Hölder regularity for porous medium systems

We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. ∂tu-div(m|u|m-1Du)=0,m>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$\end{document}As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.


Introduction and main results
Let E be an open set in R N and E T := E × (0, T ] for some T > 0. For a positive m, we are interested in the following parabolic system ∂ t u i − div(m|u| m−1 Du i ) = 0 weakly in E T , i = 1, · · · , n, u ≡ (u 1 , · · · , u n ). (I) In the scalar case, i.e. n = 1, it is commonly referred to as the porous medium equation. It presents degeneracy in the set [|u| = 0] when m > 1 and singularity when 0 < m < 1.
As is well known, everywhere regularity is in general not expected from systems of elliptic or parabolic type (cf. [8]). Nevertheless, the special quasi-diagonal structure of (I) enables us to achieve the Hölder regularity of locally bounded solutions.
Let u be a solution to the system (I). Loosely speaking, our main result reads u ∈ L ∞ loc (E T ) ⇒ u ∈ C α loc (E T ) for some α ∈ (0, 1) depending only on n, N and the exponent m; see Theorem 1.1 for details. Based on our explicit local estimates, we are able to obtain a Liouville type result, which asserts bounded global solutions must be constant; see Corollary 1.1. In addition, we explore sufficient conditions that ensure local boundedness of solutions in Sect. 6.
The norm of u refers to the standard Euclidean norm, i.e. |u| = (u 2 1 + u 2 2 + · · · + u 2 n ) 1 2 . However, it is worth pointing out that our Hölder estimate for solutions to the system (I) still holds for any norm of u. This point will be clear from the proof. Thus, in particular, we could use |u| 1 = |u 1 | + |u 2 | + · · · + |u n | in the system (I). Moreover, our results also hold for more general systems modeled on (I), which permit us to insert bounded and measurable diffusion coefficients in the system (I) (cf. (1.1) -(1.2)).
Our motivation of studying the system (I) with measurable coefficients as in (1.1) -(1.2) mainly stems from the C 1,α regularity for the parabolic p-Laplace equation, i.e. u t − div |Du| p−2 Du = 0.
The spatial gradient Du will satisfy a system with structures similar to (I). In fact, we may differentiate the above equation in x s formally to obtain Here the summation convention is evoked. Introduce the vector valued function v := Du. In terms of v the above differentiated equation can be written as where the measurable coefficients are defined by which in turn fulfills the condition min{ p − 1, 1}|ξ | 2 ≤ a i j ξ i ξ j ≤ max{ p − 1, 1}|ξ | 2 for any ξ ∈ R N .
In particular, when N = 1, v reduces to a scalar function v that satisfies the one dimensional porous medium equation Therefore Du satisfies a system with the same structure as (I) for m = p − 1; see the general system (1.1) -(1.2). Particularly, our main result implies the following well-known C 1,α regularity of a weak solution u to the parabolic p-Laplace equation (cf. [2][3][4]7,17]): Du ∈ L ∞ loc ⇒ Du ∈ C α loc . Our approach to the Hölder regularity for (I) is inspired by [5] where the C 1,α regularity for the elliptic p-Laplace equation has been investigated. The proof in [5] reflects a general idea due to De Giorgi: if the set where the equation is degenerate is confined within a small portion of a ball, then it is actually non-degenerate (cf. Sect. 3); if conversely the degenerate set occupies a seizable portion, then the solution can be compared with the radius of the ball (cf. Sect. 4).
The main new input of our argument lies in the introduction of proper intrinsically scaled cylinders, instead of balls, where the homogeneity of the parabolic systems is restored. We refer to [4,6] for an account of the theory for intrinsic scaling. It would also be helpful to refer to the recent work [14] on the Hölder regularity for porous medium type equations. Some arguments we use here are closely related to this work. This kind of idea has been used in [4,7] to treat the C 1,α regularity for parabolic systems of p-Laplacian type, via a combination of Campanato's approach and De Giorgi's iteration. Nevertheless, the quasi-diagonal structures of the system (I) allow us to implement the idea of intrinsic scaling more or less like we deal with scalar equations. Once proper energy estimates are derived from the system, we do not refer back to it any more. Thus we rely solely on De Giorgi's techniques.
Finally, we mention that there is a "cousin" of the system (I), namely The systems (I) and (II) are formally identical only in the scalar case, i.e. n = 1. The system (II) has been derived in [19] from Bean's critical-state model in the superconductivity theory.
With the norm |u| 1 , it has been established in [11] modeling a competitive adsorption process among different species, via a multi-component isotherm of Freundlich type. A study of Hölder regularity is carried out in [11] for non-negative solutions to (II) with the norm |u| 1 . The non-negativity of u turns out to be essential, since the argument relies on the fact that |u| 1 satisfies the scalar porous medium equation. See [11,19] for discussions on Dirichlet problems and [9,16] on Cauchy problems.
Similar interest in the Hölder regularity of solutions to various parabolic systems also appears in [13,15,18]. Certain degeneracy is considered in [13] for systems with different structures from ours, whereas systems in [15,18] are non-degenerate as the arguments are based on the regularity criterion in [8].

