Study On The Well-posedness And Regularity Of The K-? Equations For The Model Of Turbulent Flows

In this paper,we study the initial-boundary value problems of the k-? equations for the model of turbulent flows:?where ?ij= 0 if i ?= j,?ij= 1 if i = j,and ?,?t,?e,C1 and C2are five positive constants satisfying ? + ?t= ?e,and-?n is the unit outward normal to ??.In the above equation,?,u,h,k,? and p denote the density,velocity,total enthalpy,turbulent kinetic energy,the rate of viscous dissipation of turbulent flows and the pressure,respectively.The existence of local strong solution to the initial-boundary value problems of the k-? equations for turbulent flows in a bounded domain is studied in the third chapter.We first linearize the k-? equations.We obtain the local existence of strong solution to the linearized equations and the regularity satisfied by the strong solution by use of energy estimates to the linearized equations.Having finished the handling with the linearized equations,we construct by use of the infinite differentiability of solution to the heat equation a set of functions with good smoothness.On the basis of this set of functions,we establish,inductively,a series of approximate solution to the original nonlinear equations and prove by use of the idea of the fixed point theorem this series of approximate solution do converge to the strong solution to the original nonlinear equations.Thus we have proved the local existence of the strong solution to the original equations.Having had local strong solution in hand,a natural question is whether this solution can be extended to a global solution.Therefore,we provide a sufficient condition ensuring the local strong solution to the initial boundary value problem of the k-? equations in a 2 dimensional bounded domain can be extended,or equivalently,the blowup criterion to the strong solution.In the process of deriving this blowup criterion,the inequality(2.2.2)plays a key role.The inequality(2.2.2)becomes(1.1.15)in 3 dimensional space.The regularity index of ?f ?L2(0,T;H32)in the inequality(1.1.15)is32,however,the energy estimates could only provide us with the regularity index 1,which implies that we could not by use of the inequality(1.1.15)derive the blowup criterion in the 3 dimensional space similar to that in the 2 dimensional space.That is the reason why the blowup criterion could not be generalized to the 3 dimensional space for the moment.Our main result reads: Suppose(?,u,h,k,?)is the strong solution.Denote by T*the maximal time of existence for the strong solution.If T*< ?,then...