A class of blowup and global analytical solutions of the viscoelastic Burgersʼ equations

Physics Letters A 377 (2013) 2275–2279

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Physics Letters A www.elsevier.com/locate/pla

A class of blowup and global analytical solutions of the viscoelastic Burgers’ equations Hongli An a , Ka-Luen Cheung b , Manwai Yuen b,∗ a b

College of Science, Nanjing Agricultural University, Nanjing 210095, PR China Department of Mathematics and Information Technology, The Hong Kong Institute of Education, 10 Po Ling Road, Tai Po, New Territories, Hong Kong

a r t i c l e

i n f o

Article history: Received 16 March 2013 Received in revised form 17 May 2013 Accepted 19 May 2013 Available online 16 July 2013 Communicated by R. Wu Keywords: Viscoelastic Burgers Analytical solutions Non-Newtonian fluid Perturbational method Blowup Global

a b s t r a c t In this Letter, by employing the perturbational method, we obtain a class of analytical self-similar solutions of the viscoelastic Burgers’ equations. These solutions are of polynomial-type whose forms, remarkably, coincide with that given by Yuen for the other physical models, such as the compressible Euler or Navier–Stokes equations and two-component Camassa–Holm equations. Furthermore, we classify the initial conditions into several groups and then discuss the properties on blowup and global existence of the corresponding solutions, which may be readily seen from the phase diagram. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The viscoelastic Burgers’ equations [1] are expressed by



ut + uu x = σx + μu xx , σt + u σx − σ u x = α u x − β σ

(1)

where the function u stands for the velocity, σ is the stress and the constants μ, α , β are the viscosity, the elastic modulus of the material and the decay rate of the stress, respectively. This model has its origin in the work of Burgers [2] and resembles a one-dimensional version of the upper convected Maxwell model [3]. Such a system provides fundamental pedagogical examples for many important topics in partial differential equations such as traveling waves, shock formation, similarity solutions and singular perturbation (see e.g. [4–7]). In recent years, much attention has been paid to the construction and analysis of travelling waves solutions of the viscoelastic Burgers’ equations [1,3,8]. Different from the work mentioned in [1,3], here we plan to study the blowup solutions of viscoelastic Burgers’ equations via the perturbational method [9,10]. Recently, the blowup-type solutions have been studied extensively and intensively, which is manifested by a large number of related papers [9–17]. Particularly, it was shown by Yuen in Ref. [9] that the compressible Euler or Navier–Stokes equations in R 1+1 :



ρt + ρx u + ρ u x = 0, ρ (ut + uu x ) + K (ρ γ )x = μu xx

admit the perturbational blowup solutions (ρ , u ):

*

Corresponding author. Tel.: +852 68213432. E-mail addresses: [email protected] (H. An), [email protected] (K.-L. Cheung), [email protected] (M. Yuen).

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.05.061

(2)

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H. An et al. / Physics Letters A 377 (2013) 2275–2279

⎧ ⎨ ρ γ −1 (x, t ) = max{ρ γ −1 (0, t ) − γ −1 [b˙ (t ) + b(t ) a˙ (t ) ]x − Kγ

⎩ u (x, t ) =

a(t )

(γ −1)ξ x2 , 2K γ aγ +1 (t )

0},

(3)

a˙ (t ) x + b(t ) a(t )

where a(t ), b(t ) and c (t ) satisfy the following ordinary differential equations:

⎧ ξ ⎪ a¨ (t ) = aγ (t ) , a(0) = a0 > 0, a˙ (0) = a1 , ⎪ ⎪ ⎨ (1+γ )˙a(t ) ˙ 2ξ a˙ 2 (t ) b¨ (t ) + b(t ) + [ aγ +1 (t ) + (γ − 1) a2 (t ) ]b(t ) = 0, b(0) = b0 , b˙ (0) = b1 , a ( t ) ⎪ ⎪ ⎪ ⎩ ∂ ρ γ −1 (0, t ) + ρ γ −1 (0, t ) a˙ (t ) − γ −1 [b˙ (t ) + b(t ) a˙ (t ) ]b(t ) = 0, ρ (0, 0) = α ∂t a(t ) Kγ a(t )

(4)

where the quantities K and γ are given by Eq. (2) and a0 , a1 , b0 , b1 , α are arbitrary constants. Interestingly, the above solutions can provide the mathematical explanations for the drifting phenomena of some propagation wave like Tsunamis which are generated by the earthquakes [9]. We observe that the viscoelastic Burgers’ equations (1) share similarities with the (1 + 1)-dimensional compressible Euler or Navier–Stokes model (2) if we take γ = 2 and roughly treat σ as ρ in some cases. Therefore, it is natural to inquire whether we can construct the perturbational blowup solutions for the Burgers’ equations (1). In this Letter, we successfully apply the perturbational method to obtain the self-similar solutions. Theorem 1. For the Burgers’ equations (1), there exists a class of perturbational solutions:

⎧ ⎨u = ⎩σ = with



k2 −β e−β t k1 +k2 t +e−β t

x+

k3 (k2 +β k1 +β k2 t )+k4 (eβ t k2 −β) , 1+(k1 +tk2 )eβ t

β 2 e−β t x2 2(k1 +k2 t +e−β t )

+

(5)

β 2 (k4 −k1 k3 −k2 k3 t ) x + E ∗ (t ) 1+(k1 +tk2 )eβ t

k −β e−β t

E˙ ∗ (t ) + β E ∗ (t ) − (α + E ∗ (t )) k +2 k t +e−β t + 1 2 E ∗ (0) = E 0∗

k4 β 2 (eβ t k2 −β)(k4 −k1 k3 −k2 k3 t ) [1+(k1 +tk2 )eβ t ]2

+

k3 β 2 (k2 +β k1 +β k2 t )(k4 −k1 k3 −k2 k3 t ) [1+(k1 +tk2 )eβ t ]2

= 0,

(6)

where k1 , k2 , k3 , k4 and E 0∗ are constants. Remark 2. It is noticed that if we take

a = k1 + k2 t + e−β t ,

b = 1 + (k1 + k2 t )eβ t = aeβ t ,

(7)

the above theorem may be readily rewritten into

u=

a˙ a

x + B ∗ (t ) =

a˙ a

x+

k3 ab˙ b2

+

k4 (k2 b − aβ) ab

,

σ=

a¨ 2a

x2 +

β 2 (k4 − k1k3 − k2k3t ) b

x + E ∗ (t )

(8)

with E ∗ given by (6). Interestingly, the form of the velocity u coincides with that given in [9,10,16], which fully shows the efficiency of the method adopted here. 2. Reduction of the viscoelastic Burgers’ equations Our construction process of the polynomial-type solutions follows the method of under-determined coefficients in classical differential equations. For convenience, we give the following proposition for solving the viscoelastic Burgers’ equations (1). Proposition 3. For the Burgers’ equations (1), there exists a class of analytical solutions:

u = A (t )x + B (t ), with

σ = C (t )x2 + D (t )x + E (t )

⎧ ˙ (t ) + A 2 (t ) − 2C (t ) = 0, A (0) = a0 , A ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B˙ (t ) + A (t ) B (t ) − D (t ) = 0, B (0) = b0 , C˙ (t ) + A (t )C (t ) + β C (t ) = 0, C (0) = c 0 , ⎪ ⎪ ⎪ ˙ (t ) + 2B (t )C (t ) + β D (t ) = 0, D (0) = d0 , D ⎪ ⎪ ⎩ E˙ (t ) − A (t ) E (t ) + B (t ) D (t ) − α A (t ) + β E (t ) = 0,

(9)

(10) E (0) = e 0 ,

where a0 , b0 , c 0 , d0 and e 0 are constants. Proof. We just plug the functions

u = A (t )x + B (t ),

σ = C (t )x2 + D (t )x + E (t ),

into the viscoelastic Burgers’ equations to obtain the corresponding results. For the first equation, we have

(11)

H. An et al. / Physics Letters A 377 (2013) 2275–2279

ut + uu x − σx − μu xx

2277

(12)

 ∂  ∂ ∂  = A (t )x + B (t ) + A (t )x + B (t ) A (t )x + B (t ) − C (t )x2 + D (t )x + E (t ) − 0 ∂t ∂x ∂x = A˙ (t )x + B˙ (t ) + A 2 (t )x + A (t ) B (t ) − 2C (t )x − D (t )  = A˙ (t ) + A 2 (t ) − 2C (t ) x + B˙ (t ) + A (t ) B (t ) − D (t ) =0

(13) (14) (15) (16)

by requiring



˙ (t ) + A 2 (t ) − 2C (t ) = 0, A

(17)

B˙ (t ) + A (t ) B (t ) − D (t ) = 0. For the second equation, we have

σt + u σx − σ u x − α u x + β σ

(18)