Main results
We postpone the formal definition of weak solutions to Sect. 1.3 and refer to Sect. 1.2 for the use of notations. Now we proceed to state in a precise manner our main results concerning the interior Hölder continuity of weak solutions to the system (I).
Theorem 1.1 Let u be a bounded, local, weak solution to the system (I) in E T . Then u is locally Hölder continuous in E T . More precisely, setting M := u ∞,E T , there exist constants γ > 1 and α ∈ (0, 1) that can be determined a priori only in terms of m, n and N , such that for every compact set K ⊂ E T , included in E T , we will show the following oscillation decay for 0 < r < : See Sect. 1.2 for the meaning of these cylinders. The conclusion of Theorem 1.1 can be derived from this oscillation estimate via a standard covering argument. The oscillation decay in Remark 1.1, while local in nature, has a global implication. Indeed, let u be a bounded, local weak solution to the system (I) in the semi-infinite strip S T := R N × (−∞, T ) for some T ∈ R. Then when m > 1, we have for 0 < r < that max 1≤i≤n ess osc for any cylinder (x o , t o )+ Q r (θ ) compactly included in S T . Now fixing r and letting → ∞, we immediately arrive at a Liouville-type result. The case 0 < m < 1 can be analyzed similarly. Thus we arrive at the following Liouville type result.

Notations
Suppose u is a bounded solution to (I) in E T . Let := ∂ E T − E × {T } be the parabolic boundary of E T , and for a compact set K ⊂ E T introduce the intrinsic, parabolic m-distance from K to by For > 0 let K (x o ) be the cube with center at x o ∈ R N and edge 2 . We define backward cylinders scaled by a positive parameter θ by We also need another type of cylinders for some positive parameter ϑ. In Sect. 6, we will use the following type of cylinders When (x o , t o ) = (0, 0) or θ = ϑ = 1, we will omit them from the notation.
In what follows, we will use γ as a generic positive constant in our estimates, which may change from line to line. Nevertheless, such γ can be determined a priori in terms of given data.

Notion of solution
It is always very important to pinpoint what we mean by a weak solution, as different notions may lead to different implications. In the following we may consider a general system modeled on the system (I): where the functions j (x, t, u, z) is measurable in (x, t) for all (u, z) ∈ R N × R nN and continuous in (u, z) for a.e. (x, t) ∈ E T , and satisfy the structure conditions for all i = 1, · · · , n and for some positive m, C o and C 1 . Together with n and N , they will be referred to as the data in the sequel. A vector valued function u satisfying is a local, weak sub(super)-solution to (1.1) with the structure conditions (1.2), if for every compact set K ⊂ E and every sub- for all non-negative test functions φ ∈ W 1,2 loc 0, T ; L 2 (K ) ∩ L 2 loc 0, T ; W 1,2 o (K ) .

Remark 1.5
The notion of solution may generate discrepancies, as it is not entirely clear even in the scalar case. An interesting study on the equivalence of different notions of solution for the prototype porous medium equation is carried out in [1].
In what follows, special care needs to be taken for the time variable of a solution since we will use test functions that involve the solution itself. Lack of regularity in the time variable notwithstanding, they become admissible provided we employ a proper mollification in the time variable.
To this end, we introduce for Properties of this mollification can for instance be found in [10, Lemma 2.2]. In particular, we need the following identities: From the weak form (1.3) of the differential inequality for sub(super)-solutions we may deduce the mollified version: with compact support in E T . Indeed, we may take the test function φ h in the weak formulation (1.3) of sub(super)solutions to (I). Since φ is compactly supported in E T , we have after an application of Fubini's theorem On the other hand, we calculate the term involving the time derivative using (1.4) This yields the desired claim in (1.5).

