A Non-newtonian Flow With Variable Viscosity Coefficient

In this paper,we consider the following one-dimensional compressible non-newtonian with variable viscosity where t≥ 0,x∈R,p>2,the unknown functions ρ=ρ(x,t),u=u(x,t),μ=μ(x,t)andπ(ρ)=Apy(A>0,y>1)denote the density,the velocity,the coefficient of viscosity and the pressure.Without loss of generality,we set A=1.We consider the Cauchy problem with(p,u)vanishing at infinity.For given initial functions,we require thatρ(x,0)=ρ0(x),u(x,0)=u0(x),x∈R.(0.2)We assume f(t,x,y),f(t,x,y)∈C∞((0,1]×(-∞,+∞)×(-∞,+∞)).We denote that h1(t,x)is the derivative of f(t,x,u)to t,namely h1(t,x,y)=(?)[f(t,x,y)]/(?)t,;h2(t x)is the derivative of f(t,x,u)to x,namely h2(t,x,y)=(?)[f(t,x,y)]/(?)x;h3(t,x)is the derivative of f(t,x,u)to y,namely h3(t,x,y)=(?)[f(t,x,y)]/(?)y.Assume f=f(t,x,y)for all the A and(t,x,y)∈(0,1]x(-∞,+∞)x(-∞,+∞),satis-fying Where c1,c2,c3,c4,c5,c6 are positive constant,and there exists g(t,x).Assume H1(t,x)≥0,H2(t,x)≥0,H3(t,x)≥0,for all(t,x)∈(0,1]×(-∞,+∞),satis-fying Where q≥ p is a positive constant and c7,c8,c9,c10 are positive constant.Function μ(t,x),(t,x)(0,1]×(-∞,+∞))satisfying Where c11,c12 are positive constant.We get the following results Theorem 1 Assuming that f(t,x,y)as(0.3)-(0.4)and assume that the initial data(p0,uo)satisfies:ρ0≥0,u0∈L2(R),u0x∈L2(R)∩Lp(R),ρ0 1/2 u0∈L2(R),(0.6)Where(t,x)∈(0,T0]x R.Further,for constant p>2,q≥ p.Assume that ρ0 also satisfiesΦρ0∈L1(R)n H1(R)n W1,q(R),WhereΦ(?)(e+x2)1+τ0,(0.7)and τ0 is a positive constant.Then there exists a positive time T0(T0≤1)such that the problem(0.1)-(0.2)has a unique strong solution(p,u)on R x(0,T0]satisfying Moreover,inf 0≤t≤T0∫ΩM ρ(x,t)dx≥1/4 ∫Rρ0(x)dx,for some constant M>0 and ΩM(?){x∈R||x|<M}.First,we need to carry out the definition introduced by prior estimation for the solution of(0.1)-(0.2)initial boundary value problem,namely,the solution of the approximation equationλ(t)(?)1+‖u‖L2(ΩR)+‖ux‖L2(ΩR)+‖ux‖Lp(ΩR)+‖ρ1/2u‖L2(ΩR)+‖Φρ‖L1(ΩR)∩H1(ΩR)∩W1,q(ΩR)·Secondly,it can be known from the calculation that there are normal numbers T0 and N,so that sup 0≤s≤T0(‖Φρ‖L1(ΩR)∩H1(ΩR)∩W1,q(ΩR))≤NFinally,according to the above estimation,combined with truncation technique and stan-dardized proof method,the result key words of theorem 1 can be obtained.