ai (t)ai (t) −κ

]ai (t)

(N�

k=1ak (t)

)θ−1 , for i = 1, 2, · · · , N

ai (0) = ai0 > 0, ai (0) = ai1

(16)

where ai0 and ai1 are initial conditions.In particular, for ξ < 0,(1) if all ai1 < 0, the solutions (13) blow up on or before the finite time

T = min(−ai0/ai1 : ai1 < 0, i = 1, 2, · · · , N ); (17)

(2) if all ai1 ≥ 0 the solutions (13) exist globally.

Remark 2: We emphasize that the intrusion of the viscosity coefficients h(ρ) and g(ρ) not onlymakes the solutions are quite different from what were discussed by Guo and Xin14 and Yuen,11 butalso makes the occurrence of a generalized Emden dynamical system.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-4 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

Remark 3: It is known that the Navier-Stokes equations can be used to describe the driftingphenomena of the propagation wave like Tsunamis in oceans. Interestingly, numerical simulationsfully exhibit such drifting behaviors. Therefore, we call (13) the drifting solution and the lineartime-dependent functions di are the drifting terms. When these functions di degenerate to constants,namely di1 = 0 and di0 = const, they coincide with the case that was discussed by Yuen in Ref. 12.

Proof of Theorem 1. According to the lemma, it is easy to check that the function (13) satisfiesthe continuity equation (4). In the following, we shall validate that the function (13) also holds forthe momentum equation (5).

For the i-th momentum equation of the Navier-Stokes equations (5), by defining an ellipticallysymmetric variable via

s =N∑

k=1

(xk − dk)2

a2k (t)

. (18)

Now we proceed with γ = 2, then we obtain

ρ

[∂ui

∂t+

N∑k=1

uk∂ui

∂xk

]− κ1

∂

∂xi

(ρθ∇ · �u) − κ2∇ · (

ρθ∇ui) + κ

∂

∂xiρθ (19)

= ρ

{∂

∂t

[ai

ai(xi − di ) + di

]+

[ai

ai(xi − di ) + di

]∂

∂xi

[ai

ai(xi − di ) + di

]}

− κ1θ

N∑k=1

ak

akρθ−1 ∂ρ

∂xi− κ2θρ

θ−1 ai

ai

∂ρ

∂xi+ κθρθ−1 ∂ρ

∂xi

= ρ

{[(ai

ai− a2

i

a2i

)(xi − di ) + di + a2

i

a2i

(xi − di )

]− θρθ−2

(κ1

N∑k=1

ak

ak+ κ2

ai

ai− κ

)∂ρ

∂xi

}

= ρ

{[(ai

ai− a2

i

a2i

)(xi − di ) + di + a2

i

a2i

(xi − di )

]

−θ

(κ1

N∑k=1

ak

ak+ κ2

ai

ai− κ

)f (s)θ−2(N�

k=1ak

)θ−2

∂

∂xi

f (s)N�

k=1ak

⎫⎪⎪⎪⎬⎪⎪⎪⎭

= ρ

⎧⎪⎪⎪⎨⎪⎪⎪⎩

ai

ai(xi − di ) + di − 2θ

(κ1

N∑k=1

ak

ak+ κ2

ai

ai− κ

)f (s)θ−2 f (s)(

N�

k=1ak

)θ−1

(xi − di

a2i

)⎫⎪⎪⎪⎬⎪⎪⎪⎭

= ρ(xi − di )

a2i

⎧⎪⎪⎪⎨⎪⎪⎪⎩

ai ai − 2θ f (s)θ−2 f (s)(N�

k=1ak

)θ−1

(κ1

N∑k=1

ak

ak+ κ2

ai

ai− κ

)⎫⎪⎪⎪⎬⎪⎪⎪⎭

+ ρdi

= ρ(xi − di )

a2i

{ξ + 2θ f (s)θ−2 f (s)

} + ρdi (20)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-5 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