Energy estimates
This section is devoted to local energy estimates for the general system (1.1) -(1.2). Due to the degenerate or singular nature of the system, we need to consider the case 0 < m < 1 (Proposition 2.1) and the case m > 1 (Proposition 2.2) separately.
, · · · , n}, and every non-negative, piecewise smooth cutoff where we have defined Proof We prove the statement for sub-solutions only, as the other case is similar. For fixed t o − S < t 1 < t 2 < t o and ε > 0 small enough we define the cutoff function in time by It is an admissible test function thanks to the notion of solution and the choice of cutoff functions. In the following we omit the reference to the vertex (x o , t o ) in the notation. For the integral in (1.5) containing the time derivative we compute Here we have used in the second line the identity (1.4) 1 and the fact that the map s → (ϕ(s) − ϕ(k)) + is a monotone increasing function, which implies that the term in the second line is non-negative. Since , we can pass to the limit as h → 0 in the integral on the right-hand side. We therefore get with the obvious meaning of I ε and II ε . We now pass to the limit ε → 0. For the term I ε we obtain for any Next, we observe that the term on the right-hand side of (1.5) disappears as h → 0, since by It remains to consider the diffusion term. After passing to the limit as h → 0, we use the structure conditions (1.2) for the vector-field A (i) , and subsequently Young's inequality to estimate Combining the preceding estimates and letting ε → 0 we arrive at Remembering 0 < m < 1, the terms in the bracket are estimated by Recalling u i ∈ C 0, T ; L m+1 loc (E) , then one easily obtains the desired estimate.

Proposition 2.2 Let m > 1 and u be a local weak sub(super)-solution to (1.1) -(1.2) in E T .
There exists a constant γ > 0, such that for all cylinders Q R,S : every k ∈ R, every i ∈ {1, · · · , n}, and every non-negative, piecewise smooth cutoff function Proof In the weak formulation of the sub(super)-solution to the i-th equation of the system

De Giorgi-type Lemmas
Moreover, for some γ > 0 depending only on the data.

Proof of Lemma when m > 1
Assume (x o , t o ) = (0, 0). We will present the proof for super-solutions only, sub-solutions being similar. In order to employ the energy estimate in Proposition 2.2, we set Introduce the test function ζ vanishing on the parabolic boundary of Q n and equal to identity in Q n+1 , such that |Dζ | ≤ γ 2 n and |ζ t | ≤ γ 4 n θ 2 .
In this setting, the energy estimate may be written as where we have set A n := [u i < k n ] ∩ Q n . To estimate the left-hand side from below we introduceũ i := max u i , 1 4 M . Then for the time part we estimate For the space part, we first observe that Here in the last line we have used the fact that [u i < k n ] and [ũ i < k n ] coincide. Collecting all the estimates above gives us that ess sup In terms of Y n = |A n |/|Q n |, this can be rewritten as Hence by [4, Chapter I, The proof is concluded.

Proof of Lemma 3.1 when 0 < m < 1
Assume (x o , t o ) = (0, 0). In order to employ the energy estimate in Proposition 2.1, we notice first that, fork < k we have As a result, the energy estimate reads In order to use this energy estimate, we set for n = 0, 1, · · · , Introduce the test function ζ vanishing on the parabolic boundary of Q n and equal to identity in Q n , such that |Dζ | ≤ γ 2 n and |ζ t | ≤ γ 4 n θ 2 .
In this setting, the energy estimate may be written as Noting that k n ≤ M and ϕ (s) = m|s| m−1 , we may estimate the terms on the left-hand side of the energy estimate from below by In the last line we have used (3.1), and consequently |u| ≤ 2n Now setting φ to be a cutoff function which vanishes on the parabolic boundary of Q n and equals identity in Q n+1 , an application of the Sobolev imbedding [4, Chapter I, Proposition 3.1] gives that In terms of Y n = |A n |/|Q n |, this can be rewritten as The proof is concluded.

