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 ?=?(x,t),u=u(x,t),?=?(x,t)and?(?)=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 that?(x,0)=?0(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 ?(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:?0?0,u0?L2(R),u0x?L2(R)?Lp(R),?0 1/2 u0?L2(R),(0.6)Where(t,x)?(0,T0]x R.Further,for constant p>2,q? p.Assume that ?0 also satisfies??0?L1(R)n H1(R)n W1,q(R),Where?(?)(e+x2)1+?0,(0.7)and ?0 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??M ?(x,t)dx?1/4 ?R?0(x)dx,for some constant M>0 and ?M(?){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 equation?(t)(?)1+?u?L2(?R)+?ux?L2(?R)+?ux?Lp(?R)+??1/2u?L2(?R)+????L1(?R)?H1(?R)?W1,q(?R)·Secondly,it can be known from the calculation that there are normal numbers T0 and N,so that sup 0?s?T0(????L1(?R)?H1(?R)?W1,q(?R))?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.