 ∂   ∂  ∂ = C (t )x2 + D (t )x + E (t ) + A (t )x + B (t ) C (t )x2 + D (t )x + E (t ) − C (t )x2 + D (t )x + E (t ) A (t )x + B (t ) ∂t ∂x ∂x  ∂  2 A (t )x + B (t ) + β C (t )x + D (t )x + E (t ) (19) −α ∂x    = C˙ (t )x2 + D˙ (t )x + E˙ (t ) + A (t )x + B (t ) 2C (t )x + D (t ) − C (t )x2 + D (t )x + E (t ) A (t ) − α A (t )  (20) + β C (t )x2 + D (t )x + E (t )  2 2 2 = C˙ (t )x + D˙ (t )x + E˙ (t ) + 2 A (t )C (t )x + A (t ) D (t ) + 2B (t )C (t ) x + B (t ) D (t ) − A (t )C (t )x − A (t ) D (t )x − A (t ) E (t ) − α A (t )  (21) + β C (t )x2 + D (t )x + E (t ) = C˙ (t )x2 + D˙ (t )x + E˙ (t ) + 2 A (t )C (t )x2 + 2B (t )C (t )x + B (t ) D (t ) − A (t )C (t )x2 − A (t ) E (t ) − α A (t ) + β[C (t )x2 + D (t )x + E (t )]   = C˙ (t ) + A (t )C (t ) + β C (t ) x2 + D˙ (t ) + 2B (t )C (t ) + β D (t ) x + E˙ (t ) + B (t ) D (t ) − A (t ) E (t ) − α A (t ) + β E (t )

(22)

=0

(24)

(23)

by requiring

⎧ ⎨ C˙ (t ) + A (t )C (t ) + β C (t ) = 0, C (0) = c0 , ˙ (t ) + 2B (t )C (t ) + β D (t ) = 0, D (0) = d0 , D ⎩ ˙E (t ) − A (t ) E (t ) + B (t ) D (t ) − α A (t ) + β E (t ) = 0,

The proof is completed.

(25) E (0) = e 0 .

2

The following task is to further solve the dynamical system (10) for obtaining Theorem 1. For this purpose, the technique employed by Yuen in [9] will be adopted. Proof of Theorem 1. We observe that the functions (5) take the same form as that given in Proposition 3 for the Burgers’ equations (1). According to the proposition, one just needs to prove that the corresponding conditions can be satisfied. For convenience, we take

A∗ = C∗ =

a˙

B∗ =

,

a a¨

2a

k3 ab˙ b2

D∗ =

,

+

k4 (k2 b − aβ) ab

β 2 (k4 − k1k3 − k2k3 t ) b

(26)

, ,

(27)

so we have

˙ ∗ + A ∗2 − 2C ∗ A = =

d a˙ dt a a¨ a

−

+

2 a˙ a

2 a˙ a

+

(28)

−2

a¨

2 a˙ a

−

= 0, ˙∗

a¨ a

(30) (31)

∗ ∗

C + A C + βC

=

(29)

2a

d a¨ dt 2a

+

∗

a˙ a¨ a 2a

(32)

+β

a¨ 2a

(33)

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H. An et al. / Physics Letters A 377 (2013) 2275–2279

= = =

... a

+β

a¨

(34)

2a 2a (k1 + k2 t + e−β t ) + β(k1 + k2t + e−β t )

(35)

2a (−β)3 e−β t + β(−β)2 e−β t

(36)

2a

= 0.

(37)

In the following, we check the rest three conditions:

B˙ ∗ + A ∗ B ∗ − D ∗



= =

d k3 ab˙ b2

dt k3 ab¨ b2

+

+

k4 (k2 b − aβ)

−

b2

2k3 ab˙ 2 b3

k2 2 (k3 eβ t + k4 e2β t )

−

(1 + k1 eβ t + k2 teβ t )2 β 2 (k4 − k1k3 − k2k3 t ) − 1 + (k1 + k2 t )eβ t = 0, ˙∗

∗ ∗

D + 2B C + β D



d

= 0,

+

a˙ k3 ab˙ b2

a

k4 (k2 b − aβ)

+

ab

k4 (k2 b˙ − a˙ β) ab

−

(38)

−

β (k4 − k1k3 − k2k3t )

k4 b˙ (k2 b − aβ) ab2

2

b

−

β 2 (k4 − k1k3 − k2k3 t ) b

β (k4 − k1k3 − k2k3t ) b k2 k3 β 2 e−β t

k1 + k2 t + e−β t

E˙ ∗ (t ) −

−

a˙  a

(−k2 + β e−β t )(k3 β k2t + k3 β k1 + k3k2 + k4 eβ t k2 − k4 β) (k1 + k2t + e−β t )(k1 eβ t + tk2 eβ t + 1)

+

(40) (41) (42)

(44)



+

a¨ k3 ab˙ b2

a

+

k4 (k2 b − aβ) ab

(45)

+

β (k4 − k1k3 − k2k3t ) 3

b

β 2 (k3k2 β t + k3k1 β + k3k2 − k4 β) β 3 (k3 k1 + tk3k2 − k4 ) − k1 eβ t + tk2 eβ t + 1 k1 eβ t + tk2 eβ t + 1



 k3 ab˙ k4 (k2 b − aβ) β 2 (k4 − k1 k3 − k2 k3 t ) = 0. α + E ∗ (t ) + β E ∗ (t ) + + 2 b

The proof is complete.

