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Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.


Introduction
Let Ω be a bounded domain in R N and T a positive number. Set Ω T = Ω × (0, T ). We study the behavior of weak solutions of the system (H1) S(x) ∈ L q (Ω), q > N 2 ; and (H2) D, E ∈ (0, ∞), γ ∈ ( 1 2 , ∞). This system has been proposed by Hu and Cai ([10], [11]) to describe natural network formulation. Then the scalar pressure function p = p(x, t) follows Darcy's law, while the vector-valued function m = m(x, t) is the conductance vector. The function S(x) is the time-independent source term. Values of the parameters D, E, and γ are determined by the particular physical applications one has in mind. For example, γ = 1 corresponds to leaf venation [10]. Of particular physical interest is the initial boundary value problem: in addition to (1.1) and (1.2) one requires m(x, 0) = m 0 (x), x ∈ Ω, (1. 3) p(x, t) = 0, m(x, t) = 0, (x, t) ∈ Σ T ≡ ∂Ω × (0, T ), (1.4) at least in a suitably weak sense; here the initial data should satisfy m 0 (x) = 0 on ∂Ω.
The existence of weak solutions of this initial boundary value problem was proved by Haskovec, Markowich, and Perthame [8]. However, the regularity theory remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities.
Let us call a point (x, t) ∈ Ω T singular if m is not Hölder continuous in any neighborhood of (x, t); the remaining points will be called regular points. By a partial regularity theorem, we mean an estimate for the dimension of the set S of singular points. It is well-known that weak solutions to even uniformly elliptic systems of partial differential equations are not regular everywhere. We refer the reader to [6] for counter examples. Thus it is only natural to seek partial regularity theorems for these weak solutions. The system under our consideration exhibits a rather peculiar nonlinear structure. The first equation in the system degenerates in the t-variable and the elliptic coefficients there are singular in the sense that they are not uniformly bounded above a priori, while the second equation contains the term (m · ∇p)∇p, which is a cubic nonlinearity. Thus the classical partial regularity argument developed in ( [6], [1]) does not seem to be applicable here. Our system does resemble the so-call thermistor problem considered in ( [16]- [18]). The key difference is that the elliptic coefficients in the preceding papers and also in [6] are assumed to be bounded and continuous functions of solutions. As a result, the modulus of continuity can be taken to be a bounded, continuous, and concave function. This fact is essential to the arguments in both [16] and [6]. Our elliptic coefficients here are quadratic in m, and thus a new proof must be developed. Note that the question of existence in the case where γ = 1 2 is addressed in [9]. In this case the term |m| 2(γ−1) m is not continuous at m = 0. It must be replaced by the following function Partial regularity relies on local estimates [6]. One peculiar feature about our problem (1.1)-(1.4) is that certain important global estimates have no local versions. This is another source of difficulty for our mathematical analysis. We are ready to state our main result: Theorem 1.1. Let (H1)-(H3) be satisfied. Assume that N ≤ 3. Then the initial boundary value problem (1.1)- (1.4) has a weak solution on Ω T whose singular set S satisfies (1.5) P N +ε (S) = 0 for each ε > 0.
Here P s , s ≥ 0, denotes the s-dimensional parabolic Hausdorff measure. Recall that the sdimensional parabolic Hausdorff measure of a set E ⊂ R N × R is defined as follows: where Q r j (z j ) are parabolic cylinders with geometric centers at z j = (y j , τ j ), i.e., one has It is not difficult to see that P s is an outer measure, for which all Borel sets are measurable; on its σ-algebra of measurable sets, P k is a Borel regular measure (cf. [5], Chap.2.10). If P s (E) < ∞, then P s+ε (E) = 0 for each ε > 0. We define the parabolic Hausdorff dimension dim P E of a set E to be dim P E = inf{s ∈ R + : P s (E) = 0}. Then Theorem 1.1 says that Hausdorff measure H s is defined in an entirely similar manner, but with Q r j (z j ) replaced by an arbitrary closed subset of R N × R of diameter at most r j . (One usually normalizes H s for integer s so that it agrees with surface area on smooth s-dimensional surfaces.) Clearly, To characterize the singular set S, we will need to invoke the following known result.
The proof of this lemma is essentially contained in [1].
