On the Non-Newtonian Fluid Equation with a Source Term and a Damping Term
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.
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 . However, the damping term and the source term in (1) may change the situation.
Bertsh et. al.  and Zhou et. al.  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.  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 , .
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 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.
(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 .
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.
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
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),
By (52), by the arbitrary of , we know thatThus in .
3. The Proof of Theorem 4
For small , let Obviously , and
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
4. The Proof of Theorem 5
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. Then