South Asian Journal of Mathematics 2012, Vol. 2 2): 148 153 www.sajm-online.com ISSN 2251-1512 RESEARCH ARTICLE Decay rate of the compressible Navier-Stokes equations with density-dependent viscosity coefficients Ruxu LIAN 1, Xuemei SHEN 2 1 Department of Mathematics, North China University of Water Resources and Electric Power, Zhengzhou, China 2 School of Mathematics and Computer Science, Xinyang Vocational and Technical College, Xinyang, China 464000 E-mail: ruxu.lian.math@gmail.com Received: 10-03-2011; Accepted: 12-20-2011 *Corresponding author The research of R.X. Lian is partially supported by the National Natural Science Foundation of China No. 10871134) and the Program for New Century Excellent Talents in University support of the Ministry of Education of China NCET-06-0186. Abstract The main aim of this paper is to investigate the large time behavior of the smooth solutions to the three-dimensional isentropic compressible Navier-Stokes equations CNS) with density-dependent viscosity coefficients. For a constant γ γ > 1), we show the decay rate of the L γ norm of density. Key Words Navier-Stokes equations, large time behavior, decay rate MSC 2010 35Q30, 35Q35 1 Introduction We consider the general isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients ρ t + divρu) = 0, 1.1) ρu) t + divρu u) + p divµρ)du)) λρ)divu) = 0, where t > 0, x R N, N 1, ρ 0 and u denote the density and velocity respectively. Pressure function is taken as pρ) = ρ γ with γ > 1, Du) = u+ u) T )/2, µρ) and λρ) are the Lamé viscosity coefficientsdepending on the density) satisfying µρ) 0, µρ) + Nλρ) 0. 1.2) The prototype of 1.1) is the physical model of the viscous Saint-Venant system used widely in geophysical flow [16] to simulate the motion of water surface in shallow region. Without the loss of generality, we simply consider 1.1) in three dimension for the case Du) = u, µρ) = ρ, λρ) = 0, 1.3) Citation: R. Lian, X. Shen, Decay rate of the compressible Navier-Stokes equations with density-dependent viscosity coefficients, South Asian J Math, 2012, 22), 148-153.
South Asian J. Math. Vol. 2 No. 2 namely the systems ρ t + divρu) = 0, ρu) t + divρu u) + p divρ u) = 0, x, t > 0. 1.4) Note here that the case γ = 2 in 1.4) corresponds to the viscous Saint-Venant system. When there is no external or internal force involved, there is huge literature on the problem of large time behavior of the global smooth solutions to the compressible Navier-Stokes equations. For multi-dimensional Navier-Stokes equations with constant viscosity coefficients, Matsumura and Nishida [13, 14] obtain the H s global existence and time-decay rate of strong solutions. The large time decay rate of the global solution in multi-dimensional half space or exterior domain is also considered for the compressible Navier-Stokes equations by Kagei and Kobayashi [7, 8], Kobayashi and Shibata [9], and Kobayashi [10], in which under small initial perturbation in Sobolev space, the optimal L 2 time-decay rate in three dimension is established. The Green s function of an artificial viscosity system associated with the isentropic Navier-Stokes equations is investigated by Hoff and Zumbrun [5, 6] to study the wave propagation for compressible fluids, and the L time-decay rate of the diffusive waves is deduced. Liu and Wang [12] is also concerned with the Green s function for the isentropic Navier-Stokes equations and derive a pointwise convergence of solution to diffusive waves with the optimal time-decay rate in odd dimension. When exterior and internal potential forces are considered, the global existence and the large time behavior of a strong solution to the Navier-Stokes equations are obtained by Matsumura and Nishida [15]. The large time decay rate of the global solutions to the isentropic compressible flow is established by Deckelnick [3, 4], Shibata and Tanaka [17] and Ukai et al [18]. For the non-isentropic compressible flow, Duan [1] investigates the optimal L p convergence rate in R 3 and Duan [2] studies the optimal L p L q convergence rate with 1 p < 6/5 and 2 q 6 in R 3 for isentropic compressible flow. The main purpose of this paper is concerned with the large time behavior of the smooth solutions for the initial boundary value problem to the three-dimensional compressible Navier-Stokes equations where the existence of the global smooth solutions is assumed. The method used in [11] is applied here to obtain the decay rate of the pressure by means of the function introduced in [19]. The rest of the paper is as follows. In Section 2, the main results about the decay rate of ρ Lγ ) are stated in details. The a-priori estimates of the compressible Navier-Stokes can be acquired in section 3. The main results of the theorem are proved in section 4 finally. 2 Main Results We present the main results about the decay rate of the L γ norm of density for the initial boundary value problem to the equations 1.4) in this section. We consider the Cauchy problem for the three-dimensional isentropic compressible Navier-Stokes equations 1.4) with the initial data and boundary conditions 149
X. Lian, et al: Decay rate of the compressible Navier-Stokes equations with density-dependent viscosity coefficients ρ, u)x, 0) = ρ 0, u 0 ), x, ux, t) = 0, x, t 0, 2.1) and the initial data satisfies ρ 0 0 L γ ), ρ 1 2 0 ) L 2 ), ρ0 u 0 L 2 ). 2.2) Then we will show the theorem Theorem 2.1. Let γ > 1. Assume that the initial data satisfies 2.2) and ρ, u) is the global smooth solution to the Cauchy problem 1.4) and 2.1), then it holds for ν 0, 1) that C1 + t) 1 γ, γ 4/3, ρx, t) Lγ ) C1 + t) 1 γ +ν, γ = 4/3, 2.3) C1 + t) 3γ 1) γ, 1 < γ < 4/3. Remark 2.1. For general viscosity coefficients µρ) = ρ α, λρ) = α 1)ρ α with α > 0, the decay rate of the density can be obtained too. Since the proof of Theorem 2.1 for α = 1 in the present paper can also be applied after some modification. 3 The a-priori estimates We will deduce the a-priori estimates about the Cauchy problem 1.4)-2.1). Similarly to the arguments used in [11], we can establish the following the a-priori estimates, we omit the details here. Lemma 3.1. Let T > 0. Under the assumptions of Theorem 2.1, it holds for any smooth solution ρ, u) to the Cauchy problem 1.4) and 2.1) that 1 2 ρu 2 + 1 T 1 γ 1 ργ) dx + ρ u 2 dxdt = 0 2 ρ 0 u 0 2 + 1 ) γ 1 ργ 0 dx, 3.1) ρu 2 + ρ 1 2 ) 2 dx + 1 γ 1 ργ) dx + C ρ 0 u 0 2 + ρ 1 2 0 ) 2 + 1 γ 1 ργ 0 where C > 0 is a positive constant independent of time. T 0 ρ u 2 + ρ γ 2 ρ 2) dxdt ) dx, 3.2) 4 Proof of main results We will prove the Theorems 2.1 in this section based on the results obtained in previous section. 150
South Asian J. Math. Vol. 2 No. 2 Proof. Step 1. The estimate of H t). We introduce the following functional in Eulerian form as [19]. Denote Ht) = = ) 2ρx, 2 x 1 + t)ux, t) t)dx + 1 + t)2 γ 1 x 2 ρdx 21 + t) x ρu)dx + 1 + t) 2 ρ γ x, t)dx ρu 2 + 2 γ 1 ργ) dx :=I 1 + I 2 + I 3 > 0. 4.1) Differentiating 4.1) with respect to t, using 1.4), 2.1) and 2.2), we have I 1 t) = 2 x ρu)dx, 4.2) I 2t) = 2 x ρu)dx 21 + t) ρu 2 + 3ρ γ )dx + 21 + t) ρ udx, 4.3) I 3 t) = 21 + t) ρu 2 + 2 γ 1 ργ) dx 21 + t) 2 ρ u 2 dx. 4.4) Finally, combining 4.2), 4.3) and 4.4), we deduce H 25 3γ) t) = 1 + t) ρ γ dx + 21 + t) ρ udx 21 + t) 2 ρ u 2 dx γ 1 25 3γ) 1 + t) ρ γ dx + 1 ρdx. 4.5) γ 1 2 Step 2. The large time behavior of the pressure. After finding the conclusion 4.5), we will deduce that if γ 5/3, it holds that then we obtain If 1 < γ < 5/3, we have from 4.1) and 4.5) that H t) C, 4.6) ρ γ dx C1 + t) 1. 4.7) H t) 5 3γ)1 + t) 1 Ht) + C. 4.8) In the case 1 < γ < 4/3, the application of Gronwall s inequality to 4.8) gives rise to ) Ht) C 1 + t) 5 3γ + 1 + t), 4.9) namely Ht) C1 + t) 5 3γ, 4.10) then we have ρ γ dx C1 + t) 3γ 1). 4.11) In the case 4/3 < γ < 5/3, in a similar argument, we deduce Ht) C1 + t), 4.12) 151
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