A key observation about our weak solutions in the study of partial regularity is the following proposition. The proof of this proposition will be given at the end of Section 2. Let (m, p) be a weak solution. In view of ( [3], [16]), to establish Theorem 1.1, we will need to define a suitable scaled energy E r (z) for our system. For this purpose, let z = (y, τ ) ∈ Ω T , r > 0 with Q r (z) ⊂ Ω T and pick where q is given as in (H1). We consider the following quantities: The right choice for E r (z) seems to be The last term in E r (z) accounts for the non-homogeneous term S(x) in (1.1). Due to the fact that the first equation (1.1) does not have the ∂ t p term, we are forced to use the term A r (z) instead of 1 r N+2 Qr(z) |p − p z,r | 2 dxdt in E r (z). This will cause two problems: one is that in our application of the classical blow-up argument ( [3], [6], [16]), the resulting blow-up sequence is not compact in the desired function space; the other is the characterization of the singular set S. That is, it is not immediately clear how one can describe the set (1.14) in terms of the parabolic Hausdorff measure. (Note that this issue is rather simple in the context of [16].) To overcome these two problems, we find a suitable decomposition of p. This enables us to show that the lack of compactness in the blow-up sequence does not really matter. To be more specific, we obtain that the blow-up sequence can be decomposed into the sum of two other sequences, one of which converges strongly while the terms of the other are very smooth in the space variables, and this is good enough for our purpose. This idea was first employed in [16]. However, as we mentioned earlier, the nature of our mathematical difficulty here is totally different. A similar decomposition technique can also be used to derive the parabolic Hausdorff dimension of the set in (1.14).
The key to our development is this assertion about energy: for all z ∈ Ω T and r > 0 with Q r (z) ⊂ Ω T .
The proof of this proposition is given in Section 4. It relies on the decomposition of the function p we mentioned earlier. An immediate consequence of this proposition is: Corollary 1.1. Let the assumptions of Theorem 1.1 hold. To each M > 0 there corresponds a pair of numbers δ 1 , ε 1 in (0, 1) such that whenever (1.17) |m z,r | < M 2 and E r (z) < ε 1 we have Proof. We essentially follow the proof of Corollary 3.8 in [16] (also see [6]). Let M > 0 be given. By Proposition 1.2, there exist 0 < ε, δ < 1 such that (1.15) and (1.16) hold. We claim that we can take where ω N is the volume of the unit ball in R N . To see this, let (1.17) hold. Obviously, (1.18) is satisfied for k = 1. Now for each positive integer j suppose (1.18) is true for all k ≤ j. We will show that it is also true for k = j + 1. To this end, we integrate the inequality This corollary combined with the argument in ( [6], p.86) asserts that there exist c = c(δ 1 , ε 1 , r) ∈ (0, 1), γ = γ(δ 1 ) > 0 such that Obviously, m z,r , − Qr(z) |m − m z,r | 2 dxdt are both continuous functions of z. By Proposition 1. This puts us in a position to apply a result in [12]. To state the result, we define, for µ ∈ (0, 1), Parabolic Hölder spaces can be characterized by the following version of Campanato's theorem ( [12], Theorem 1).
Lemma 1.2. Let u ∈ L 2 (Ω T ). If there exist α ∈ (0, 1) and R 0 > 0 such that for all z in an open subset O of Ω T and all ρ ≤ R 0 with Q ρ (z) ⊂ Ω T , then we have That is, u is Hölder continuous in O.
To describe the singular set S, we set Here and in what follows lim r→0 means lim r→0 + because we always have r > 0. If z ∈ R, we take M > 2 sup r>0 |m z,r |. By Corollary 1.1, there exist δ 1 , ε 1 ∈ (0, 1) such that (1.17) and (1.18) hold. We can find a r such that For the same r we obviously have Consequently, m is Hölder continuous in a neighborhood of z. That is, R is a set of regular points. Obviously, R is an open set. Note that since we have the term A r (z) instead of 1 r N+2 Qr(z) |p − p z,r | 2 dxdt in E r (z) Proposition 1.2 does not imply that p is locally Hölder continuous in the space-time domain R. The difference between the two quantities can be seen from the following calculation: |p − p y,r (t)| 2 dxdt Obviously, the last term above causes the problem. Of course, for each t = t 0 , p(x, t 0 ) is locally To estimate the parabolic Hausdorff dimension of the singular set S ⊆ Ω T \ R, we have the following proposition.
The proof of this proposition relies on almost the same decomposition of p as that in the proof of Proposition 1.2. The details will be given in Section 3.
