A Class Of Non-Newtonian Fluids With Non-Linear External Forces

Fluid mechanics is the branch of mechanics. With the development of science and tech-nology, there are a lot of non-Newtonian fluids in the process of product and life. The prob-lem of the fluid dynamics, especially the nonlinear problems appeal to many mathematicians. The fluids obey conservation lows of mass and momentum conservation. The fluid dynamics study different models, which are the ideal fluid dynamics, the viscous fluid dynamics, the non-Newton fluid dynamics, the compressible fluid dynamics and the incompressible fluid dynamics.In this paper, we study existence and uniqueness of solutions to the following non-Newtonian fluids with non-Linear external forces: with the following initial-boundary value: where p, u, π is the density, velocity, and pressure of the non-Newtonian fluids,f=f(u, x, t) is an external force, p>2,μ0>0, ρ0≥0, ΩT=I×(0, T),I=(0,1).In this paper,we consider the following definition of solutions.Definition1The pair (p,u) is called a solution to the initial boundary value problem (0.1)-(0.2), if the following conditions are satisfied: (ⅰ)(ⅱ) For allφ∈C ([0, T];H1(I)),φt∈L∞(0, T; L2(I)), for a.e. t∈(0, T), we have(ⅲ) For all φ∈C([0, T];H101(I))∩L∞(0, T;H2(I)), φt∈L2(0, T;H01(I)), for a.e. t∈(0, T), we have:Because of vacuum may appear, momentum equation with singularities, and equations has a strong coupling. So it is difficult for us to prove the existence and uniqueness of solutions directly. So we study the problem in two steps:firstly, we consider the case of non-vacuum non-Newtonian fluids with non-Linear external forces; then, we prove vacuum non-Newtonian fluids with non-Linear external forces. So we first consider the following form: with the following initial-boundary value: where p, u, π is the density, velocity, and pressure,f=f(u, x, t) is an external force of fluids, p>2, μ0>0, ρ0≥δ>0,8is a positive constant, ΩT=I×(0, T),I=(0,1).We prove the existence and uniqueness of local strong solutions to the initial boundary value problem, we have the following theorem:Theorem1Assume that ρ0is sufficiently smooth, and ρ0≥δ>0,δ is a positive constant,f is a continuous function that satisfies the following conditions: u0∈H01(I)∩H2(I), if there is a function g∈L2(Ω2), such that the following identity holds: Then there exists a T*∈(0,+∞), such that the initial and boundary problem (0.3)-(0.4) has a unique strong solution (ρ, u) in ΩT*, satisfying the following properties:Then we apply this theorem to get the existence and uniqueness of the solution to the vacuum non-Newtonian fluid with an external force. We obtain the following theorem:Theorem2Assume that ρ0is sufficiently smooth and0≤p0∈H1(I),f is a continuous function that satisfies the following conditions: u0∈H01(I)∩H2(I), if there is a function g∈L2(Ω), such that the following identity holds: Then there exists a T*∈(0,+∞), such that the initial and boundary problem (0.1)-(0.2) has a unique strong solution (ρ, u) in ΩT*, satisfying the following properties:...