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 ?,33(53)(1998),199-208.][1](?)where v=1/?,u,p,e,? and ? represent the specific volume,velocity,pressure,internal energy density,temperature and microrotation velocity,respectively.p and? stand for viscosity coefficient and heat conductivity.Here we consider a polytropic gas that admits p(v,?)=R?/v=Av-?exp(?-1/Rs),e(?)=cv?,(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 ?=0 for x?R,t?0 with initial data(v,u,?,?)(x,0)=(v0,u0,?0,?0)(x),x?R,inf x?R v0(x)>0,inf x?R ?0(x)>0.(0.3)The corresponding initial data at far field x=±? are given by limx?±?(v0,u0,?0,?0)(x)=(v±,u±,?±,?±),(0.4)where we assume ?-=?+=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_,?_,?_)?(v+,u+,?+,?+)whereas they studied the case(v_,u_,?_,?_)=(v+,u+,?+,?+)=(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,?,?)(x,0)=(v0,u0,?0,?0)(x),x?[0,1](0.5)and non-slip and heat insulated boundary condition u(d,t)=0,?(d,t)=0,?x(d,t)=0,d=0,1,t?0.(0.6)Here,we consider viscosity coefficient and heat conductivity take the forms?=1,?=?(?)=??,(?)??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(?=1).