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,π stand for the density,velocity and pressure,respectively.Where,ρ0≥ 0,A>0,γ>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 u0ε∈H01 ∩ H2.(?)here ρ0δ=Jδ*ρ0+δ.We can obtain a sequence(ρk,uk)of smooth solutions.Finally,the following uniform estimates are obtained.(?)where C is a positive constant only depending onM0.M0=1+|ρ0|H1+|g|L2+|u0|H1∩H2.From the uniform estimate,we obtain the following proposition by taking the limit of k,ε.As k→∞uk→uε in L∞(0,T*;L2)∩ L2(0,T*;H01),ρk→ρε in L∞(0,T*;L2).As ε→0 uε→u in L∞(0,T*;L2)∩ L2(0,T*;H01),ρε→ρ in L∞(0,T*;L2).Theorem 0.1 Assumes 0<δ ≤ρ0∈H1,u0 ∈ H01 ∩ H2.If there is a smooth function g ∈L2(Ⅰ),such that 1-(|u0x|p-2 u0x)x+πx(P0)=ρ01/2g,a.e.x ∈I,then there exists small time T*∈(0,+∞)and a unique strong solution(ρ,u)to the initial boundary value problem,such thatρ∈C([0,T*];H1),ρt∈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<δ<<1,we obtain ρ0δ=Jδ*ρ0+δ,u0δ∈H01 ∩ H2 is a solution of Lpu0δ=(ρ0δ)1/2gδ-πx(ρ0δ).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 δ,as δ→0 uδ→u in L∞(0,T*;L2)∩ L2(0,T*;H01),ρδ→ρ in L∞(0,T*,L2).we can get the following theorem.Theorem 0.2 Assumes 0≤ρ0∈H1,u0 ∈ H01 ∩ H2.If there is a function g ∈ L2(Ⅰ),such that-(|u0x|p-2u0x)x+πx(ρ0)=ρ01/2g a.e.x ∈I then there exists small time T*∈(0,+∞)and a unique strong solution(ρ,u)to the initial boundary value problem,such thatρ∈C([0,T*];H1),ρt ∈ 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).