The well-posedness problem of an anisotropic porous medium equation with a convection term

The initial boundary value problem of an anisotropic porous medium equation is considered in this paper. The existence of a weak solution is proved by the monotone convergent method. By showing that ∇u∈L∞(0,T;Lloc2(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla u\in L^{\infty}(0,T; L^{2}_{\mathrm{loc}}(\Omega ))$\end{document}, according to different boundary value conditions, some stability theorems of weak solutions are obtained. The unusual thing is that the partial boundary value condition is based on a submanifold Σ of ∂Ω×(0,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\partial \Omega \times (0,T)$\end{document} and, in some special cases, Σ={(x,t)∈∂Ω×(0,T):∏ai(x,t)>0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Sigma = \{(x,t)\in \partial \Omega \times (0,T): \prod a_{i}(x,t)>0 \}$\end{document}.


Introduction
The well-posedness and regularity of weak solutions to the porous medium equation were addressed from the sixties to eighties in the twentieth century by many mathematicians, one can refer to [3,6,17,18,21,22] and the references therein. Later, DiBenedetto [2] and Ziemer [28] studied the regularity to the more general equation considering suitable assumptions on a and b. The proofs followed different approaches: DiBenedetto's proof was based on a parabolic version of De Giorgi's technique, while Ziemer's approach was related to Moser's iteration technique. But, since many reactiondiffusion processes depend on different environments, one should consider a reactiondiffusion equation with anisotropic characteristic, then the anisotropic porous medium equation modeled by was introduced and studied since 1980s. Actually, Song [19,20] studied the existence and uniqueness of the very weak solution of the anisotropic porous medium equation with singular advections and absorptions. Henriques [7] established an interior regularity result for the solutions of (1.4). Li [11] developed the finite element method to derive a special analytical solution for anisotropic porous medium equation for time-independent diffusion. Also, several applied models related to an anisotropic porous medium have been introduced recently. The first one is the flow diverter model. Since the explicit modeling of thin wires of simulation of flow diverter (FD) imposes extremely high demand of computational resources and time, such a fact limits its use in time-sensitive presurgical planning. One alternative approach is to model as a homogenous porous medium, which saves time but with compromise in accuracy. Then, Ou et al. [13] proposed a new method to model FD as a heterogeneous and anisotropic porous medium whose properties were determined from local porosity. The second one is a multiple-relaxation-time lattice Boltzmann model for the flow and heat transfer in a hydrodynamically and thermally anisotropic porous medium [8]. The third one arises from computational fluid dynamics (CFD). Doumbia et al. [5] gave a CFD modelling of an animal occupied zone using an anisotropic porous medium model with velocity-dependent resistance parameters. Another model comes from the physical characteristics of cracked rocks. By testing elastic velocities and Thomsen parameters-as a function of crack density for fixed values of aspect ratio-predicted by the model with data acquired from synthetic rock samples, Nascimento et al. [12] introduced a new ultrasonic physical model in an anisotropic porous cracked medium. Moreover, in the theory of PDE, the anisotropic equation has provoked more people's attention in recent time. For example, the existence and multiplicity of nontrivial solutions to the anisotropic elliptic equation has been an active topic in recent years (see [4,15,16], etc.), while the anisotropic parabolic equation was studied in [1,14], etc.
In this paper, we consider the well-posedness of weak solutions to the following initial boundary value problem: and We found that if one wants to prove the uniqueness (or the stability) of weak solutions to this equation, the homogeneous boundary value condition (1.9) can be replaced by that a(x) = 0, x ∈ ∂ . Even much earlier, Yin and Wang [23,24] studied the following equation: divided the boundary value condition into three parts, and in particular they showed that if a(x) = 0, f i (x) = 0 when x ∈ ∂ , then the uniqueness of a weak solution to equation (1.11) can be proved independent of the boundary value condition (1.9). The optimal boundary value condition matching up with equation (1.11) was studied by the author recently in [26]. Instead of a(x)| x∈∂ = 0 in [27], we only assume that a i (x, t) > 0, (x, t) ∈ × (0, T) and do not emphasize that So, for any given t ∈ [0, T], both x ∈ ∂ : a i (x, t) = 0 and x ∈ ∂ : a i (x, t) > 0 may have a positive (N -1)-dimensional Hausdorff measure in ∂ . Naturally, based on past experience [23,24,27], we guess that a partial boundary value condition u(x, t) = 0, (x, t) ∈ ⊆ ∂ × (0, T), (1.12) is enough to ensure the well-posedness of weak solutions to equation (1.7). The further work is to specify the explicit expression of in (1.12). Different from other related references [23,24,27] in which is just a cylinder, we found that appearing in (1.12) is a submanifold of ∂ × (0, T) and, in some special cases, Actually, compared with [7,19,20], the degeneracy of diffusion coefficient a i (x, t) has brought more essential difficulties. For example, maybe it is not difficult to construct the fundamental solution of equation (1.4) by Barenblatt's method, but it is impossible to construct the corresponding fundamental solution of the simplest anisotropic porous medium equation by a similar method. However, the main aim of this paper is to study the well-posedness of weak solutions to equation (1.7), and we pay no attention to the fundamental solution for the time being. The local integrability ∇u ∈ L ∞ (0, T; L 2 loc ( )), which was found for the first time in this paper, acts as an important role to overcome the above difficulties.
The remainder of this paper is structured as follows. In Sect. 2, we present the definition of weak solution and the main results. In Sect. 3, the existence of a weak solution is proved. In Sect. 4, the stability of a weak solution to the usual initial boundary value problem is studied. In Sect. 5, the local integrability of ∇u is found and the uniqueness of a weak solution to the usual initial boundary value problem is obtained. In Sect. 6, when N i=1 a i (x, t)| x∈∂ = 0, the stability of a weak solution based on a partial boundary value condition is proved. In Sect. 7, we prove the stability of weak solutions under the general condition N i=1 a i (x, t) ≥ 0.