Thus Theorem 1.1 is a consequence of Propositions 1.1-1.3. The rest of the paper is organized as follows. In Section 2, we develop some new global estimates. They serve as a motivation for our local estimates. The section will end with the proof of Proposition 1.1. In Section 3, we will first establish some local estimates and then proceed to prove Proposition 1.3. Section 4 is devoted to the proof of Proposition 1.2. Note that the three propositions are independent, and thus the order of their proofs is not important.

Global Estimates
In this section, we first summarize the main a priori estimates already established in [8]. Then we present our new global estimates. The proof of Proposition 1.1 is given at the end.
To begin with, we use p(x, t) as a test function in (1.1) to obtain Here and in what follows we suppress the dependence of p, m on (x, t) for simplicity of notation if no confusion arises. Let τ ∈ (0, T ), Ω τ = Ω × (0, τ ). Take the dot product of both sides of (1.2) with m, integrate the resulting equation over Ω τ , and thereby yield Multiply through (2.1) by 2E 2 , integrate over (0, τ ), and then add it to (2.2) to arrive at Take the dot product of (1.2) with ∂ t m and integrate the resulting equation over Ω to obtain Multiply through this equation by−E 2 and add the resulting one to (2.4) to obtain Differentiate (2.1) with respect to t, multiply through the resulting equation by E 2 , then add it to the above equation, and thereby deduce where p 0 is the solution of the boundary value problem Local versions of (2.1) and (2.3) will be established in Section 3. Unfortunately, they are not enough to yield a partial regularity result. Naturally, one tries to seek a local version of (2.7). But this cannot be done because we have no control over ∂ t p. To partially circumvent this, we have developed some new estimates.

Proposition 2.1. Let (H1) and (H2) be satisfied and (m, p) a weak solution of (1.1)-(1.4). Then:
where · q,Ω denotes the norm in L q (Ω). We shall write · s for · s.Ω for simplicity; (C2) For each K > 0 we can choose β ∈ (0, 1) suitably small such that By the Sobolev embedding theorem, we have Thus the first integral on the right-hand side of the above inequality is finite.
Proof. The proof of (C1) is standard. See, e.g., ( [2], p. 131). For the reader's convenience, we shall reproduce the proof here. Let κ be a positive number to be determined. Write This implies (C1).
Let K > 0, β > 0 be given and v be defined as in (2.10). For L > K, define Set v L = θ L (|m| 2 ). Then the function v β L m is a legitimate test function for (1.2). Upon using it, we arrive at In the derivation of the third term above, we have used the fact that (2.15) ∇v L = 0 on the set where |m| 2 > L 2 or |m| 2 < K 2 .
Use v β L p as a test function in (1.1) to deduce . Multiply through the above inequality by 2E 2 , add the resulting inequality to (2.14), thereby obtain Choosing β sufficiently small so that the second term on the right-hand in the above inequality can be absorbed into the third term on the left-hand side there, integrating the resulting inequality with respect to t, and then taking L → ∞ yields (C2). The proof is complete.
It turns out that a local version of (C2) is possible only if N ≤ 3. This accounts for the restriction on the space dimension in Theorem 1.1.
At the end of this section, we present the proof of Proposition 1.1.
Set m j = m(x, t j ) and denote by p j the solution of (1.1) with m being replaced by m j . Obviously, we have We claim that we also have (2.20) p j → p * ≡ p(x, t * ), the solution of (1.1) corresponding to t = t * , strongly in L 2 (Ω) as j → ∞, and this will be enough to imply the proposition. To see this, note that m j ⊗m j ∇p j = (m j ·∇p j )m j , and thus we have the equation Using p j as a test function, we can easily derive Thus we may assume that (2.23) p j ⇀ p weakly in W 1,2 0 (Ω) and strongly in L 2 (Ω) (passing to a subsequence if need be.) This together with (2.19) implies m j · ∇p j ⇀ m * · ∇p weakly in L 1 (Ω), and therefore also weakly in L 2 (Ω).
Thus we can take j → ∞ in (2.21) to obtain The solution to this equation is unique in W 1,2 0 (Ω), and therefore p = p * and the whole sequence {p j } tends to p * strongly in L 2 (Ω). The proof is complete.

Local estimates
In this section we begin with a derivation of local versions of (2.1) and (2.3). Then we proceed to prove Proposition 1.3.
