A Class Of Compressible Non-Newtonian Fluids With Vacuum

With the development of the technology, we find that there are a lot of fluids which belong to non-Newtonian fluids in the process of product and nature. The classical non-Newtonian fluids are macromolecule melt and macromolecule liquor, and the all kinds of slurry and suspend liquor, paint, dope, palette and biology fluids, for example, in the body of people and animals, the blood, the synovia of arthrosis cavity, lymph liquor, cell liquor, brain liquor etc, which are provided with the property of non-Newtonian fluids. So the non-Newtonian fluids exist widely in nature.In this paper, we study, in one dimension, local existence of a unique solution to a class of compressible non-Newtonian fluids with initial and boundary conditions. We choose Γ (denote viscous terms of the momentum equation) with the following formwhere 1 < p < 2, then μ = (p -1)|u_x|^p-2. This model with this kind viscous term captures the shear thinning fluid. Because of -1 < p - 2<0, the momentum equation has singularity, which brings us many difficulties. Moreover, Vacuum may appear , which brings more difficulties. This model with the viscous term captures the shear thinning fluid. Then, we choose another kind of Γ with the following formwhere p ≥ 2, μ₀> 0. We study local existence of a unique solution to the class of compressible non-Newtonian fluids with initial and boundary conditions and vacuum. Thismodel with the viscous term captures the shear thickening fluid. Firstly, we study the following initial boundary value problemPt + (pu)x = 0,{pu)t + (pu2)x - {\ux\p-2 ux)x + irx = pf (x,t)enT) (1)ir = tt(p) = Ap7, A > 0, 7 > 1,withf (P.u)Lo = (Po.tto) x e [0,1],1where QT = I x (0,T) = (0,1) x (0,T). p e (1, 2), p# ^ 0, and for some g e L2(/), such that(|uOx|p"2uox)x = tti(po) - (po)' 9- (3)Since the momentum equation has singularities, and vacuum may appear, we consider the problem in two steps. We firstly consider the case of non-vacuum and regularizing the viscous term. Then we study the original case with vacuum .For the case of non-vacuum, we consider the following initial boundary problemPt + {pu)x = 0,(pu)t + (Pu2)x-(\ux\p-2ux)x + ?rx = pf (x,t)enT, (4)7T = Ap^, A>0, 7 > 1,withf x €[0,1],(5) { u\x=o=u\x=1 = 0 te[O,T],where fj > 0. Due to the singlarities, we need regularize the equation, then we construct the following approximating problem: Let u° = 0, and for k = 1, 2, ? ? ? ,pk + uk^lpkx + uk^lpk = 0, (6)okuk + okuk^luk - ( "K"'XJ_<sub>— \ uk + 7rft = oK f â–  (x t) eQ,t (7)p ut -1- p u ux \ f_<sub>kA2 | ^ ] ux -|-7rx — p j, ^,ijt iiTi \()pk\t=o = Po, uk\t=o = ueo, x € [0, 1]; uk(0,t) =uk(l,t) = 0, te [0,r],where it§ € Hq(I)(MI2{I) is the smooth solution of the following boundary value problemwhere but it is difficult for p e (1, 2), we must separately discuss the cases for 1 < p ^ |, | < p < 2, then we will finally get the following uniform estimate to the approximate solutions:ess sup ( \pk{t)\Hi{I) + \uk{t)\ liP ff2rn + Iv^^I^Il2^) + \pk(t)\L2{I) )(9)where C is a positive constant, depending only on M0 = 1 + \po\Hl(I) + lfliL2(7) + |/Il°°(0,According to (9), by taking limit about k and e, then we obtain the following theorem:Theorem 1 Assume that po>/ are both sufficiently smooth, po ^ 8 for some given constant 8 > 0, and f(O,t) = /(l,i) = 0, u0 € H^(I) DH2(I). If there exists some g e L2(I), g(0) = g(l) = 0, such that (J\p2) + kx(pq) = (po)2P a.e. in I.Then there exists a T* 6 (0, +oo), such that the initial boundary problem (4)⁵ has aunique solution (p, u) in ï¿¡It, satisfyingpt 6 C([O,T*];L2(I)), ut € L2(0, 2ux) eC([0,T*};L2(I)).According to Theorem 1, we may prove existence and uniqueness of the solution to the initial and boundary problem with vacuum. So we need regularize the initial value of the original problem, such that they satisfy the conditions in Theorem 1.Since p0 is smooth, for small 0 < 8 ? 1> ps0 = p0 + 6, let us0 e H^(I) (1 H2(I) be a unique solution of the following boundary value problem' * e (0, i),where g$ € C?(I) satisfies\9s\l*(i) < Mi2(/), ]im |&$ - #|z,2(/) = 0.So we obtain that (ps, us) is a unique solution of the problem (lO)-(ll), and the following uniform estimate holds \p6(t)\HHl) nwhere C is a positive constant, depending only on Mq. According to the uniform estimate, we take limit about <5, then obtain the following theorem:Theorem 2 Assume that po IS sufficiently smooth, f e Loo(0, T; L2(I)), ft e L°°(0, T; L2(J)), po > 0, /(0,t) = /(l,t) = 0, u0 € Hj(I)n52(;). // tfiere ewis some g ï¿¡ L2(I), such that , + 7rx(p0) = (po)5 5 a. e. in I.Then there exists a time T* G (0, +00) and a unique strong solution (p, u) satisfying (l)-(2) such that,T*];H\l)), pteC([0,T*};L2(I)), ut e L2(0,%;H(KrH e c([o, r,]Finally, we consider the following compressible shear thichening fluidPt + (pu)x = 0,{pu)t + {Pu2)x - \(u2x + mo) ^ ux I +nx = pf (x, t) € fir, (10)7T = ^p7, ^4 > 0, 7 > 1,with the initial and boundary conditionsf {p,u)\t=o = (po1uo) xe [0,1],I ?|B=o = <=l = ° te[°'Tlwhere p ^ 2, /xq > 0.Firstly, we prove existence and uniqueness of the solution to the initial boundary value problem with non-vacuum, and we may use the same technique as Theoreml, it is different from it that we don't need to regularize the viscous term. Then we obtain the following result:Theorem 3 Assume that po> / o^e both sufficiently smooth, po ^ 5 for some given constant 5 > 0, and f(Q,t) = f(l,t) — 0, no € Hq(I) D H2(I). If there exists some g e L2(/), 0(0) = 9(1) = 0, such that 5 g.Then there exists a time T* e (0,T) and a unique solution {p,u) satisfying (10)-(ll) such that } ut € L2(0,T*; ^...