The definition of the weak solution and the main results
The definition of weak solution and the main results of this paper are listed below.
and for any function ϕ ∈ C 1 0 (Q T ), there holds The initial value condition is satisfied in the sense of that where φ(x) ∈ C ∞ 0 ( ). The boundary value condition (1.9) or the partial boundary value condition (1.12) is satisfied in the sense of trace.
From Theorem 2.3 to Theorem 2.6, we all assume that a i (x, t) > 0, x ∈ and denote that (1.7) with the initial value u 0 (x), v 0 (x) respectively, and with the boundary value condition (1.9). If α i ≥ 0, and there is a constant α > 1

Theorem 2.3 Let u(x, t) and v(x, t) be two solutions of equation
then the solution of equation (1.7) is unique. (1.7) with the initial value u 0 (x), v 0 (x) respectively, with the same boundary value condition (1.9). If

Theorem 2.4 Let u(x, t) and v(x, t) be two nonnegative solutions of equation
Theorem 2.4 implies that we only can show that the stability of weak solutions is true for a kind of solutions which satisfy (2.7). The following stability theorems are established on a partial boundary value condition.

Theorem 2.5 Let u(x, t) and v(x, t) be two solutions of equation
with the initial value u 0 (x), v 0 (x) respectively, and with a partial boundary value condition (2.14) Here, Theorem 2.5 is based on the fact that we can show the first order partial derivative to the solution u is with the local integrability The weakness of Theorem 2.5 is that the expression of , (2.15) seems too complicated. By choosing another test function, we can prove another stability theorem based on a simpler partial boundary value condition.
respectively, but without the boundary value condition. If (2.13) is true and We find that the partial boundary value conditions with (2.12) are submanifold of ∂ × (0, T), while in the previous works the corresponding partial boundary value conditions are the cylinder domains 1 × (0, T), where 1 ⊆ ∂ is a relatively open subset [9,10,23,25,27], etc.
Last but not least, once the well-posedness problem has been solved, we can consider the extinction, blow-up phenomena, the positivity, and the large time behavior of the weak solutions of anisotropic porous medium equation (1.7) in the future. However, different from the porous medium equation, because of the anisotropy, these problems are not so easy to be solved, the methods used in the usual porous medium equation (1.1) cannot be extended to the anisotropic porous medium equation (1.13) directly.

