Abstract
The aim of this paper is to discuss the model for a class of shear thickening fluids with non-Newtonian potential and heat-conducting. Existence and uniqueness of local strong solutions for the model are proved. In this paper, there exist two difficulties we have to overcome. One is the strong nonlinearity of the system. The other is that the state function is not fixed.
1. Introduction
With the development of the technology, we find that there are a lot of fluids which belong to non-Newtonian fluids in the process of product and nature. The classical non-Newtonian fluids are macromolecule melt and macromolecule liquor and 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, and brain liquor, which are provided with the property of non-Newtonian fluids. So the non-Newtonian fluids exist widely in nature (see [1–6]).
In the absence of gravitational potential term, the mathematical model of non-Newtonian fluids reduces to the Navier-Stokes equations. There have been many results concerning the existence and nonexistence of solutions for such equations (such as [7–10]). In [11], Xin studied the blowup of smooth solutions to the compressible Navier-Stokes equations in arbitrary space dimensions with initial density of compact support. Choe and Kim in [12, 13] proved an existence result when is either a bounded domain or the whole space with the compatibility condition. For other results one can refer to [14–17] and the references cited therein.
In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state. A state function describes the equilibrium state of a system. For example, internal energy , temperature , density , pressure , and so on are state quantities because they describe quantitatively an equilibrium state of a thermodynamic system, irrespective of how the system arrived in that state. If two quantities are known, then we can get other quantities. For example, temperature and density are known, and we can get internal energy , pressure , and so on.
In mathematics, a state function has the following properties:
(i) A state function has continuous first partial derivatives.
(ii) There are the tiny variable and the limited variable . The integral value depends on the size of the integral upper limit and lower limit and has nothing to do with other factors. Thus,
(iii) Consider
The following are some examples of state equations:
(i) Polytropic gases: ( is a constant and is gas thermal insulation coefficient and is a constant) (see [15–17]).
(ii) State equation of ideal gas: ( is a constant) (see [18]).
(iii) Isentropic process: (see [18]).
In this paper, we consider generalized state equation. Energy increases with the increase of temperature. In mathematics, and (we define , , , and ). Ignore the molecular potential energy and all of the gas fluid, for example, methane, propane, and butane from the oil.
In this paper, we consider a class of heat-conducting non-Newtonian fluids of Navier-Stokes system:with the initial and boundary conditionswhere , , the initial density , , and are given constants. The unknown variables , , and denote the density, velocity, temperature, and the non-Newtonian gravitational potential, respectively.
Definition 1. The triple is called a strong solution to the initial boundary value problem (3)-(4), if the following conditions are satisfied:
(i) Consider(ii) For all , , for a.e. , we have(iii) For all , , for a.e. , we have
(iv) For all , , for a.e. , we have
1.1. Main Results
Theorem 2. Assume that satisfies the following conditions:and further that there exists a constant , such thatThen there exist a time and a unique strong solution to (3)-(4) such that
The rest of this paper is organized as follows. In Section 2, we present some elementary lemmas. In Section 3, we devote ourselves to the study of the problem with positive density. In Section 4 we give the proof of the main theorem.
2. Preliminaries
We first give some known facts for later use.
Lemma 3 (see [15]). Assume that on , where is bounded and open, , . Then where denotes the length of .
Lemma 4 (see [15]). Let be a Hilbert space with a scalar product and let be a Banach space such that and is dense in , . Then
Lemma 5. and is state function. According to the mathematical properties of state function, we can get that ,
3. Existence of Solutions with Positive Density
3.1. A Priori Estimates for Smooth Solutions
We first construct approximate solutions of problem (3)-(4). By using the iterative scheme, inductively, consider the following:
(i) Firstly define , .
(ii) Assuming that , were defined for , we can obtain as the unique smooth solution to the following system:where is the initial mass, for a given function and numbers , ,
With this process, the nonlinear coupled system has been deduced into a sequence of decoupled problems and each problem admits a smooth solution. Moreover, we find from (16), (21) with the smooth function ; that is,
Obviously there is a unique solution to the above initial value problem and also by a standard argument, we could get
Then by (19) and (21) we get . Lastly, with , , we obtain from (20) and (21).