(39)

(43)

∗

2

dt

=−

+

ab

2k3 a˙ b˙



(k2 t + k1 )(k3k2 t − k4 + k3k1 )eβ t β 2 [k3 − (k3 k2t + k3k1 − 2k4 )eβ t ]k2 β + (1 + k1 eβ t + k2 teβ t )2 (1 + k1 eβ t + k2teβ t )2

=−

=

a

b

(46) (47) (48) (49)

2

In the following, we would like to qualitatively discuss the properties on the blowup and global existence of the perturbational solutions (5). With the aid of a = k1 + k2 t + e−β t , a˙ = k2 − β e−β t and a¨ = β 2 e−β t > 0, the first and second derivative tests will be employed to find the extremum of a(t ), t ∈ [0, +∞). After that, by comparing the extremum, we may readily determine whether there exists a finite time T ∈ (0, +∞), such that a( T ) = 0. Therefore, we can obtain the corresponding blowup and global results for the analytical solutions (5). Proposition 4. (1) For β > 0, (1a) if



k1 < −1,



k 2 > 0,

−1 < k1 < 0, k 2  0,

solutions (5) blow up at a finite time T ; (1b) if



k1 < −1, k 2  0,



−1 < k1 < 0, k2  β,







−1 < k1 < 0, k1 + kβ2 (1 − ln kβ2 )  0

or

−1 < k1 < 0, k1 + kβ2 (1 − ln kβ2 ) > 0

or

k 1  0, k 2 < 0,



k 1  0, k 2  0,

(50)

(51)

solutions (5) globally exist. (2) For β = 0, (2a) if k1 > 0, k2  0 or k1 < 0, k2  0, solutions (5) globally exist; (2b) if k1 < 0, k2 > 0 or k1 < 0, k2 > 0, solutions (5) blow up at T = − kk1 . 2

In order to intuitively exhibit the properties on the blowup and global existence of solutions (5), we give the phase diagram (details can be seen in Fig. 1). Remark 5. For accurately estimating the blowup time for β > 0, interested readers may refer to the classical article of Lambert function [18].

H. An et al. / Physics Letters A 377 (2013) 2275–2279

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Fig. 1. Phase diagram of k1 and k2 .

Acknowledgements The authors would like to thank the referees for their helpful comments and suggestions. This work is partially supported by Start-up Research Grant RG 58/2011-2012 of Hong Kong Institute of Education, the Fundamental Research Funds for the Central Universities KJ2013036 and the foundation of Nanjing Agricultural University LXYQ201201112. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

V. Camacho, R.D. Guy, J. Jacobsen, Traveling waves and shocks in a viscoelastic generalized of Burgers’ equation, SIAM J. Appl. Math. 68 (2008) 1316–1332. J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948) 171–199. D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990. L.C. Evans, Partial Differential Equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. R.J. LeVeque, Numerical Methods for Conservation Laws, second ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. J.J. Stoker, Water Waves, Wiley Classics Library, John Wiley & Sons Inc., New York, 1992. G.B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original, A Wiley– Interscience Publication. P. Popivanov, A. Slavova, Nonlinear Waves. An Introduction, Series on Analysis, Applications and Computation, vol. 4, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011, x+168 pp. M.W. Yuen, Perturbational blowup solutions to the compressible 1-dimensional Euler equations, Phys. Lett. A 375 (2011) 3821–3825. M.W. Yuen, Perturbational blowup solutions to the 2-component Camassa–Holm equations, J. Math. Anal. Appl. 390 (2012) 596–602. T. Makino, Exact solutions for the compressible Euler equation, J. Osaka Sangyo Univ. Nat. Sci. 95 (1993) 21–35. T.H. Li, Some special solutions of the multidimensional Euler equations in R N , Commun. Pure Appl. Anal. 4 (2005) 757–762. M.W. Yuen, Self-similar blowup solutions to the 2-component Camassa–Holm equations, J. Math. Phys. 51 (2010) 093524, 14 pp. M.W. Yuen, Analytical solutions to the Navier–Stokes equations, J. Math. Phys. 49 (2008) 113102, 10 pp. Z.G. Guo, Blow-up and global solutions to a new integrable model with two components, J. Math. Anal. Appl. 372 (2010) 316–327. M.W. Yuen, Self-similar solutions with elliptic symmetry for the compressible Euler and Navier–Stokes equations in R N , Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4524–4528. X.J. Zong, Properties of the solutions to the two-component B-family systems, Nonlinear Anal. 75 (2012) 6250–6259. R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996) 329–359.