Existence Of The Unique Strong Solution For A Class Of Full Non-Newtonian Fluids With Vaccum

In nature, all things obey the invariable order of nature, for instance, conservation laws(mass, momentum and energy), universal gravitation etc. We are opening out, conquering and transforming the nature by these foundational laws. Of course, it is out of question that the flowing gas or liquid (fluid) obey the order of nature, for example, the fluid obey conversation lows of mass and momentum. If we want to know the state of fluids, we may solve equations that are constructed by these conversation laws. With the development of the technology, we find that there are a lot of fluids which belong to non-Newtonian fluid in the process of product and nature. The classical non-Newtonian fluid is Macromolecule melt and macromolecule liquor, and the all kinds of slurry and suspend liquor, paint, dope, palette and biology fluids, for example, in the body of people and animals, the blood, the synovia of arthrosis cavity, lymph liquor, cell liquor, brain liquor etc, which are provided with the property of non-Newtonian fluid. So the non-Newtonian fluid widely exists in nature, it is necessary to studying these dynamics equations system.In this paper, we mainly disscuss the existence and uniqueness of the following problem: with following initial-boundary value:whereρ, u andθdenote the density, velocity and absolute temperature respectively, . We assume that:Here we consider its solution denned by:Definition 1.1 The (ρ,u,θ) is called a strong solution to the initial boundary value problem (1.1)-(1.2), if the following conditions are satisfied: Now we can state our main result in this paper.Theorem 1.1 Assume thatρ0≥0 is sufficiently smooth and . If there are two functions g, h∈L~2(I), such that the following identity holds:and is small enough, such thatThen there exists , such that the initial boundary problem (2.1)-(2.2) has a unique strong solution (ρ, u,θ) inΩT_*, satisfying the following properties:It is difficult to prove the existence and uniqueness directly to our problem for the existing of the vacuum and the singularity . So we take two steps: firstly, we consider the problem with positive density. In dealing with this, we use the method of iterative to construct the approximate solution, and get the uniform estimate, then get the existence and uniqueness of it. At last, we consider the original problem with the help of the last result.So, we first consider the following:with initial boundary value : where is an given positive number. For the singularity, we need to regularize it: Let u~0 = 0,whereand is smooth is smooth, satisfy the following problemWe estimate the approximate solution in the case and respec-tively, and finally we obtain :using (1.5), we could take limits with respect to k,e and get the following Theorem.Theorem 1.2 Assume thatρ0≥δ≥0 is sufficiently smooth, whereδis a given constant. If there are two smooth functions g, h∈L~2(I), such that the following identity holds: and small enough, such that . Then there exists a T_*∈(0, +∞), such that the initial boundary problem (1.3)-(1.4) has a unique strong solution (ρ,u,θ) inΩT_*, satisfying the following properties:From Theorem 1.2, we could prove the existence and uniqueness problem with vacuum. Here we only need to regularize the initial functions in (1.1) to satisfy the conditions in Theorem 1.2.For every small , and is the only solution of :where ,andSo ,we get is the solution of (1.3)-(1.4),such thatess supUsing this, we could take limit in terms ofδ, then finally obtain the Theorem 1.1.