Existence And Uniqueness Of The Strong Solutions For Some Models Of Compressible Full Non-Newtonian Fluid

In nature, the flow of the fluid generally obey the law of conservation of mass, the law of conservation of momentum and balance of energy. In mathematics, we could describe those laws through the differential equations. Moreover, we could know the state of the fluid in the future by studying the quality of the solutions of the equations. Now, the one which has been extensively studied is the Newtonian fluid, and in this class of fluid, the stress tensor is in pro-portion to the shear rate (linear relationship). There have been lots of results related to this class of fluid, and some of these results are still very in-depth. Correspondingly with the Newtonian fluid is another one, whose stress tensor is not proportion to the shear rate. It is usually called the non-Newtonian fluid. The study on the non-Newtonian fluid mechanics is of great significance in two respects. One is that the study of the uniqueness of the weak solutions to the Navier-Stokes equations in three dimension needs us to consider a more general Navier-Stokes equations, and this give rise to the research on the in-compressible or compressible non-Newtonian fluid with variety of non-linear constitutive relations. On the other hand, in the area of the geology, glaciology, hemorheology, bio-mechanics,and chemical industry, people may meet with a large number of problems of the non-Newtonian fluid, which has sparked the increasing interest in the study of the non-Newtonian fluid. Currently, the known results about the non-Newtonian fluid are quite few, and most of those focused on the incompressible ones.In this paper, we mainly study two class of compressible non-Newtonian fluid models. First of all, in chapter two, we discuss the following model in one space dimensional bounded interval with initial and boundary condtions here,ρis the density, u the velocity,θthe energy. R> 0 is the universal constant,4/30,δis an given constant, and alsoρ0,g,h are sufficiently smooth.We construct the approximate solutions as follows:(i). firstly define u0=0;(ii). assume uk-1 has been obtained, then we search (ρk,θk, uk) as the solution of the following systems here, (u0ε,θ0ε) is the smooth solution of and in above, we denoteFor the above approximate solutions (ρk,θk,uk), by some computations, we could obtain that there exists an (?)05(μ,T) such that if|ρ0|H1≤(?)05(μ,T), for k≥1, for some constant C depending onμ. In fact, we have obtained for some constant C depending on p.With the uniform estimates (10)and(11), we could prove the convergence for the full sequence of approximate solutions (ρk,θk,uk). And then, by taking limit aboutκandε, we obtain the following propositionProposition 1. Suppose the conditions in Theorem 1 hold. Assume further thatρ0≥δ>0 for a given constantδ, andρ0,g, h are smooth enough, then the conclusion of Theorem 1 hold.Having Proposition 1 in hand, we could then prove the existence and unique-ness of strong solutions in the vacuum case. For every smallδ,0<δ<<1, define satisfy where gδ,hδ∈C0∞(I) andThen by using propositon 0.0.1, we could obtain the strong solutions (ρδ,θδ, uδ), such thatIn fact, we still have for some constant C depending onμ.Since the above uniform estimates do not depend on the lower bound ofρ0, we could then taking limit aboutδ, and then obtain the conclusion ofTheorem 1.In the third chapter of this paper, we consider the following one space di-mensional problem, for (x, t)∈IT coupled the initial and boundary conditions We still allow the initial vacuum, i.e.ρ0≥0.For above system, we provedTheorem 2. Let p, q> 2,μ0,μ1>0.ρ0≥0,θ0,u0 are smooth enough, if there are two smooth functions h, g, such that ten there exists an small time T*∈(0,+∞), such that (15)-(16) admit an unique strong solution (ρ,θ,u) in IT*.Similar as the previous part, we still divide the proof into two parts. We firstly discuss the no vacuum case and then by virtue of these results to prove Theorem 2.More precisely, we construct the approximate solutions as follows:(i) let u0=0;(ii)assume uk-1 has been obtained, we search (ρk,θk,uk) as the solution of where,ρ0≥δ>0, (u0,θ0) satify (17) and in above, we denoteFor the above approximate solutions, after some complicated computation, we could obtain that for 00 for some small constantδ. Then there exists an small time T*∈(0,+∞), such that (15)-(16) admit a unique strong solution in IT*.Next, for everyδ>0, defineρ0δ=ρ0+δ, by (17) we have where For the initial values (ρ0δ,θ0,u0), by using Proposition 0.0.2, there have a unique strong solution (ρδ,θδ, uδ) such that where C is an positive constant not depending onδ. Finally, by taking limit aboutδ, we could prove Theorem 2.