Reduction of the supreme of solutions
Let M be defined in (3.1) and ν be the constant determined in Lemma 3.1 by taking θ = M 1−m . Suppose for all i ∈ {1, · · · , n}, the following measure theoretical information holds We claim that if (4.1) 1 holds, then there exists η ∈ (0, 1) depending only on the data, such that Similarly, if (4.1) 2 holds, then there exists η ∈ (0, 1) depending only on the data, such that Combining the two estimates, the supreme of |u| is reduced in a smaller cylinder, i.e.  To start with, we observe that there exists some s ∈ [−θ 2 , − 1 2 νθ 2 ], such that Indeed, if the above inequality were not to hold for any s in the given interval, then we would have arrived at a contradiction to (4.1) 1 :

Propagation of measure information
Now we introduce a lemma concerning the time propagation of measure theoretical information in (4.4).
In estimating the term containing |u| m−1 we have evoked (4.3). The left-hand side of the energy estimate can be bounded from below by where we have defined Notice that Collecting all the above estimates yields that Finally, this allows us to choose the various parameters quantitatively. Indeed, we may assume ε is so small that (1 − ε) 2 ≥ 1 2 and choose Then the above estimate is simplified as To conclude, we may choose ε such that This completes the proof.

A shrinking lemma
Based on the measure theoretical information obtained in Lemma 4.1 for each slice K × {t}, where t ∈ s, s + δθ 2 , we may prove the following shrinking lemma.

Lemma 4.2 Assume (4.4) holds.
There exists γ > 0 depending only on the data, such that for any positive integer j * , we have Proof We employ the energy estimate of Proposition 2.2 in a larger cylinder Q = K 2 × (s, s + δθ 2 ], with levels k j = 2M 1 − ε 2 j+1 for j ≥ 0 and a cutoff function ζ in K 2 that is equal to 1 in K and vanishes on ∂ K 2 , such that |Dζ | ≤ 2 −1 . Then, we obtain Recalling ( , together with an application of Hölder's inequality, this gives Set A j = [u i > k j ] ∩ Q and integrate the above estimate in dt over (s, s + δθ 2 ]; we obtain by using the energy estimate and the Fubini theorem that Now square both sides of the above inequality to obtain Add these inequalities from 0 to j * − 1 to obtain From this we conclude This is exactly the desired conclusion.

A De Giorgi-type lemma near the supreme
Recall that Q = K × s, s + δθ 2 . We need the following De Giorgi-type lemma in the set Q. The proof is analogous to the one for Lemma 3.1. ξ ∈ (0, 1). There exists a positive constant ν o depending only on the data and δ, but independent of ξ , such that if

Lemma 4.3 Let
Proof For simplicity, we may fix the top of the cylinder Q, i.e. s +δθ 2 , at 0. For n = 0, 1, . . . we set n , 0 . Write down the energy estimate in Proposition 2.2 over the cylinder Q n . Taking also into account the set [u i > k n ], we are working in the set (4.3), i.e. there holds Here γ o = 2n 1 2 and n represents the number of equations in the system (1.1). We also use n as a generic integral index in the proof here.
This gives the energy estimate ess sup Applying Hölder's inequality and the Sobolev embedding [4, Chapter I, Proposition 3.1], and recalling that ζ = 1 on Q n+1 , we get for a constant γ depending only upon N . Combine this with the energy estimate to get In terms of Y n = |A n |/|Q n | this can be rewritten as By [4, Chapter I, Lemma 4.1], Y n → 0 as n → ∞, provided This finishes the proof.
Next we may apply Lemma 4.1 with and ν replaced by 2 and 1 4 N ν respectively. With δ and ε fixed in terms of these parameters as in Lemma 4.1, we may fix ν o in Lemma 4.3 depending on the data and also on δ. Then we may choose ξ = ε2 − j * , whereas j * is chosen according to Lemma 4.2, such that Thus Lemma 4.3 yields that After L steps, we arrive at The quantitative information as in (4.6) will reach the top of Q (θ ) when L is so large that 4(δ + Lδ) ≥ 1. Therefore, we may conclude with η =ξ L ξ .

