Finite Velocity Of The Propagation Of Perturbations For A Class Of Non-Newtonian Fluids With Variable Pressure

Fluid mechanics is a branch of mechanics,it is the study of fluid phenomenon and the related mechanical behavior,it according to the different"mechanical model is divided into"ideal fluid mechanics,compressible fluid mechanics,non-newtonian fluid mechanics,etc.Nowadays,non-newtonian fluid mechanics has become part of the system of basic science,so the study of non-newtonian fluid,it is very necessary.In recent years,due to the rapid development of science and technology,the research and application of non-Newtonian flow has become more and more important.In addition to mathematics,non-Newtonian fluids are also very common in our daily life,such as blood and lymph in humans.A variety of body fluids,Chemically applied mud,oil,paint and coating,as well as a large number of non-Newtonian problems in Biomechanics,Hydrology,Hemorheology,Geology,etc.At present,there are not many research results related to non-newtonian fluids,and most of the existing results mainly study local solutions,which stimulates people's interest and understanding of the research on non-newtonian fluids.In the future,people need to know more about and study the important characteristics of non-newtonian fluids.In this paper,we will consider the following one-dimensional compressible non-Newtonian fluid equations with variable pressure(?)With the following initial value conditions(?)? and u respectively the density and speed of the fluid,and unknown functions ?=?(x,t),u=u(x,t),QT=I×(O,T),/=(-1,1).? indicates the pressure of the fluid,and we assume ?=?(x,t),?(·,t)?C02(-1.1),(?)t?(0,T),?x(·,T0)? C01(-1.1),T0?[0,+?)meet the following conditions:(?)Where C1,C2 is the given positive constant.We assume that the initial value(?0,u0)satisfies the following regularity conditions:?0??0?H1(I),u0?H2(I),(0.4)where ?0 is a positive constant.Firstly,we define the weak solution as follows.Definition 1 The pair of(?,u)is called a weak solution to the initial boundary value problem(0.1)-(0.2),if the following conditions are satisfied:(i)0?????C([0,T];L2(I))?L?(0,T;H1(I)),(|ux|p-2ux)x?L2(0,T;L2(I)),?t?L2(0,T;L2(I)),u?C([0,T];L2(I)?L2(0,T;H2(I)),ut?L2(0,T;L2(I)).(ii)For all ?? C([0,T];L2(I))? L?(0,T;H1(I)),?t?L2(0,T;L2(I)),for a.e.t ?(0,T),we can get:(?)(iii)For all ??C0?(QT),for a.e.t ?(0,T),we can get:(?)(?)Then we give the main theorems of this paper by(0.1),(0.2),(0.4)as follows:Theorem 1(Finite Propagation of Perturbations)Let ?=?(x,t)is continuous dif-ferentiable fun+tion,satisying(0.3).Let 2<p<3,In QT(T ?(0,+?)),(?,u)is the initial value a weak solution to the problem(0.1)-(0.2),and satisfy the initial values of the definitions 1 and(0.4).Further assume that u0 and ? Meet the following conditions supp u0(?)IR0,?=?(x,t).(x,t)e(I\IR1)×(0,T),(0.7)Where R0 and R1 are fixed constants,IR0 means that I is centered at the origin and has a radius of R0,and IRo=(-R0,R0),where-1<R0<1;IR1=(-R1,R1),Where-1<R1<1.There is a normal number R*associated with p,T,M*,and-1<R*<1,supp u(·,t)[(-R+,R*),a.e.t?(0,T),(0.8)M*is defined in Lemma 3.7.Remark 1:Theorem 1 indicates that the velocity support of the disturbance propagating to the initial boundary value problem(0.1)-(0.2)is finite.In other words,if we give a small disturbance to a fluid that is initially stable,then for a finite amount of time,the velocity has a finite boundary.Remark 2:Theorem 1 solution exists,and the following theorem can be introduced.Theorem 2(Local Existence of Solution)Assume that 2<p<3,the initial data(?0,u0)satisfy(0.4),In I x(0,T),pressure ?=?(x,t),then for the initial value problem(0.1)-(0.2)There is a small time T*?(0,+?)and a weak solution(?,u)meets the following conditions(?).