The Stability of the Solutions for a Porous Medium Equation with a Convection Term

Abstract

This paper studies the initial-boundary value problem of a porous medium equation with a convection term. If the equation is degenerate on the boundary, then only a partial boundary condition is needed generally. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of the solutions is studied. In some special cases, the stability can be proved without any boundary value condition.

1. Introduction

Consider the motion of the ideal barotropic gas through a porous medium. Let be the gas density, the velocity, and the pressure. The motion is governed by the mass conservation law the Darcy law and the equation of stage . Here, is a given matrix. We usually assume that with . The above laws then lead to a semilinear parabolic equation for the density : If , where is a function and is the unit matrix, then (3) becomes Also, (4) can be regarded as the generalization of the nonlinear heat equation where the function has the meaning of nonlinear thermal conductivity dependent on the temperature . If in (4) or in (5), that is, which is called the porous medium equation, there are well-known monographs or textbooks devoting to the well-posedness problem of (6); one can refer to [1–6] and the references therein. If in (4) or depending on in (5), the situation may be different from that of (6). For example, if , we consider the equation and suppose that there are two classical solutions and of (7) with the initial values and , respectively. Then it is easy to show that which implies that the classical solutions (if there are) of (7) are controlled by the initial value completely. In other words, the stability of the classical solutions of (7) is true without any boundary value condition. Yin and Wang [7] also showed that the non-Newtonian fluid equation with the type has similar properties, where is a bounded domain in with appropriately smooth boundary, , and is a constant. Since the diffusion coefficient vanishes on the boundary, it seems that there is no heat flux across the boundary. However, Yin and Wang [7] showed that the fact might not coincide with what we image. In fact, the exponent , which characterizes the vanishing ratio of the diffusion coefficient near the boundary, does determine the behavior of the heat transfer near the boundary. They proved that, if , the solution of (9), for some constant , and the trace of on the boundary can be defined in the traditional way; then, in physics sense, there is no heat flux across the boundary actually, while, if , the existence and uniqueness of solutions were proved without any boundary conditions, which means that whether there is heat flux across the boundary is uncertain. Later, Yin and Wang [8] had shown that only a partial boundary value condition matches up with the equation

Inspired by Yin and Wang [7, 8], we will study the porous medium equation with a convection term, with the initial value and with the partial boundary condition where is defined as follows. When , ; when , and is the inner normal vector of . The expression of is derived in [9], we do not repeat the details here.

We suppose that is a function, and

Definition 1. A nonnegative function is said to be the weak solution of (11) with the initial value (12), if for any function , , , there holds and the initial condition is satisfied in the sense that If satisfies (13) in the sense of the trace in addition, then we say it is a weak solution of the initial-boundary value problem of (11).

First of all, we will study the well-posedness problem of (11).

Theorem 2. If , , satisfy (15), then (11) with initial value (12) has a nonnegative solution. Moreover, if , then ; the solution is unique.

Then, we will study the stability of the solutions.

Theorem 3. If , i.e. equation (11) is not with the convection term, and are two solutions of equation (11) with the initial value , respectively, , then

Since in Theorem 3, there are some regrets more or less. For (11) itself, we can not prove the same conclusion for the time being. However, as compensation, we can consider a more complicate equation than (11),

Theorem 4. Let and be two solutions of (19) with the initial values , , respectively, if , and satisfies then the stability of the weak solutions is true in the sense of (18).

It is more or less strange that the case is not included in Theorems 3 and 4.

At last, we will probe the stability of the weak solutions based on the partial boundary value condition.

Theorem 5. Let , be two solutions of (11) with the initial values , , respectively. If , , and the partial boundary condition (12) is satisfied in the sense of trace, then where .

Theorem 6. If is a domain, , and , then (11) with the initial value and the partial boundary condition (13) has a BV solution. Moreover, let be two solutions of (11) with the different initial values , respectively. Then where , , and is the inner normal vector of .

If , Theorem 6 has been included in Theorem 3, while, if is not true, then Theorem 6 has its independent sense. Such phenomena that the solution of a degenerate parabolic equation may be free from the limitation of the boundary condition also can be found in [7–14]. We will use some ideas in [9, 14]. The uniqueness of the weak solutions when had been proved in [14]. Since [14] was written in Chinese, for the completeness of the paper, we still give its proof in what follows. In addition, how to obtain the stability (23) without condition (22) is a very interesting problem. Last but not least, roughly speaking, in this paper, we can show that if or , then the weak solution can be defined the trace on the boundary in the traditional sense; it is surprising that if , whether can be defined the trace on the boundary is unknown for the time being.

2. The Well-Posedness Problem

We consider the following regularized problem:

According to the standard parabolic equation theory, there is a weak solution which satisfies by the maximum principle.

Theorem 7. If , , and satisfy (14), then (11) with initial value (12) has a nonnegative solution.

Proof. First we suppose that and , and consider the following normalized problem Here, , and Thus, the solution of the problem is also a solution of problem (25). Moreover, by comparison theorem, we clearly have which yields Now, we can prove that the limit function is a weak solution of (6) with the initial value (8).
Multiplying both sides of the first equation in (25) by and integrating it over , we have By the fact then we have Thus, we obtain By choosing a subsequence, we can assume that weakly in . We need to prove that For any , denoting that , we have Let . The left hand side is while on the right hand side, by and by the condition , using the control convergent theorem, we have Thus we obtain (37).
At the same time, since , by (31), we have Thus, is a solution of (11) with the initial value (12).
If only satisfies (14), by considering the problem of (25) with the initial value which is the mollified function of , then we can get the conclusion by a process of limitation. Certainly, the solution generally is not continuous at , but satisfies (15) and (17). Theorem 7 is proved.