FIG. 1. (a)–(g) Time evolutions of the density function ρ at a regular time intervals t = 3. Here the parameters are chosenas γ = 1, ξ = 2, k = k1 = 1, k2 = 2, α = 10.

with the N-dimensional generalized Emden dynamical system given by

⎧⎪⎪⎨⎪⎪⎩

ai (t) = −ξ[κ1

∑Nk=1

ak (t)ak (t) +κ2

ai (t)ai (t) −κ

]ai (t)

(N�

k=1ak (t)

)θ−1 , for i = 1, 2, · · · , N

ai (0) = ai0 > 0, ai (0) = ai1

(21)

with arbitrary constants ξ , ai0, and ai1.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-6 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

(a) (b)

(c) (d)

(e)

t=0 t=3

t=6 t=9

t=12t=12

(g)

t=18

(f)

t=15

FIG. 2. (a)–(g) Contour figures of the density function ρ with the time interval t = 3. While the parameters are chosen asγ = 1, ξ = 2, k = k1 = 1, k2 = 2, α = 10.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-7 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

FIG. 3. (a)–(g) Time Evolutions of the density function ρ at time intervals given by t = 4. Here the parameters are chosenas γ = 2, ξ = 2, k = k1 = 1, k2 = 2, α = 10.

If we require the function f (s) satisfies the following differential equation:

{ ξ

2θ+ f (s)θ−2 f (s) = 0

f (0) = α ≥ 0,or ρ = 0 (22)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-8 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

(a) (b)

(c) (d)

(e)

t=0 t=4

t=8 t=12

t=16 t=16

(g)

t=24

(f)

t=20

FIG. 4. (a)–(g) Contour plots of the density function ρ with t = 4. Here the parameters are chosen as γ = 2, ξ = 2, k = k1

= 1, k2 = 2, α = 10.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-9 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

then we can have

f (s) =⎧⎨⎩

αe− ξ

2θs for θ = 1

max

((− ξ (θ−1)

2θs + α

) 1θ−1

, 0

)for θ �= 1.

(23)

Therefore, the function (13) is the drifting solution with elliptic symmetry of the compressibleNavier-Stokes equations with dependent-density viscosity. �

In order to shed some light on the possible behaviors that the solutions we obtained exhibit, wepresent the numerical simulations. Here we take N = 2. Figures 1(a)–1(g) show the time evolutionsof density function ρ given by (13) at regular time intervals when θ = 1. From the figures, one caneasily see that the elliptical eddy continuously move forward with time changing. Such phenomenacan also be clearly noticed from the contour Figures 2(a)–2(g). It is known that the Navier-Stokesequations can be used to describe the drifting phenomena of the propagation wave like Tsunamis inoceans. Therefore, we conclude that the moving behaviors our solution reveals are nothing but thedrifting phenomena in nature. When θ �= 1, the density function ρ is expressed by another formula,namely, (14). The similar behaviors are exhibited in Figs. 3 and 4.

III. CONCLUSION AND DISCUSSION

Due to the importance of the Naiver-Stokes equations in various branches of physics, manyexperts have paid great attention to them, especially to the constructions of analytical solutions.For example, when the viscosity coefficients are chosen by h(ρ) = κ1ρ

θ , g(ρ) = 0, Yuen obtainedthe self-similar solutions in Ref. 11. Yuen also constructed some self-similar solutions with ellipticsymmetry when the viscosity variables are h(ρ) = μ and g(ρ) = 0 in Ref. 12. Guo and Xin derivedsome analytical solutions when h(ρ) = ρθ and g(ρ) = (θ − 1)ρθ in Ref. 14. Interestingly, herewe successfully derived some drifting solutions with elliptic symmetry for the compressible Navier-Stokes equations with general forms of viscosity coefficients. Numerical simulations show that theanalytical solutions obtained can be applied to explain the drifting phenomena of propagations ofwave like Tsunamis in oceans. In addition, we would like to point out that the velocity componentsai are governed by the generalized Emden dynamical system, which is quite different from that theclassical Emden equations obtained in Refs. 11 and 12. Does the generalized Emden system haveany nice properties as the classical one? What is the relation between the generalized Emden systemand the Ermakov system? These problems will be deeply considered in our future work.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (Grant Nos.11301269 and 11301266), Jiangsu Provincial Natural Science Foundation of China (Grant No.BK20130665), the Fundamental Research Funds KJ2013036 for the Central Universities andResearch Grant MIT/SRG14/12-13 of the Hong Kong Institute of Education.

