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

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.


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 ,  = const.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 ℎ(, ) has the meaning of nonlinear thermal conductivity dependent on the temperature  = (, ).If () ≡ 1 in (4) or ℎ(, ) ≡ ℎ() in (5), that is, which is called the porous medium equation, there are well-known monographs or textbooks devoting to the wellposedness problem of (6); one can refer to [1][2][3][4][5][6] and the references therein.If () ≥ 0 in (4) or ℎ(, ) depending on  in (5), the situation may be different from that of (6).For example, if ()| ∈Ω = 0, we consider the equation and suppose that there are two classical solutions  and V of (7) with the initial values  0 and V 0 , 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   = div (  () |∇| −2 ∇) , (, ) ∈ Ω × (0, ) (9) has similar properties, where Ω is a bounded domain in   with appropriately smooth boundary, () = dist(, Ω), and  > 0 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 0 <  <  − 1, the solution of (9),  ∈   0 for some constant  > 1, 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  ≥  − 1, 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 Σ  is defined as follows.When 0 <  < 1, Σ  = Ω; when  ≥ 1, Σ  = { ∈ Ω :    (0)  () < 0} and {  } is the inner normal vector of Ω.The expression of Σ  is derived in [9], we do not repeat the details here.
We suppose that   () is a  1 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  ∈  1 (  ), | = = 0, | Ω = 0, there holds and the initial condition is satisfied in the sense that lim 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).
Then, we will study the stability of the solutions.Theorem 3. If   () ≡ 0, i.e. equation (11) is not with the convection term,  and V are two solutions of equation (11) with the initial value  0 (), V 0 () respectively,  > 1, then Since   () ≡ 0 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 V be two solutions of (19) with the initial values  0 (), V 0 (), respectively, if 1 <  < 2, and then the stability of the weak solutions is true in the sense of (18).
It is more or less strange that the case  = 1 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.
If   ≡ 0, Theorem 6 has been included in Theorem 3, while, if   ≡ 0 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][8][9][10][11][12][13][14].We will use some ideas in [9,14].The uniqueness of the weak solutions when Σ  = Ω 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  <  < 1 or  ≥ 2, then the weak solution  can be defined the trace on the boundary in the traditional sense; it is surprising that if 1 ≤  < 2, whether  can be defined the trace on the boundary is unknown for the time being.

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.
Proof.First we suppose that  0 ∈  ∞ 0 (Ω) and 0 ≤  0 ≤ , and consider the following normalized problem Here,   () ⩾ () > 0, 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  =    − (1/)  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  2 .We need to prove that For any ∀ ∈  ∞ 0 (Ω), denoting that   ≜ +1/, we have Let  → ∞.The left hand side is while on the right hand side, by and by the condition 0 <  < 2, using the control convergent theorem, we have lim Thus we obtain (37).At the same time, since   ∈  1 , by (31), we have lim Thus,  is a solution of (11) with the initial value (12).If  0 only satisfies (14), by considering the problem of (25) with the initial value  0 which is the mollified function of  0 , then we can get the conclusion by a process of limitation.Certainly, the solution (, ) generally is not continuous at  = 0, but satisfies (15) and (17).Theorem 7 is proved.Lemma 8. Let  0 satisfy (14).If 0 <  < 1 and  is a solution of (11) with the initial value (12), then there exists a constant  > 1 such that Proof.Since  < 1, there exists constant  ∈ (, 1),  < ( + 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  > 0, 1 >  > 0 and  0 () ≥ 0 satisfies (14), then Σ  = Ω, 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 , V be two solutions of equation (11) with For all 0 ≤  ∈  1 0 (  ), Since 0 <  < 1, by Lemma 8, we can define the traces of , V on the boundary.By a process of limit, we can choose   (  − V  ) as the test function; then Moreover, we can prove that lim In detail, the limitation (51) is established by the following calculations.
In (52 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.

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  > 0, let and let Proof of Theorem 3. Suppose  0 , V 0 only satisfy (7),  > 1.Let , V be two solutions of (58) with the initial-boundary values  0 , V 0 , respectively.For all 0 ≤  ∈  1 0 (  ), By a process of limit, we can choose   (  − V  ) as the test function; then Clearly, we have As for the term we have lim The last equality of (65) is due to that since  > 1, we have lim Now, after letting  → 0, 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  0 , V 0 only satisfy (7), 1 <  < 2. Let , V be two solutions of equation ( 11) with the initialboundary values  0 , V 0 , respectively.For all 0 ≤  ∈  1 0 (  ), (69) By a process of limit, we can choose   ((  − V  )) as the test function; then Let us analyze every term in the left hand side of (70).For the first term, we clearly have lim For the second term, we have For the third term, since lim by 0 >  − 2 > −1, we have lim by lim →∞    () = 0. Now, we deal with the terms related to the convection function   in (70).In the first place, by (20),       (  , , ) −   (V  , , )     ≤ 2 () ; according to the definition of the trace, we have lim Moreover, we can prove that lim In detail, the limitation (77) is established by the following calculations.

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  0 , V 0 only satisfy (7), 1 ≤  < 2. Let , V be two solutions of (11) with the initial-boundary values  0 , V 0 , respectively, and with the same homogeneous partial boundary value condition For all 0 ≤  ∈  1 0 (  ), By a process of limit, we can choose   ((  − V  )) as the test function as in Theorem 4; then Therefore, in both cases, the right hand side of inequality (88) goes to 0 as  → ∞.
At the same time, then, lim and by ( 22) then we have lim Clearly, lim Now, after letting  → 0, let  → ∞ in (85).Then By Gronwall Lemma, the stability (23) is true.Theorem 5 is proved.
Theorem 12 (see [9]).Let   be the solution of ( 105) with ( 106) and ( 107 By the theorem, we can prove the existence of the entropy solution  ∈ BV(  ) of equation (98) in the sense of Definition 11.