Regularity and Exponential Stability of the pth Power Newtonian Fluid in One Space Dimension
In this chapter, we are interested in the regularity and exponential stability of solutions in H i (i = 2,4) for a pth power Newtonian fluid undergoing one-dimensional longitudinal motions.
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Regularity and Exponential Stability of the πth Power Newtonian Fluid in One Space Dimension 3.1 Introduction In this chapter, we are interested in the regularity and exponential stability of solutions in π» π (π = 2, 4) for a πth power Newtonian ο¬uid undergoing one-dimensional longitudinal motions. We assume that the pressure π«, in terms of the absolute temperature π and the speciο¬c volume π’, is given by π«=
π π’π
(3.1.1)
with the pressure exponent π β₯ 1. The balance laws of mass, momentum, and energy in Lagrangian form are as follows: π’π‘ = π£π₯ , ( π£π₯ ) , π£π‘ = βπ« + π π’ π₯ ( ) ) ( π£π₯ ππ₯ ππ£ ππ‘ = βπ« + π π£π₯ + π
. π’ π’ π₯
(3.1.2) (3.1.3) (3.1.4)
Here, π’, π£, π are speciο¬c volume, velocity, and absolute temperature, respectively. The positive constants ππ£ , π, π
represent speciο¬c heat, viscosity and conductivity, respectively. Since the magnitude of the speciο¬c heat ππ£ plays no role in the mathematical analysis of the system, in what follows we will assume the scaling ππ£ = 1.
Y. Qin and L. Huang, Global Well-posedness of Nonlinear Parabolic-Hyperbolic Coupled Systems, Frontiers in Mathematics, DOI 10.1007/978-3-0348-0280-2_3, Β© Springer Basel AG 2012
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Chapter 3. Regularity and Exponential Stability
We consider a typical initial boundary value problem for (3.1.2)β(3.1.4) in the reference domain {(π₯, π‘) : 0 < π₯ < 1, π‘ β₯ 0} under the initial conditions π’(π₯, 0) = π’0 (π₯), π£(π₯, 0) = π£0 (π₯), π(π₯, 0) = π0 (π₯),
π₯ β [0, 1]
(3.1.5)
and boundary conditions π£(0, π‘) = π£(1, π‘) = 0, ππ₯ (0, π‘) = ππ₯ (1, π‘) = 0.
(3.1.6)
Obviously, when π = 1, (3.1.1) reduces to the case of a polytropic ideal gas (see (3.4.1)). This chapter mainly continues to the case of π > 1, which was selected from [61]. The notation in this chapter is standard. We put β₯ β
β₯ = β₯ β
β₯πΏ2 [0,1] . Subscripts π‘ and π₯ denote the (partial) derivatives with respect to π‘ and π₯, respectively. We use πΆπ (π = 1, 2, 4) to denote a generic positive constant depending on the π» π [0, 1] norm of initial data (π’0 , π£0 , π0 ), min π’0 (π₯) and min π0 (π₯), but independent of π₯β[0,1]
π₯β[0,1]
time variable π‘. β«1 For convenience and without loss of generality, we may assume 0 π’0 (π₯) ππ₯ = 1. Then from conservation of mass and boundary condition (3.1.6), we have β« 1 π’(π₯, π‘) ππ₯ = 1. (3.1.7) 0
We deο¬ne two spaces as { 2 π»+ = (π’, π£, π) β π» 2 [0, 1] Γ π» 2 [0, 1] Γ π» 2 [0, 1] : π’(π₯) > 0, π(π₯) > 0, } βπ₯ β [0, 1], π£(0) = π£(1) = 0, πβ² (0) = πβ² (1) = 0 and
{ 4 = (π’, π£, π) β π» 4 [0, 1] Γ π» 4 [0, 1] Γ π» 4 [0, 1] : π’(π₯) > 0, π(π₯) > 0, π»+ } βπ₯ β [0, 1], π£(0) = π£(1) = 0, πβ² (0) = πβ² (1) = 0 . Now our main results in this chapter read as follows.
2 and the compatibility conditions Theorem 3.1.1. Suppose that (π’0 , π£0 , π0 ) β π»+ 2 hold. Then there exists a unique generalized global solution (π’(π‘), π£(π‘), π(π‘)) β π»+ to the problem (3.1.2)β(3.1.6) verifying that for any (π₯, π‘) β [0, 1] Γ [0, +β),
0 < πΆ1β1 β€ π’(π₯, π‘),
π(π₯, π‘) β€ πΆ1
(3.1.8)
and for any π‘ > 0, β₯π’(π‘) β 1β₯2π» 2 + β₯π£(π‘)β₯2π» 2 + β₯π(π‘) β πβ₯2π» 2 + β₯π£π‘ (π‘)β₯2 + β₯ππ‘ (π‘)β₯2 (3.1.9) β« π‘( ) β₯π’ β 1β₯2π» 2 + β₯π£β₯2π» 3 + β₯π β πβ₯2π» 3 + β₯π£π‘ β₯2π» 1 + β₯ππ‘ β₯
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