Research On The Well-posedness Of Solutions For A Class Of Compressible Non-newtonian Fluids And Variational Exponential Evolution Equations

Fluid mechanics is a branch of science which studies the phenomenon and behavior of fluids in the associate field.At present,Newtonian fluid in which the stress tensor is proportional to the shear rate has been widely studied,on this basis,one can get the famous Navier-Stokes equation.Correspondingly with the Newtonian fluid is another kind of one,whose stress tensor is not proportion to the shear rate.It is usually called the non-Newtonian fluid.Non-newtonian flow exists widely in aerospace,energy,ocean,chemistry,biological medicine,geology,and other fields,it also makes people increasing research interest in non-newtonian fluid system.At present,there is few results about non-newtonian fluid studies,and most of those focused on the local solution·In this paper,we mainly discuss the compressible non-newtonian flow equation and variable exponential evolution equation.In the third chapter of this paper,we consider the following compressible non-newtonian fluid equation on a one-dimensional bounded interval:with initial and boundary conations where the unknown variables ? = ?(x,t),u = u(x,t),?(?)=a??(a>0,?>1)denote the density,velocity and pressure,respectively.?:=(0,1),p?(7/6,2),the initial density ?0?0.For the above systems,we provedTheorem 1 let 5/3<p<2 and(p0,u0)satisfies the following conditions 0??0?H1(?),u0?H01(?)?H2(?),?0??,(3)and the following compatibility condition where g ? L2.Then there exist ? = ?(?,?,?)>0,if the initial energy E0:=satisfies E0??,(5)the initial boundary value problem(1)-(2)exists a unique strong solution(?,u)such that and for 0<T<?,where q?3-2/p.Theorem 2 let 7/6<p?5/3 and(?0,u0)satisfies the following conditions 0??0?H1(?),u0? H01(?)?H01(?)?H2(?),?0??,||u0x||pp?M,(6)and the following compatibility condition where g ? L2.Then there exist ?=?(a,?,?)>0,if the initial energy E0:=?(1/2?0u02+a?0?/?-1)satisfies E0??,(8)the initial boundary value problem(1)-(2)exists a unique strong solution(?,u)such that and for 0<T<?,where q?3-2/P.It could be seen that the problem we studied has strongly nonlinearities and singu-larities,in addition,the initial density contains vacuum.All of these bring difficulties for us.Because of the particularity of non-newtonian flow,it is difficult to deal with the effective viscous flux F which has been extensively used in[36],we have to use the new skills to overcome the difficulties which F brings,and obtain L2 estimate of the gradient of velocity and L? estimate of the density,and the following prior estimates can be obtained:0?t?T sup||?||??7?/4 and and the existence of global strong solutions are obtained.In addition,we also study the long time behavior of the solution.In the fourth chapter of this paper,we consider a class of variatioral exponen-tial p(x)-Laplace equation with boundary degeneracy.when p(x)is a measurable function,it derived from the theory of electrorheological equations,when p(x)= p is a constant,it is the non-newtonian polytropic filtration equation.with initial condtion u|t=0=u0(x),x??,and boundary condtion u|?T=0,(x,t)??T=(?)?×(0,T),(11)where ?(?)RN,p(x)is a measurable function,and p(x)>1.a(x)? C1(?),and a(x)>0,x??,a(x)=0,x?(?)?.f(s,x,t)is a suitably smooth function and satisfies|f(u,x,t)-f(v,x,t)|?c|u-v|,(x,t)? QT.For the above problems,we prove the following results:Theorem 3 If a(x)satisfies the following conditions Moreover,if u0?L?(?),u0?W1,p(x)(a,?),(12)then there is a solution of equation(9)with the initial value(10).Theorem 4 If a satisfies w1)-w2),let u,u be two solutions of equation(8)with the initial values u0,v0 respectively,f(u,x,t)be a Lipschitz function on u.If u0,v0 satisfy(12),and it is supposed that?? a(x)d(x)-p(x)dx?c,(13)then??|u(x,t)-v(x,t)|2dx?c??|u(x,0)-v(x,0)|2dx.(14)here d(x)= dist(x,(?)?)is the distance function from the boundary.Since the variable exponentials p(x)and a(x)in the equation(9)would cause degeneracy,it would be difficult to prove the well-posedness of the solution.We prove the existence and stability of the solution in which we use the Fichera-nleinik linear equation theory with second order nonregative eigenvalues,classic p-Laplace research methods and properties of the Sobolev-Orlicz space.According to the above theorem,we conclude that the solutions of the equation(9)is controlled by the initial value(10)completely.This provides a reference for us to discuss the stability of the equation which needs boundary conditions or only partial boundary conditions.