Finite Velocity of the Propagation of Perturbations for a Class of Non-Newtonian Fluids
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Finite Velocity of the Propagation of Perturbations for a Class of Non-Newtonian Fluids Hongjun Yuan · Huapeng Li
Received: 5 July 2011 / Accepted: 3 September 2012 / Published online: 12 September 2012 © Springer Science+Business Media B.V. 2012
Abstract In this paper, we consider the initial boundary value problem of a class of nonNewtonian fluids. We obtain that finite velocity of the propagation of perturbations. Keywords Existence · Non-Newtonian fluids · Finite propagation · Morser’s iteration
1 Introduction and Main Results We are concerned with the one-dimensional equations of non-Newtonian fluids of the form (x, t) ∈ QT , ρt + (ρu)x = 0, (1.1) (ρu)t + (ρu2 )x − (|ux |p−2 ux )x + πx = 0, (x, t) ∈ QT , with the initial and boundary conditions (ρ, u)|t=0 = (ρ0 , u0 ), u|x=−1 = u|x=1 = 0,
x ∈ [−1, 1],
(1.2)
where ρ and u stand for the density and the velocity of the fluid, respectively, while π = π(ρ) denotes the pressure, QT = I × (0, T ), I = (−1, 1). Mathematically, the study of non-Newtonian fluid mechanics is of great significance. In the field of chemistry, bio-mechanics, glaciology, geology, and hemorheology, a large number of problems may arise with non-Newtonian fluids, which has sparked the increasing interest in the study of the non-Newtonian fluids, see [1–3]. Up to now, the results about non-Newtonian fluids are quite few. Recently, H. Yuan and X. Xu [4] established an existence result on the local solutions with nonnegative densities. H. Yuan · H. Li () Institute of Mathematics, Jilin University, Jilin, Changchun 130012, P.R. China e-mail: [email protected] H. Yuan e-mail: [email protected]
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H. Yuan, H. Li
They assumed that the initial data satisfy a compatibility condition which was, roughly √ speaking, equivalent to the boundedness of ρut (0)L2 (I ) . Then using a classical energy method, they obtained local existence and uniqueness of solution. For related results we refer the reader to [5–10] and the references therein. For the Newtonian fluids, there are huge literatures on the existence and uniqueness of solutions. If vacuum is taken into account, Huang, Li and Xin [11] established the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or non-vacuum. For other results we refer the reader to [12–17] and the references cited therein. In this paper, we establish finite velocity of the propagation of perturbations by Moser’s iteration techniques (see [20, 21]) and the local (in time) existence of weak solution for the system (1.1)–(1.2). It is well known that if 2 < p < 3, the second equation of (1.1) is always with degeneration, which is a difficult point. Moreover, the system (1.1) is with strong nonlinearity, so we are facing another difficulty. We will assume that the initial data (ρ0 , u0 ) satisfies the following condition
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