The existence of weak solution
Proof of Theorem 2.2 We consider the following normalized problem: Similar to the porous medium equation (1.1), we can show that problem (3.1) has a nonnegative solution u n , which is called as a viscous solution generally and satisfies and by comparison theorem, we have is well defined. Now, we will prove u is a weak solution of (1.7). First, multiplying both sides of the first equation in (3.1) by φ = u n -1 n , denoting that Thus, from (3.5), we can find that which implies By choosing a subsequence, we may assume that weakly in L 2 (Q T ).
In the second step, we want to show that Let n → ∞. Then For the right-hand side of (3.9), by the assumption using the dominated convergent theorem, we have (3.11) From (3.9)-(3.11), we obtain (3.8).
In the third step, since b i ∈ C 1 , by (3.4), we have Then u t ∈ L 1 (Q T ) and (2.1) is true, and we can define the trace of u on the boundary ∂ .
The initial value condition true in the sense of (2.3) can be found in [22] etc.
Actually, in the next section, we will show that |u x i | ∈ L 2 0, T; L loc ( ) .

The stability of weak solution of the usual initial boundary value problem
For any given positive integer n, let g n (s) = As we have pointed out in the introduction section, for the classical porous medium equation t) and v(x, t) are two nonnegative solutions of the initial boundary value problem, by choosing g n (u mv m ) as the test function, we easily show that Now, for the anisotropic diffusion equation (1.7) considered in this paper, since α i may be different from one to another, though for every i we cannot choose g n (u 1+α iv 1+α i ) as a test function. If we insist on using a similar method to obtain the stability (5.3), then only for a kind of weak solution we can achieve the requirement. Proof By a process of limit, we can choose g n (uv) as the test function, then

Theorem 5.1 Let u(x, t) and v(x, t) be two nonnegative solutions of the initial boundary
Obviously, By (2.7) and α i ≥ 1, using the Lebesgue dominated theorem, we have From (5.8)-(5.9), we obtain We now prove that In detail, by (5.4), we have Here, we have used the notation Let n → ∞ in (5.12). Since (5.5), While D 0 = {x ∈ : |u -v| = 0} has a positive measure, by that Thus, in both cases, the right-hand side of inequality (5.12) goes to 0 as n → ∞. Moreover, (5.14) At last, let n → ∞ in (5.6). Then

The partial boundary value condition based on a submanifold
In this section, we consider equation (1.7) with the initial value condition (1.8) and with a partial boundary value condition (2.12). For a small positive constant λ > 0 and any t ∈ [0, t), let Proof of Theorem 2.5 If we choose φg n (uv) as the test function, then and from Q T |u t | dx dt ≤ c, we deduce Since by that α i ≥ 1, using the Lebesgue dominated theorem, we have At the same time, if we denote that then by (2.10) we have For the convection term, by (6.5) and (2.13), we have By (2.13), using the homogeneous boundary value condition (2.12), we have Now, after letting n → ∞, let λ → 0 in (6.2). Then Theorem 2.5 is proved.

The proof of Theorem 2.6
In this section, we prove Theorem 2.6. For a small positive constant λ > 0 and any t ∈ [0, t), set λt and φ(x) as (6.1).
Proof of Theorem 2.6 Let u(x, t), v(x, t) be two solutions of equation (1.7) with the initial boundary values u 0 (x), v 0 (x) respectively, but without the partial boundary value condition (2.12). By assumption (2.17), we can choose g n (φ(uv)) as the test function and get In the first place, we have Clearly, Since a i (x, t) satisfies (2.17) and φ(x, t) = 0, x ∈ \ λt by that α -≥ 1 and |∇u| ∈ L ∞ 0, T; L 2 loc ( ) , |∇v| ∈ L ∞ 0, T; L 2 loc ( ) , using the Lebesgue dominated convergence theorem, we have In the second place, we have While by (2.18) Similarly, we have In the third place, since (1.13) |b i (u, x, t)b i (v, x, t)| ≤ c √ a i (x, t), we have Moreover, since (4.1), we clearly have Now, let n → ∞ in (7.1). Then Theorem 2.6 is proved.