Strong Solutions of Navier-Stokes-Poisson Equations for Compressible Non-Newtonian Fluids
Yukun Song^{*}, Yang Chen
College of Science, Liaoning University of Technology, Jinzhou, P. R. China
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To cite this article:
Yukun Song, Yang Chen. Strong Solutions of Navier-Stokes-Poisson Equations for Compressible Non-Newtonian Fluids. Science Journal of Applied Mathematics and Statistics. Vol. 4, No. 4, 2016, pp. 134-140. doi: 10.11648/j.sjams.20160404.13
Received: June 12, 2016; Accepted: June 25, 2016; Published: June 30, 2016
Abstract: This paper study the Navier-Stoke-Poisson equations for compressible non-Newtonian fluids in one dimensional bounded intervals. The motion of the fluid is driven by the compressible viscous isentropic flow under the self-gravitational and an external force. The local existence and uniqueness of strong solutions was proved based on some compatibility condition. The main condition is that the initial density vacuum is allowed.
Keywords: Strong Solutions, Navier-Stokes-Poisson Equations, Non-Newtonian Fluids, Vacuum
1. Introduction
In this paper we study a class of one-dimensional isentropic compressible non-Newtonian fluids of Navier Stokes Poisson system:
(1)
With the initial and boundary conditions
(2)
Where
denote the unknown
density, velocity, geopotential and pressure, respectively.
In the sense of physics, the motion of the fluid is driven by the compressible viscous isentropic flow under the self-gravitational and an external force , the initial density are given constants.
During the past decades, fluid dynamics has attracted the attention of many mathematicians and engineers. The study of non-Newtonian fluid mechanics is of great significant because of the non-Newtonian fluids are widely used as up to date ones in various fields of applied sciences, such as the models for the flow of glacier, the flow of blood through arteries be proposed in blood rheology, the dynamics of tectonic plates in the earth’s mantle in geology etc. ([1-3]).
Up to now, the results on non-Newtonian fluids are quite few. In [4], the local existence and uniqueness results of non-Newtonian fluids were given in the case of and by assuming a similar compatibility condition as (3). [5] studied the global existence and uniqueness results of heat-conducting fluids if and the initial density in norm is small enough. The results on fluid particle interaction non-Newtonian models, see [6-7].
For the Newtonian fluids without considering the energy consevation equation term have been studied by many authors ([8-15]). For detail, [9] applied the weak convergence method showed the existence of global weak solutions under the assumption that if and if . Later, this method was improved to reduce more general results ([10-12]). In [13-15], we can find some local existence results on strong solutions in three dimensional space followed the compatibility condition
for some (3)
As for the Navier-Stokes-Poisson equations for Newtonian fluids with density dependent or independent with viscosity, the existence of strong solutions, regularity and large time behavior of solutions were investigated, for these results, see [16-20]. In this paper, we discuss the system (1)-(2) with , we prove the local existence and uniqueness of strong solutions under some conditions. As we know that when , the second equation of (1) is always with degeneration. Moreover, the initial density is allowed with vacuum and the strong nonlinearity of equations bring us another difficulty.
2. Main Results
2.1. Main Theorem
Theorem 2.1.1 Assume thatsatisfies the following conditions
, and if there is a function , such that the initial data satisfy the following compatibility condition:
for a.e. (4)
Then there exist a time and a unique strong solution to (1)-(2) such that
2.2. Preliminaries
First of all, some known facts are given for latter use.
Lemma 2.2.1 Assume thaton, where is bounded and open, ,. Then
Where denotes the length of
Lemma 2.2.2 Let H be a Hilbert space with a scalar product (.,.)_{H} and let X be a Banach space such that
↪↪andis dense in ,. Then
↪
3. Existence of Solutions
In this section, we will prove the local existence of strong solutions. To get the existence of strong solutions, some more regular estimates are required. Provide that is a smooth solutions of (1)-(2) and , where
is a positive number, as we can deal with approximate system, we only consider initial nonvacuum. Combining the classical results of (1)_{3} with our correlated uniform estimates, we can get the existence of strong solutions of our system. Throughout the paper, we denote by
In the following sections, we will use simplified notations for standard Sobolev spaces and Bochner spaces, such as etc.
A priori Estimates for Smooth Solutions
We construct an anxiliary function
Then we will prove that is local bounded (in time). Next, each terms of will be estimated as follows:
Estimate for
Firstly, By (1)_{2}
(5)
Then
Taking it by L^{2}-norm, Young’s inequality, we get
(6)
We deal with the term of , Multiplying (1)_{3} by
and integrating over (0, 1), we get
Consequently,
(7)
where is the initial mass.