Lemma 8. Let satisfy (14). If and is a solution of (11) with the initial value (12), then there exists a constant such that

Proof. Since , there exists constant , such that . Therefore, there exists such that . Therefore, Thus can be defined the trace on the boundary in the traditional way. By the definition of the trace, we also know that can be defined as the trace on the boundary in the traditional way. The lemma is proved.

Theorem 9. If and satisfies (14), then , and the solution of the initial-boundary value problem (11)–(13) is unique.

Proof. First of all, by Theorem 7 and Lemma 8, there is a nonnegative solution of the initial-boundary value problem (11)–(13). Then, we prove its uniqueness. Let be two solutions of equation (11) with For all , For any given positive integer , let , . Then , and we have Since , by Lemma 8, we can define the traces of on the boundary. By a process of limit, we can choose as the test function; then Moreover, we can prove that In detail, the limitation (51) is established by the following calculations.Since , In (52), let . If is a set with measure, by , we have If the set has a positive measure, then, Therefore, in both cases, the right hand side of inequality (52) goes to 0 as .
Clearly, Now, let in (50). Then We have the conclusion.

By Theorems 7 and 9, we clearly have the following.

Corollary 10. Theorem 2 is true.

3. The Stability without the Boundary Value Condition

Consider a simpler equation than (11). with the initial value (12), but without any boundary value condition. For a small positive constant , let and let

Proof of Theorem 3. Suppose , only satisfy (7), . Let , be two solutions of (58) with the initial-boundary values , , respectively. For all , By a process of limit, we can choose as the test function; then Clearly, we haveAs for the term we have The last equality of (65) is due to that since , we have Now, after letting , let in (62). Then Theorem 3 is proved.

Consider a more complicated equation than (11).with the initial value (12), but without any boundary value condition.

Proof of Theorem 4. Suppose , only satisfy (7), . Let be two solutions of equation (11) with the initial-boundary values , , respectively. For all ,By a process of limit, we can choose as the test function; then Let us analyze every term in the left hand side of (70). For the first term, we clearly have For the second term, we have For the third term, since by , we have by .
Now, we deal with the terms related to the convection function in (70). In the first place, by (20), according to the definition of the trace, we have Moreover, we can prove that In detail, the limitation (77) is established by the following calculations. By (21), Let in (78). If is a set with measure, then If the set has a positive measure, then, Therefore, in both cases, the right hand side of inequality (74) goes to 0 as .
At last, Now, after letting , let in (71). By (72), (73), (74), (76), (77), (78), (79), (80), and (81), then Theorem 4 is proved.

4. The Stability Based on the Partial Boundary Value Condition

In this section, we will prove Theorem 5; the proof is similar as that of Theorem 4

Proof of Theorem 5. Suppose , only satisfy (7), . Let , be two solutions of (11) with the initial-boundary values , , respectively, and with the same homogeneous partial boundary value condition For all , By a process of limit, we can choose as the test function as in Theorem 4; thenLet us analyze every term in (85). By and then according to the definition of the trace, by (83), we have Moreover, we can prove that In detail, the limitation (87) is established by the following calculations. Since ,In (88), let . If is a set with measures, then If the set has a positive measure, then, by (22), Therefore, in both cases, the right hand side of inequality (88) goes to 0 as .
At the same time, then, Since and by (22) then we have Clearly, Now, after letting , let in (85). Then By Gronwall Lemma, the stability (23) is true. Theorem 5 is proved.

5. The BV Solution of Equation

Recently, Zhan considered the initial-boundary value problem of the following equation in [9] with where

Definition 11. A function is said to be the entropy solution of equation (98) with the initial-boundary values (99)-(100), if we have the following.
(1) There exists , , such that, for any , where is composite mean value of .
(2) For any , , for any , and for any small , satisfies (3) The boundary condition (100) is true in the sense of trace. The initial value is true in the sense of The existence of the BV solution of equation (98) is by considering the following regularized problem: with the initial-boundary conditions If there is a constant such that where , then we have the following important estimate.

Theorem 12 (see [9]). Let be the solution of (105) with (106) and (107). If and are all bounded, assumption (108) is true; then where and is independent of .

By the theorem, we can prove the existence of the entropy solution of equation (98) in the sense of Definition 11.

Theorem 13 (see [9]). Suppose that is , is , , and there is a constant such that (108) is true. Then (98) with the initial condition (99) has an entropy solution in the sense of Definition 11. Moreover, let , be two solutions of (6) with the initial value , satisfying (7). Then where , , and is the inner normal vector of . In particular, if , then ; we have

If , Theorem 13 implies that the solution of (98) is controlled by the initial condition. In other words, no boundary value condition is needed.

Now, let Then, for any , by the fact that a.e. in , If , , there exists such that inequality (108) is true. But, in general, the distance function only is a continuous function and is differential for almost everywhere in ; then does not belong to , so we can not have Theorems 12 and 13 directly. However, if we check the proof of Theorems 12 and 13, only if we assume that is a appropriately smooth such that is integral on , then similar to the proof of Theorems 12 and 13, we can prove Theorem 4; we omit the details here.

Remark 14. If is a domain, then is differential near the boundary , so is a continuous function on and is integral on .

Remark 15. If , , by Theorem 12, , we can define the trace of on the boundary .

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The paper is supported by NSF of China (no. 11371297), supported by NSF of Fujian Province (no. 2015J01592), China.