Moreover, assume that for a given , For some, will be fixed later. We will prove thatand some more useful uniform estimate about .
(i) Estimation for .
Then multiplying (16) by and integrating it over with respect to , we have
Integrating by parts, using the Sobolev inequality and Lemma 3, we deduce that
Then, considering differential equation (16) with respect to and multiplying it by , integrating it over on , and using the Sobolev inequality, we have
From (28) and (29), by Gronwall’s inequality, it follows thatand, taking , we getAnd using (16) we can also obtainwhere is a positive constant.
(ii) Estimation for .
Differentiating (19) with respect to , multiplying it by , and integrating it over on , we derive
Using the Sobolev inequality, Young’s inequality, Lemma 3, and (25), (30) and (31), we obtain Here,
In the same way, we have Here,
Following the same method, we obtain Here,
In a similar way, Here,
Similarly,
Substituting into (43), after choosing , we getand, integrating it over , we havewhere is a positive constant.
We estimate as follows: multiplying (17) by and integrating it over , we obtain Next, we get and, according to the smoothness of , we havewhere is a positive constant.
We haveby Gronwall’s inequality, it follows thatfor ,
Due to (44), (47), and (50), we get
Denote ; we getby Gronwall’s inequality, ( has nothing to do with ), it follows that
Note that, for a given function , by a simple computation, we have
Firstly, we need to do the following estimates: by (19)and we have taking it by -norm, we get
(iii) Estimation for .
Differentiating (20) with respect to , multiplying it by , and integrating it over (0,1) on , we derive
Then, using the Sobolev inequality, Young’s inequality, Lemma 3, and (25), (30), and (31), we obtain
Substituting into (33), after choosing
We estimate as follows: multiplying (20) by and integrating it over , we obtain
According to the smoothness of , we havewhere is a positive constant.
In the same way,
Integrating (60) over , combining with (62) and (63) we haveand for , , we getand, by Gronwall’s inequality, ( has nothing to do with ), it follows that
Secondly, we need to do the following estimates. By (20)we have and, taking it by -norm, using Lemma 3, we getwhere is a positive constant. For , we get
3.2. Convergence
In this subsection, the approximate solutions strongly converging to the solution of (3)-(4) will be proved. To this end, we define and then we can easily verify that , , satisfy the system of equationswith the initial and boundary conditions
Multiplying (72) by and integrating it over , we obtainwhere , for all and .
Multiplying (73) by and integrating it over , we obtain
Using the Sobolev inequality, Young’s inequality, and Lemma 3, we obtain Here,
In the same way, we have Here,
Following the same method, we obtain Here,
In a similar way, we have Here,
Similarly,
Using the same method, we have Here,
Substituting into (105), after choosing , we getThus, we getThere , .
Multiplying (74) by and integrating it over , we obtain
We use the Sobolev inequality, Young’s inequality, and Lemma 3, and we obtain
Substituting into (91), after choosing ,Thus, we getThere , . Using (94), denoteThere , .
Thus, we getBy Gronwall’s inequality, noting that , we get
We can obtainand thus we can getThere, .
Therefore, we conclude that the full sequence converges to a limit in the following strong sense:for , and we get
4. Proof of the Main Theorem
4.1. Existence ()
In this part, we need to prove that is a solution of problemwith the initial and boundary conditions and also satisfy the integral identity after the process. Finally, we need to show that the solution satisfies the regular properties in Definition 1. Since the abovementioned can be obtained by the same method as in [15], we omit it for brevity.
4.2. Uniqueness
Let , be two solutions of problem (3)-(4). After substituting into (7), respectively, we define and then we can obtainsince Using the Sobolev inequality, Young’s inequality, and Lemma 3, we obtain Here,
In the same way, Here,
By the same token, Here,
In the similar way, we have Here,
Similarly,
Using the same method, we obtain Here, Thus, we haveThere , .
In the same way,There , . Consider There , .
With (119) multiplied by , combining with (118) and (120),We can getThere , .
By Gronwall’s inequality, it yields
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research is supported by the National Natural Science Foundation of China Grant no. 11271153. The research is supported by Specialized Research Fund for the Doctoral Program of Higher Education no. 20140101-20161231.