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

In this paper, we consider a class of compressible non-Newtonian fluids in one-dimensional bounded intervals:with the initial and boundary conditionswhereρ, u andÏ€denote the unknown density, velocity and pressure, respectively. The motion of the fluid is driven by an external force f,Ï…is a given funtion, the initial densityρ₀≥ρ> 0, p>2,Ω_T = I×(0,T),I = (0,1).In the recent three decades, fluid dynamics has attracted the attention of many mathematiciansand engineers. The Navier-Stokes equations are generally accepted as right governingequations for the compressible or motion of viscous fluid, which is usually described by the principle of conservations of mass and momentum, and we deduce finally whereΓdenotes the viscous stress tensor, which depends on E_ij((?)u), andis the rate of strain. If the relation between the stress tensorΓ_ij and rate of strain E_ij is linear, then the fluid is called Newtonian. That is, Newtonian fluids satisfy the following linear relationThe coefficient of proportionalityμis called the viscosity coefficient, and it is a characteristic material quantity for the fluid concerned, which in general depends on density, temperature and pressure. Generally speaking, air, other gases, water, motor oil, alcohols, simple hydrocarboncompounds and others tend to be Newtonian fluids. The governing equations of motionsof them can be described by Navier-Stokes equations. If the relation between the stress tensorΓ_ij and the rate of stain E_ij is nonlinear, then the fluid is called to be non-Newtonian. For instance, molten plastics, polymer solutions and paints tend to be non-Newtonian fluids. The simplest model of the stress-strain relation for such fluids is given by the power laws, which states thatfor 0 < q < 1 (see B(?)hme [1]).Definition 1.1. The pair (ρ, u) is called a strong solution to the initial boundary value problem (1.1 )-(1 .2), if the following conditions are satisfied:(i)0 < (?) <ρ∈C ([0,T]; H²(I)),ρ_t∈L^∞(0, T; H¹ (I)),u_t∈L^∞(0, T; L²(I)) ,u∈C ([0, T]; L²{I))∩L^∞(0, T; W₀^1,∞(I)). (ii) For allφ∈C ([0, T]; H²(I)),φ_t∈L^∞(0, T; H¹(I)), for a.e. t∈(0, T), we have:(iii)For allφ∈C([0,T];L²(I))∩L^∞(0, T; W₀^1,∞(I)),φ_t∈L^∞(0,T; L²(I)), for a.e.∈(0, T), we have:Now we can state our main result in this paper. Theorem 1.1. Assume that (ρ₀, u₀, f,Ï…) satisfies the following conditions0<ρ<ρ₀∈H²(I), u₀∈H₀¹âˆ©H²(I),f∈L^∞(0, T; H¹ (I)), f_t∈L^∞(0, T; L²(I)),v∈L^∞(0, T; H₀¹âˆ©H³(I)), v_t∈L² (0, T; H₀¹(I)).Then there exists a unique solution (ρ,u) to the initial boundary value problem (1.1)-(1.2) such that, for all T∈(0, +∞),ρ∈C ([0, T]; H²(I)),ρ_t∈L^∞(0, T; H¹(I)),u_t∈L^∞(0,T;L²(I)), u∈C([0T];L²(I))∩L^∞(0,T;W₀^1,∞(I)), (1.6)We take this problem for two steps. Firstly, we consider the following:with the initial and boundary conditions:where 0 <δ?? 1, 0 <μ< 1, v^δ= J_δ* (?),ρ₀^δ= J_δ* (>), (?), (?) is the extension of v,ρ₀, J_δis a mollifier on R. f^δ∈C^∞(?). f^δ(0, t) = f^δ(1, t) = 0.In this part, we get the uniform estimates of solution for (1.1)-(1.2).Where C is a positive constant, depending only on M⁰. We define thatM⁰ =1 + |ρ₀|_H²(I) + |u₀|_H²(I) + |f|_{L^∞(0,T;H¹ (I))} + |f_t|_{L^∞(0,T;L²(I))}For the uniform estimates of (1.9), we can take limit forδ→0, we obtain Theorem 1.2. Assume that (ρ₀, u₀, f, v) satisfies the following conditions:0<ρ<ρ₀∈H²(I), u₀∈H₀¹(I)∩H²(I),f∈L^∞(0, T; H¹(I)), f_t∈L^∞(0, T; L²(I)),v∈L^∞(0, T; H₀¹ n H³(I)), v_t∈L² (0,T; H₀¹(I)).Then there exists a unique solution (ρ_μ, u_μ) to the initial boundary value problem (1.7)-(1.8) such that, for all T∈(0,∞):ρ_μ∈C([0,T];H'(I)),ρ_μt∈L^∞(0, T; H¹ (I)),u_μt∈L^∞(0, T; L²(I)), u_μ∈C ([0, T]; L²(I))∩L^∞(0, T; W₀^1,∞(I)),(|u_μx|^ρ-2u_μx)_x∈L^∞(0,T;L²(I)).Since C is a positive constant, depending only on M⁰, we can get (ρ_μ, u_μ) satisfies the following uniform estimatesHaving the above estimates in hand, analogously to the process forδ→0, we could take limit forμ→0, which completes the proof of Theorem 1.1.