Abstract
A kind of non-Newtonian fluid equation with a damping term and a source term is considered. After giving a result of the existence, if the diffusion coefficient is degenerate on the boundary, the local stability of the weak solutions is established without any boundary condition. If the diffusion coefficient is degenerate on a part of the boundary, by imposing the homogeneous value condition on the other part of the boundary, the local stability of the weak solutions is proved. Moreover, if the equation is with a damping term, other than the finite propagation property, the results of this paper reveal the essential differences between the non-Newtonian fluid equation and the heat conduction equation in a new way.
1. Introduction
Consider the parabolic equationrelated to the Laplacian, with the initial valueand the usual boundary valuewhere and , , is a continuous function, is a bounded domain with a smooth boundary , , . The equation comes from a host of applied fields such as the theory of non-Newtonian fluid, the water infiltration through porous media, and the oil combustion process; one can refer to [1–4] and the references therein. For the evolutionary Laplacian equationand with the initial-boundary value conditions (2) and (3), the weak solution is unique and has finite propagation property [3]. However, the damping term and the source term in (1) may change the situation.
Bertsh et. al. [5] and Zhou et. al. [6] had discussed the existence and the properties of the viscosity solutions for the equationand shown that the uniqueness of the weak solution is not true, where is a positive constant. Zhang et. al. [7] had discussed the existence and the properties of the viscosity solution for the equationand shown that the uniqueness of the weak solution is not true, where and at least there exists a point such that , .
Meanwhile, Ji Benedikt et. al. [8, 9] had shown that the uniqueness of the solution of the following equation is trueis not true provided that , and at least there exists a point such that .
In this paper, we first assume thatand then (1) is degenerate on the boundary. Such a degeneracy may have a substantial influence on the solutions. If one considers the well-posedness of (1), one expects that such a degeneracy may counteract the effects from the damp term and the source term. A typical example is the equation which was studied by Yin-Wang [10, 11]. Here, is the distance function from the boundary and satisfies (8). Yin-Wang showed that, if , although the weak solution may lack the regularity to be defined and the trace on the boundary and the boundary value condition (3) cannot be imposed in the trace sense, the uniqueness of the weak solution is still true. Moreover, the author had studied the equation and shown that condition (8) may act as the role as the boundary value condition (3) and ensure the well-posedness of the solutions [12–15].
Coming back to (1). On the one hand, based on the knowledge of (5) and (6), if , , and in (1), then the uniqueness of the solution is not true. Accordingly, in this paper, we consider the well-posedness of (1) whether or on the boundary . On the other hand, based on the knowledge of [10–15], when and satisfies (8), we can expect that the uniqueness of the weak solution to (1) is still true, even if as (6).
Definition 1. A function is said to be a weak solution of (1) with the initial value (2), ifand for any function ,The initial value is satisfied in the sense
Definition 2. The function is said to be the weak solution of (1) with the initial value (2) and the boundary value condition (3), if satisfies Definition 1, and the boundary value condition (3) is satisfied in the sense of trace.
Theorem 3. Let satisfy (8), , be a Lipschitz function,If , andthen (1) with initial value (2) has a nonnegative weak solution. Moreover, ifthen the initial-boundary value problem (1), (2), (3) has a nonnegative solution in the sense of Definition 2.
Since when , condition (16) implies that ; hereafter, the constants may depend on . We think the existence of the weak solutions can be proved only if , and the condition is just a makeshift. Also condition (16) may not be necessary, but we are not ready to pay so much attentions to the existence. We will focus on the uniqueness of the weak solution.
Theorem 4. Let be two weak solutions of (1) with the initial values , respectively. If satisfies (8), , is a continuous function, , then there is a constant such that
Since satisfies (8), Theorem 4 implies the uniqueness of the weak solution to (1) is true even without the boundary value condition. Moreover, we have the following two simple comments.
(1) Theorem 4 includes the case of and ; in other words, the uniqueness of the weak solution to the following equation is true, where and .
(2) Theorem 4 includes the case of and and at least there exists a point such that . In other words, the uniqueness of the weak solution to the following equation is true, where and .
