Generalized Solution Of Non-Newtonian Viscous Compressible Fluid In 3D Bounded Domains

In this paper, we mainly investigate the following non-linear partial differential equations of non-Newtonian viscous compressible fluid in 3D bounded domains WhereΩ∈R³ is a smooth bounded domain, T is a certain point of time, x=(x₁, X₂, X₃),ρ=ρ(x,t) is the density, u=(u₁, u₂, u₃)(x,t) is the velocity,Ï„is the partial stress tensor and its componentÏ„_ij depends on the velocity gradient tensore whose component is e_ij=1/2〔αu_i/αx_j+αu_j/αx_i〕satisfies the following compulsory condition and the growth conditionε₁ andε₂ are positive constants in (2) and (3). f = (f₁, f₂, f₃)(x, t) is the volume power density, p = p(p, u) is the pressure. In this paper, we consider the isothermal gas, so its state equation isρ=β_ρ,常数β＞0. (4) and the boundary condition and initial condition are u|αΩ×(0,T) = 0, (ρ,u)|_t=0=(ρ₀, u_o)(X) (5) The two conditions satisfy the compatibility condition.The main results and methods are as follows:The existence of the generalized solution of the equation (1)First we construct a group of orthogonal bases and make an approximate solution, then we use energy estimate method to the solution and finally get the existence of the generalized solution of the discussed equation by the limit process.