Existence And Uniqueness Of Global Strong Solution For A Class Of Compressible Non-Newtonian Fluids

Fluid mechanics,which is a branch of mechanics,focus on the study of the law of flow motion and its interaction with surrounding objects.Newtonian fluids represented by air and water have been widely studied and a relatively complete theoretical system has been formed.The main characteristic of Newtonian fluid is that the relationship between shear stress and shear strain rate satisfies the linear relationship,while the non-Newtonian fluid does not satisfy the linear relationship.The study of non-Newtonian fluid not only raises challenging new problems for fluid mechanics,but also has a wide application background.It exists in all fields closely related to national economic development and daily life,such as petroleum industry,chemical industry,food industry and biomedical engineering.This makes people increasingly interested in the study of non-Newtonian fluids.At present,the research results of non-Newtonian fluids are few and most of them focus on the study of local solutions.In this paper,we mainly consider two class of compressible non-newtonian fluid equation.In Chapter 3,we are concerned with the following compressible shear thin-ning flow in one-dimensional bounded intervals:(?)subject to initial and boundary conditions(?) where the unknown functions ?=?(x,t),u=u(x,t),?(?)=a??(a>0,?>1)and f=f(x,t,u)denote the density,velocity,pressure and external force respectively;the initial density ?0?0;?:=(0,1),p?(5/3,2).For the above systems,if the external force f=f(x,t,y)? C1([0,1]×[0,+?)×(-?,+?))satisfies the following conditions(?) where C1,C2 and C3 are constants,E0:=?(1/2?0u02+a?0?/?-1)dx,then we can obtain the next theorem:Theorem 1 Assume that 5/3<p<2 and(?0,u0)satisfies the following condi-tions 0??0?H1(?),u0?H01(?)?H2(?),p0??,and the following compatibility condition-(|u0x|p-2u0x)x+px(?0)=?01/2g,a.e.x ??,where g?L2.Then there exist ?=?(a,?,?)>0,if the initial energy E0 satisfies Eo<?,the initial boundary value problem(1)-(2)exists a unique strong solution(?,u)such that (?)and for 0<T<?,(?)where q?3-2/p.Under the conditions that the external force term was relatively small,the priori estimates were given by the method of hypothesis closure,weighting and energy esti-mates.Hence we proved the existence and uniqueness of the global strong solutions and at the same time obtained the long time behavior.Using this method,in Chapter 4,we are concerned with the following compressible shear thickening flow in one-dimensional bounded intervals:(?)subject to the initial and boundary conditions (?)Here the unknown functions ?=?(x,t),u=u(x,t);the initial density p0?0;?:=(0,1);p>2 and ?0>0 are given constants.We now demonstrate our main result as follows:Theorem 2 Assume p>2 and the initial value(?0,u0)satisfies?0?H1(?),u0?H01(?)?H2(?),0??0??,and the compatibility condition-((u0x2+?0)p-2/2u0x)x+?x(?0)=?01/2g,a.e.x??,wherc g ?L2(?).Then there exists a constant ?=?(a,?,?)>0,such that if the initial energy E0:=?01(1/2?0u02+a?0?/?-1)dx satisfies E0??,then the problem(3)-(4)has a unique global strong solution(?,u),which satisfies (?)for 0<T<?.