1 D. Bresch and B. Desjardins, “Existence of global weak solutions for a 2D viscous shallow water equations and convergenceto the quasi-geostrophic model,” Commun. Math. Phys. 238, 211–223 (2003).

2 D. Bresch and B. Desjardins, “On the construction of approximate solutions for the 2D viscous shallow water model andfor compressible Navier-Stokes models,” J. Math. Pures Appl. 86, 362–368 (2006).

3 J. F. Gerbeau and B. Perthame, “Derivation of viscous Saint-Venant system for laminar shallow water, numerical validation,”Discrete Contin. Dyn. Syst. Ser. B 1, 89–102 (2001).

4 H. J. Choe and H. Kim, “Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,” J. Differ.Equations 190, 504–523 (2003).

5 P. L. Lions, Mathematical Topics in Fluid Dynamics 2, Compressible Models (Oxford Science Publication, Oxford,1998).

6 E. Feireisl, A. Novotny, and H. Petzeltova, “On the existence of globally defined weak solutions to the Navier-Stokesequations of isentropic compressible fluids,” J. Math. Fluid Mech. 3, 358–392 (2001).

7 T. Yang, Z. Yao, and C. J. Zhu, “Compressible Navier-Stokes equations with density-dependent viscosity and vacuum,”Commun. Partial Differ. Equations 26, 965–981 (2001).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50

053506-10 H. An and M. Yuen J. Math. Phys. 55, 053506 (2014)

8 D. Hoff and H. K. Jenssen, “Symmetric nonbarotropic flows with large data and forces,” Arch. Ration. Mech. Anal. 173,297–343 (2004).

9 S. Jiang and P. Zhang, “Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,”Commun. Math. Phys. 215, 559–581 (2001).

10 W. Sun, S. Jiang and Z. Guo, “Helical symmetry solutions of the 3D Navier-Stokes equations for compressible isentropicfluids,” J. Differ. Equations 222(2), 263–296 (2006).

11 M. W. Yuen, “Analytical solutions to the Navier-Stokes equations,” J. Math. Phys. 49, 113102 (2008); “Erratum: “Analyticalsolutions to the Navier-Stokes equations” [J. Math. Phys. 49, 113102 (2008)],” J. Math. Phys. 50, 019901 (2009).

12 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).

13 H. L. An and M. W. Yuen, “Supplement to “Self-similar solutions with elliptic symmetry for the compressible Euler andNavier-Stokes equations in RN” [Commun. Nonlinear Sci. Numer. Simul. 17, 4524–4528 (2012)],” Commun. NonlinearSci. Numer. Simul. 18, 1558–1561 (2013).

14 Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier-Stokes equations with density-dependentviscosity coefficients and free boundaries,” J. Differ. Equations 253, 1–19 (2012).

15 M. W. Yuen, “Perturbational blowup solutions to the compressible 1-dimensional Euler Equations,” Phys. Lett. A 375,3821–3825 (2011).

16 C. Rogers and H. L. An, “On a 2+1-dimensional Madelung system with logarithmic and with Bohm quantum potentials.Ermakov reduction,” Phys. Scr. 84, 045004 (2011).

17 J. D. Logan, An Introduction to Nonlinear Partial Differential Equations, 2nd ed. (Wiley Interscience/John Wiley and Sons,2008).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

121.248.132.220 On: Fri, 09 May 2014 13:22:50