Substituting (7) into (6), we get
(8)
Then muntiplying (1)_{1} by , integrating over (0, 1) with respect to x, we have
Integrating by parts, using Sobolev inequality, we deduct that
(9)
Then differential (1)_{1} with respect to x, and multiplying it by , integrating over (0, 1) 0n x, and using Sobolev inequality, we have
(10)
From (9) and (10), by Gronwall’s inequality, it follows that
(11)
And using (1)_{1 }we can also obtain
(12)
Where C is a positive constant, depending only on M_{0}.
Estimate for
Multiplying (1)_{2} by u_{t}, and integrating over , we have
(13)
Since
(14)
and
(15)
Substituting (14), (15) into (13), by (1)_{1}, Sobolev inequality and Young’s inequality, we have
(16)
Where 0<η≤1, to estimate (16), combining with (1)_{1 }the following estimates are hold
(17)
Combining (16), (17), yields
(18)
Where C is a positive constant, depending only on M_{0}.
Estimate for
Differentiating (1)_{2} with respect to t, multiplying it by u_{t}, integrating it over (0, 1) on x, we derive
(19)
Note that
(20)
From (20) and (1)_{1}, (19) can be rewritten into
(21)
Using Sobolev inequality, Young’s inequality, (8), we get
We deal with the estimate of Φ_{xt}.
Differential (1)_{3} with respect to t, multiplying it by Φ_{t} and integrating over (0, 1), we have
Then
Thus
Substituting these estimates into (21), we obtain
(22)
Then integrating (22) over , we deduce that
(23)
We estimate as follows:
Using (1)_{2 }and according to the smooth of (ρ, u, Ф) we have
(24)
Then
(25)
Taking limit on τ for (23) as τ→0, we get
(26)
By virtue of (11),(8),(18) and (26), we deduce that
(27)
By the definition of , we have
(28)
For the inequality (28), if , then we take ; On the other hand, if , we can find , such that .
Choose , we deduce that
, where C_{1}, C_{2} is positive constant. Then, we obtain the following estimate
(29)
Where C is a positive constant, depending only on M_{0}.
4. Proof of the Existence
In this section, we will use the uniform estimates (33) to prove the existence of the main theorem. Our method that constructed approximate systems is similar to that in [11], we take a semidiscrete Galerkin scheme. We take our basic function space as and the finite-
dimensional subspaces as
.
Here is the mth eigenfunction of the strongly elliptic operator defined on X. Let satisfy the hypotheses of Theorem 2.1.1. Asume for the moment thatand in (0, 1) (for some constant ). We may construct an approximate solution for any
where and
in .
The initial and boundary conditions are
and
, ,
,
Under the hypotheses of Theorem 2.1.1, similarly, for any fixed , we may get the similar estimate
(30)
Combining the course of estimates and the initial condition of approximate system, we can easily deduce that C is dependent on T, ρ_{0}, u_{0}. Moreover, because the constants C are independent of the lower bound of ρ_{0}. Here C(T) does not depend on δ and m (for any m≥M), M is dependent on the approximate velocity of initial condition). Thus, we can deduce from the two above estimates that (ρ^{m}, u^{m}, Φ^{m}) conveges, up to an extraction of subsequences, to some limit (ρ_{δ}, u_{δ}, Φ_{δ}) in the obvious weak sense, and there are estimates: δ>0, we may get the similar estimate
(31)
Because C(T) is independent of δ, when δ→0, we can deduct that (ρ_{δ}, u_{δ}, Φ_{δ}) converges, up to an extraction of subsequences, to some limit (ρ, u, Φ) in weak sense and
(32)
From the L^{p}-strong estimates of the equation (1)_{3}, we can easily get the regularity in Theorem 2.1.1.
5. Proof of the Uniqueness
Let , be two solutions of the problem (1)-(2). After substituting into the equation respectively, choosing test function , we obtain
(33)
We denote
Since
Consequently (5.1) can be rewritten as
(34)
Similarly, choosing test function , we get
(35)
Moreover, from (1)_{1} we have
,
Similarly,
Multiplying it by and integrating over, we get
(36)
From (34)-(36), we obtain
(37)
And then, Grownwall’s inequality yields
,,
From the classical theorems of equation (1.1)_{3}, we get
.
This completes the proof of uniqueness.
6. Conclusion
This paper study the Navier-Stoke-Poisson equations for compressible non-Newtonian fluids in one dimensional bounded intervals. The motion of the fluid is driven by the compressible viscous isentropic flow under the self-gravitational and an external force. The local existence and uniqueness of strong solutions was proved based on some compatibility condition. Through the research of this paper can be for further study of the mechanism of this kind of models and will provide a theoretical basis for further practical applications.
Acknowledgements
This work is supported by the Tian Yuan Mathematical Foundation of China (No. 11526105) and the Scientific Research Foundation of Liaoning University of Technology (No. X201404).
References