Research On A Class Of Compressible Non-Newtonian Fluid

In this paper,we consider a class of compressible non-Newtonian fluids.(?)existence and uniqueness of local strong solutions.Here p,u,? stand for the density,velocity and pressure,respectively.Where,?0? 0,A>0,?>1.For this kind of problem,since we consider vacuum,this paper will be divided into two steps.First,we need to consider the non-vacuum case:let u0=0,The following iterative equation is construct(?)here(?)the following boundary value problem exists a unique solution u0??H01 ? H2.(?)here ?0?=J?*?0+?.We can obtain a sequence(?k,uk)of smooth solutions.Finally,the following uniform estimates are obtained.(?)where C is a positive constant only depending onM0.M0=1+|?0|H1+|g|L2+|u0|H1?H2.From the uniform estimate,we obtain the following proposition by taking the limit of k,?.As k??uk?u? in L?(0,T*;L2)? L2(0,T*;H01),?k??? in L?(0,T*;L2).As ??0 u??u in L?(0,T*;L2)? L2(0,T*;H01),???? in L?(0,T*;L2).Theorem 0.1 Assumes 0<? ??0?H1,u0 ? H01 ? H2.If there is a smooth function g ?L2(?),such that 1-(|u0x|p-2 u0x)x+?x(P0)=?01/2g,a.e.x ?I,then there exists small time T*?(0,+?)and a unique strong solution(?,u)to the initial boundary value problem,such that??C([0,T*];H1),?t?C([0,T*;L2),u?C([0,T*];H01)? L?(0,T*;H2),ut?L2([0,T*];H01),(?)ut?L?(0,T*;L2),(|ux|p-2 ux)x ? C([0,T*;L2).Secondly,we consider the case with vacuum,and we need to regularize the initial value of the equation.For some 0<?<<1,we obtain ?0?=J?*?0+?,u0??H01 ? H2 is a solution of Lpu0?=(?0?)1/2g?-?x(?0?).Then the equation has a unique solution and satisfies the following uniform estimate(?)where C is a positive constant only depending onMO.If we take the limit of ?,as ??0 u??u in L?(0,T*;L2)? L2(0,T*;H01),???? in L?(0,T*,L2).we can get the following theorem.Theorem 0.2 Assumes 0??0?H1,u0 ? H01 ? H2.If there is a function g ? L2(?),such that-(|u0x|p-2u0x)x+?x(?0)=?01/2g a.e.x ?I then there exists small time T*?(0,+?)and a unique strong solution(?,u)to the initial boundary value problem,such that??C([0,T*];H1),?t ? C([0,T*];L2),u ? C([0,T*];H01)? L?(0,T*;H2),ut?L2([0,T*];H01),(?)ut ? L?(0,T*;L2),(|ux|p-2 ux)x ? C([0,T*];L2).