The Asymptotic Limit Of Some Compressible Fluid Dynamics Equations

Fluid dynamics studies movement of fluid and fluid's conditions of rest. it also studies the reciprocity and flow rules between solid boundary when they are moving. Two types of flow motion are sometimes distinguished in different studies, they are incompressible fluid and compressible fluid. The Mach number usually is used to describe the compressibility of fluid.There are two parts in this paper. In chapter 2, we consider the equations for a compressible, isentropic or isothermal combustion flow as followswith initial valueHere, j = 1,2 ? ? ? n, n = 2 or 3;ε> 0 is a viscosity constant;ρ, u = (u¹, ? ? ?, uⁿ) and z are the fluid density, velocity and the per centum of the responseless gas in the gas flow, which are the unknown functions of x∈Rⁿ, n = 2 or 3, and t∈[0,∞); (?) is the pressure,γ> 1; A = A(·) satisfies(?) satisfiesWe give the incompressible combustion flowwith initial valueHere, (?) is a constant and density of the incompressible combustion flow.w, (?) and z₁ are respectively the velosity, pressure and per centum of the responseless gas in an incompressible flow.Then (5)-(6) has a corresponding solution (w,ρ₁,z₁) defined on Rⁿ×I, satisfying We letThe initial values of (1) and (5) satisfying We then obtain the following theorem :Theorem 1 Fix n = 2 or 3, let A(z) and (?)(ρ,z) satisfying (3) and (4) be given, and let (ρ₀, w,ρ₁, z₁) be a corresponding solution of (5)-(6) as described above, satisfying the conditions in (7), and with all the norms indicated in (3), (4) and (7) bounded by C₁. Let the functionÏ€= R^γ-1 A(δ²z)ρ^γbe given, and let C₀ > 0 be given. Then there are positive constantsδ₀ and C, C depending on C₀ , C₁ ,Ï€,ρ₀, and, in the case that T <∞, on T, such that: given initial data (ρ₀(x), u₀(x), z₀(x)) for (1) and (w₀(x),z₁₀(x)) for (5) satisfying (9)-(12) , there is a corresponding solution (ρ,u,z) of (1)-(2) defined on Rⁿ×[0,T). This solution satisfies To prove the theorem, first, we can prove the conditions of z₁ by the third equation of the incompressible combustion flow. Next, by the energy estimate, we can prove In addition, we estimate the right terms about (?) dxds, (?) dxdsand (?) dxds. We define F =εdivu -δ^-2[Ï€-Ï€₀] and G(t) = 1 +δ^-2B(t) + D(t) +δ²E(t), we can provesubstituting (25)-(27) into (20), (22) and (24), then we get(13)-(17).In chapter 3, we consider the full compressible Navier-Stokes equationwith initial valueHere j = 1,…, n, n = 2 or 3;ρ, u = (u¹,…, uⁿ) andθare the fluid density, velocity and temperature, which are the unknown functions of x∈Rⁿ, and t∈[0,∞) P = P(ρ) is the pressure. f = f(x, t) is the external force vector, h = h(x, t) is the density of external heat sources, e₀∈Rⁿ is the vector of the gravity force direction.We give the full incompressible N-S equationwith initial valueHere, (?) is a constant and density of the incompressible combustion flow; w, v and c²Ï₁ are respectively the velocity, temperature and pressure in an full incompressible N-S equation.Then (31)-(32)has a corresponding solution (w,ρ₁,v) defined on Rⁿ×I, satisfyingWe letThe initial values of (29) and (31) satisfying We then obtain the following theoremTheorem 2 n = 2或3, let ((?),w,ρ₁,v) be a corresponding solution of (31)-(32), the norms of (33) bounded by C₁. Let the function P = P(ρ) be given, and let C₀ > 0 be given. Then there are positive constantsδ₀ and C, C depending on C₀, C₁,Ï€, T ,ρ₀, and, in the case that T <∞, on T, such that: given initial data (ρ₀(x),u₀(x),θ₀(x)) for (29) and (w₀(x),v₀(x)) for (31) satisfying (35)-(38), there is a corresponding solution (ρ,u,θ) of (29)-(30) defined on Rⁿ×[0,T). This solution satisfies To prove the theorem, first, by the energy estimates, we getIn addition, we estimate the right terms above. We define F =εdivu -δ^-2 [P -P₀], M =(?) and G(t)=1+δ^-2B(t)+D(t)+δ²E(t) then we get Substituting the estimates (51)-(57) into(46), (48) and (50), we can prove (39)-(44) in the theorem.