Compared with (6) and (7), Theorem 4 reveals that the degeneracy of brings the new change about the property of the solutions.
In order to illustrate the problem more clearly, secondly, we assume thatIn this case, we consider the uniqueness of weak solution to (1) under a partial boundary value condition. This is the following theorem.
Theorem 5. Let satisfy (21) and be two weak solutions of (1) with the initial values , respectively, with the same partial boundary value conditionIf , , satisfies (17), (22), and (23), and is a continuous function, then there is a constant such that the local stability is true in the sense of (18).
If we notice that satisfies (22), according to [5–7], the uniqueness of the solution to the equation is not true when is near to , while Theorem 5 implies that the uniqueness of the solution to the equation is true provided that . This fact shows the differences between the heat conduction equation () and the non-Newtonian fluid equation () again. It is well-known that the heat conduction equation has the infinite propagation property, while the non-Newtonian fluid equation has the finite propagation property.
2. The Weak Solutions Depend on the Initial Value
It is supposed that satisfies Let and be uniformly bounded, and let converge to in . For simplicity, we may assume that is a function without loss the generality.
We now consider the following regularized problemSince satisfies (4), it is well-known that the above problem has a unique nonnegative classical solution [3, 16].
By the maximum principle, we have
Multiplying (28) by and integrating it over , we get Since , and by , we haveand
Multiplying (28) by , integrating it over ,Noticing that we have
Moreover, by the Young inequality and the Hölder inequality,By (36), (38), (39), and (40),By (35), (41), we know can be embedded into compactly. Then a.e. in . At the same time, since ,
Hence, by (31), (35), (40), (41), (42) there exists a function , dimensional vector function , and a function such that and Here, is the signed Radon measures on . In order to prove that satisfies (1), we notice that for any function ,Now, in the first place, we can provein a similar way to that of the usual Laplacian equation. Then, letting in (45),
What is more, by the weak convergent theorem, for any ,For any ,By (48),Combining (49) with (50), we havefor any . Clearly, for any , (50) is still true.
By (52), by the arbitrary of , we know thatThus in .
Since , by (53), for any function is clearly. Thenfor any function . Combining (46) with (55), satisfies (7).
At last, we are able to prove (13) as in [17]; thus we have Theorem 3.
3. The Proof of Theorem 4
For small , let Obviously , and
Theorem 6. Let be two weak solutions of (1) with the initial values , respectively. If satisfies (8), , there is a constant such that one of the following conditions is true:
;,, andthen
Proof. Let , be two solutions of (1) with the initial values . We can choose as the test function. ThenThusBy that in , we haveIf , since , , we haveIf , since , , we haveIf only if However, this inequality is natural since .
By (65)-(68), we havewhere .
If , then . Since we haveIf , then . Since and we haveIf , by (60), we haveMoreover, since , , is a continuous function,Now, let in (62). Thenwhere . This inequality implies that
Proof of Theorem 4. Since , , and , conditions (58) and (59) are true naturally, by Theorem 6, we have the conclusion.
4. The Proof of Theorem 5
Lemma 7. If , is a weak solution of (7) with the initial condition (2). Then the trace of on the boundary can be defined in the traditional way.
This lemma can be found in [14]. Recall that we have assumed (21)-(23), i.e., Let be a function satisfying thatand LetThen .
Theorem 8. Let be two weak solutions of (1) with the initial values , respectively, with the same partial boundary value condition If , satisfies (17), (22)-(23), and there is a constant such that one of the following conditions is true,;
, and (60) is true;
then the local stability (18) is true.
Proof. Since , then ; accordingly, we can choose as the test function. ThenAt firstIn the second place, by the fact that in , we haveThe last inequality is obtained similar to (66)-(68), where . Thirdly,Since when , and satisfies we haveFourthly, if ,where .
If , by (60), we haveLast but not least,Now, let in (85). Thenwhere . This inequality implies that
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The paper is partially supported by NSF of Fujian Province and partially supported by the Open Research Fund Program form Fujian Engineering and Research Center of Rural Sewage Treatment and Water Safety, China.