Proof of Theorem 1.1
With all the ingredients prepared so far, we are ready to prove Theorem 1.1. The proof splits into two parts according to either m > 1 or 0 < m < 1. In the first part, we will work with cylinders of the type Q (θ ), whereas in the second part we use cylinders of the type Q (ϑ).
If we introduce then the above estimate yields that (νθ ) and ess sup For all the indices j = 1, · · · , − 1, we alway assume that (4.1) holds for all i = 1, · · · , n, i.e., In this way the previous argument can be repeated and we have for all j = 1, · · · , , max 1≤i≤n ess sup Consequently, iterating the above recursive inequality we obtain for all j = 1, 2, · · · , , max 1≤i≤n ess sup

Reduction of oscillation away from zero
Let us suppose is the first index such that (4.1) is violated, that is, for some i ∈ {1, · · · , n} there holds According to Lemma 3.1, we must have In either case we end up with |u| ≥ |u i | ≥ 1 2 M a.e. in Q 1 2 (θ ). The lower bound (5.3) and the upper bound (5.4) of u permit us to realize the oscillation decay of u by an appeal to the classical parabolic theory in [12]. In fact, we may introduce new variablesx and a new function As a result, the function w satisfies where the functions satisfy the structure conditions Appealing to (5.3) -(5.4), we have 1 2 ≤ |w| ≤ 2n This becomes a non-degenerate, diagonal system about w. According to [12], there exist α 1 ∈ (0, 1) and γ > 1 depending only on the data, such that for all 0 < r < 1 2 , max 1≤i≤n ess osc Rephrasing this oscillation decay in terms of u, we have for all 0 < r < , max 1≤i≤n ess osc Combining (5.2) and (5.5), we arrive at the desired conclusion, i.e., for all 0 < r < we have max 1≤i≤n ess osc Here we have used m > 1 such that θ ≤ θ , and thus Q r (θ ) ⊂ Q r (θ ).

Reduction of oscillation near zero
Let us first suppose the following holds for all i = 1, · · · , n: Then according to Remark 4.1, we have ess sup If we introduce then the above estimate yields that For all the indices j = 1, · · · , − 1, we alway assume that for all i = 1, · · · , n, In this way the previous argument can be repeated and we have for all j = 1, · · · , , max 1≤i≤n ess sup Consequently, iterating the above recursive inequality we obtain for all j = 1, · · · , , max 1≤i≤n ess sup

Reduction of oscillation away from zero
Let us suppose is the first index such that (5.6) is violated, that is, for some i ∈ {1, · · · , n} there holds According to Lemma 3.1, we must have In either case we end up with for all i = 1, · · · , n and for some positive m, C o and C 1 . Note that (6.1) 2 is additional comparing with (1.2). Nevertheless they are all verified by the prototype system (1.1). For m > 0, a vector valued function u satisfying is a local, weak sub(super)-solution to (1.1) with the structure conditions in (6.1), if for every compact set K ⊂ E and every sub-interval [t 1 , t 2 ] ⊂ (0, T ], the integral formulation (1.3) holds.
We have the following result concerning quantitative supreme estimates of local weak solutions, whose proof we use Moser's iteration to present. Theorem 6.1 Let u be a local weak solution to (1.1) in E T with conditions (6.1). There exists γ > 1 depending only on the data, such that for any σ ∈ (0, 1) and where A is defined in (6.6) and κ := N +2 N ; whereas for 0 < m < 1, requiring that λ := N (m − 1) + 2r > 0 for some r ≥ 2 and u ∈ L r loc (E T ), we have ess sup where A is defined in (6.7).

Energy estimates
We first present an energy estimate that will serve as the starting point of Moser's iteration scheme.  In the following we omit the reference to x o and sum over the index i ∈ {1, · · · , n} tacitly. As in Proposition 2.1, the right-hand side of (1.5) vanishes as h → 0. The time part is calculated as The space part can be calculated by sending h → 0: x, t, u, Du) · u i f k (|u|)ζ Dζ dxdt =: I 1 + I 2 + I 3 .
For I 3 , we estimate using (6.1) 3 and Young's inequality: Collecting all above estimates and sending k → ∞, we may conclude.

Remark 6.1
It is not hard to see from the proof that the energy estimate in Proposition 6.1 still holds if we replace (6.1) 1,3 by the weaker conditions |A (i) (x, t, u, Du)| ≤ C 1 |u| m−1 |Du|, i = 1, · · · , n.

Proof of Theorem 6.1 when m > 1
For ease of notation, we set v := |u|. In the energy estimate of Proposition 6.1, we take f (v) = v β for β ≥ 0, apply the inequality |Dv| ≤ |Du| to the second term on the left, and dump the third term. The test function ζ is chosen to vanish on the parabolic boundary of Q R,S and equal 1 on Q ,τ , such that |Dζ | ≤ (R − ) −1 and |ζ t | ≤ (S − τ ) −1 . As a result, the energy estimate in Proposition 6.1 gives (v m+1+β + 1) dxdt.