Research On The Well-posedness Of Solutions For A Class Of Incompressible Non-Newtonian Boussinesq Equations

In this paper,we mainly study the first initial and boundary value problems of the following steady-state incompressible non-Newtonian Boussinesq Equations(?) where Ω(?)R3 and it is a smooth and bounded domain,1<p<2.The function u-(u1(x1,x2,x3),u2(x1,x2,x3),u3(x1,x2,x3))denotes the velocity of the fluids;Du=(▽u+▽uT)/2 denotes the strain rate tensor;θ=θ(x1,x2,x3)denotes the temperature of the fluids;π=π(x1,x2,x3)denotes the pressure of the fluids;f=f(x1,x2,x3)and g=g(x1,x2,x3)denote the external force.For the above-mentioned problem,we overcome the difficulties caused by the strong coupling and nonlinearity of the equations.Under the conditions that the external force term f and g are small in a suitable sense,we proved the existence and uniqueness of regularized solutions for the problem.This paper is divided into three chapters.The first chapter is the introduction,which briefly describes the research significance of the hydrodynamic model,and it shows the research background and research status of the incompressible non-Newtonian flow and the incompressible non-Newtonian Boussinesq equations.The second chapter is the preparatory knowledge,which gives some marks and theorems used in this paper.The third chapter is the main part of the paper,we prove the main results by using iterative method and smoothing function method.Specifically,we get the following two conclusions:Conclusion 1.Assume that 1<p<2,q>3,γ0=1-3/q,f∈Lq(Ω),g ∈L2(Ω).If‖f‖q≤Λ1,‖g‖2≤Λ2,where Λ1 and Λ2 are positive constants small in a suitable sense,then there exist a solution for the above problem u∈Vq∩C1,γ(Ω),π∈C0,γ(Ω),θ∈W2,2(Ω)(?)γ<γ0,and‖u‖C1,γ(Ω)+‖π‖C0,γ(Ω)+‖θ‖2,2≤c1‖g‖2(2C‖f‖q+1)where c1 and C are positive constants.Further more,if 6/5<p<2,there is a unique weak solution for this system of equations.Conclusion 2.Assume that 1<p<2,q>6,γ0=1-3/q,f∈Lq(Ω),g∈L2(Ω).If‖f‖q and ‖g‖2 are small in a suitable sense,then on the basis of conclusion one,we have u∈W2,2(Ω)∩C1,γ(Ω),π∈W1,2(Ω)∩C0,γ(Ω),θ∈W2,2(Ω),(?)γ<γ0.