Study On Solutions To Compressible Micropolar Fluid Model With Either Zero Or Non-constant Heat Conductivity

This dissertation is concerned with the properties of solutions to either Cauchy problem or initial boundary value problem(IBVP)for the one-dimensional compressible micropolar fluid model[N.Mujakovic,Glas.Mat.Ser Ⅲ,33(53)(1998),199-208.][1](?)where v=1/ρ,u,p,e,θ and ω represent the specific volume,velocity,pressure,internal energy density,temperature and microrotation velocity,respectively.p andκ stand for viscosity coefficient and heat conductivity.Here we consider a polytropic gas that admits p(v,θ)=Rθ/v=Av-γexp(γ-1/Rs),e(θ)=cvθ,(0.2)where A,R and the specific heat at constant volume cv are positive constants and 7>1 is the adiabatic constant.Two problems are studied:First,we study the asymptotic behavior of solutions to the one-dimensional compressible micropolar fluid model(0.1)when κ=0 for x∈R,t≥0 with initial data(v,u,θ,ω)(x,0)=(v0,u0,θ0,ω0)(x),x∈R,inf x∈R v0(x)>0,inf x∈R θ0(x)>0.(0.3)The corresponding initial data at far field x=±∞ are given by limx→±∞(v0,u0,ω0,θ0)(x)=(v±,u±,ω±,θ±),(0.4)where we assume ω-=ω+=0.We prove that if both the initial perturbation and the strength of the rarefaction waves are assumed to be suitably small,the Cauchy problem admits a unique global solution that tends time-asymptotically toward the combination of two rarefaction waves from different families.We note here this study is different from[J.Jin,R.Duan,J.Math.Anal.Appl.,450(2017),1123-1143.][2]in that we consider zero heat conductivity whereas they considered non-zero heat conductivity and different from[R.Duan,J.Math.Anal.Appl.,463(2)(2018),477-495.][3]in that we consider the far-field states of the initial data(v_,u_,ω_,θ_)≠(v+,u+,ω+,θ+)whereas they studied the case(v_,u_,ω_,θ_)=(v+,u+,ω+,θ+)=(1,0,0,1).Second,we study the strong solution of(0.1)for x E[0,1],t>0 with initial data(u,u,ω,θ)(x,0)=(v0,u0,ω0,θ0)(x),x∈[0,1](0.5)and non-slip and heat insulated boundary condition u(d,t)=0,ω(d,t)=0,θx(d,t)=0,d=0,1,t≥0.(0.6)Here,we consider viscosity coefficient and heat conductivity take the formsμ=1,κ=κ(θ)=θβ,(?)β≥0.(0.7)We prove the existence and uniqueness of global strong solution of the compressible micropolar fluids model(0.1),(0.5),(0.6),We note that this is different from[N.Mujakovic,Math.Inequal.Appl.:12(2009),651-662.][4]in that we consider the case of temperature-dependent heat conductivity whereas they considered constant heat conductivity(κ=1).