Let z = (y, τ ) ∈ Ω T , r > 0 with Q r (z) ⊂ Ω T be given. Pick a C ∞ function ξ on R N +1 satisfying Note that m ⊗ m∇p = (m · ∇p)m. Keep this in mind, while using ξ 2 (p − p y,r (t)) as a test function in (1.1), to obtain Br(y) |S(x)|ξ 2 |p − p y,r (t)|dx. (3.1) Set M 0 = ess sup Ω T |p(x, t)|. Then the fourth integral in (3.1) can be estimated as follows: We apply Poincaré's inequality to the last integral in (3.1) to yield for each ε > 0. Use (3.3) and (3.2) in (3.1), choose ε sufficiently small in the resulting inequality, and thereby arrive at Substitute (3.6) into (3.5), choose ε so small in the resulting inequality that the second integral in (3.6) can be absorbed into the third term in (3.5), then integrate with respect to t to yield We are ready to prove Proposition 1.3.
Proof of Proposition 1.3. For each ε > 0 we consider the set Thus it is enough for us to show where R is defined in (1.24). We divide the proof of this into several claims.

(3.12)
Here and in the remainder of the proof of Proposition 1.3 the constant c may depend on ε and z. It immediately follows that Thus the claim follows.
Note that the corollary in ( [15], p.144) is not applicable here. We offer a direct proof. To this end, we estimate from Poincaré's inequality that |m − m y,r (t)| 2+α dx.
Observe that Keeping these two inequalities in mind, we calculate that |m − m y,r (t)| 2α+2 dx Note that 2 + 2α ≤ d, where d is given in (3.8). We estimate The proof is complete.
Thus p satisfies the system −div[(I + m y,r (t) ⊗ m y,r (t))∇p] = div [(m · ∇p)w r ] + div [(w r · ∇p)m y,r (t)] + S(x) in Q r (z). (3.28) Here we have used the fact that (w r ⊗ m)∇p = (m · ∇p)w r . We decompose p into η + φ on Q r (z) as follows: η is the solution of the problem −div [(I + m y,r (t) ⊗ m y,r (t))∇η] = 0 (3.30) while φ is the solution of the problem −div [(I + m y,r (t) ⊗ m y,r (t))∇φ] = div [(m · ∇p)w r ] + div [(w r · ∇p)m y,r (t)] On the other hand, another classical regularity result [13] for linear elliptic equations with continuous coefficients asserts that for each s ∈ (1, ∞) there is a positive number c with the property Note that the constant c here is also independent of r. We remark that in general the above inequality is not true for s = 1. This is why Claim 3.3 is crucial to our development. Obviously, if we replace m z,r by m y,r (t) in (3.4), the resulting inequality still holds. This implies We can easily find a s ∈ (1, 2) so that We estimate To summarize, we have It follows from Poincaré's inequality that Remember that φ ∞ ≤ η ∞ + p ∞ ≤ 2 p ∞ . Hence we can always find a positive number ε 1 ∈ (0, 2) so that For 0 < ρ ≤ r we derive from (3.34) and (3.43) that Here we have used the fact that Bρ(y) |φ − φ y,ρ (t)| 2 dx is an increasing function of ρ. We set We easily infer from (3.44) that for all 0 < ρ ≤ r. This puts us in a position to apply Lemma 2.1 in ( [6], p.86), from whence follows for all 0 < ρ ≤ r. This gives the claim.

Proof of Proposition 1.2
In this section we present the proof of Proposition 1.2. We would like to remark that the proof of this proposition is more challenging than that of Proposition 1.3 mainly because we do not have a local estimate for ∂ t m or a local L ∞ estimate for p. This also causes us to impose the restriction N ≤ 3. Note that this restriction is not needed in Propositions 1.1 and 1.3.
Proof of Proposition 1.2. We argue by contradiction. Suppose that the proposition is false. Then for some M > 0 (1.15) and (1.16) fail to hold no matter how we pick numbers ε, δ from the interval (0, 1). In particular, we can choose a sequence {ε k } ⊂ (0, 1) with the property The selection of δ from (0, 1) is more delicate, and it will be made clear later. Let δ be chosen as below. For each k there exist cylinders Q r k (z k ) ⊂ Ω T such that Set λ 2 k = E r k (z k ). Then (4.1) asserts λ k → 0 as k → ∞. We rescale our variables to the unit cylinder Q 1 (0), as follows. If z = (y, τ ) ∈ Q 1 (0), write ψ k (y, τ ) = p(y k + r k y, τ k + r 2 k τ ) − p y k ,r k (τ k + r 2 k τ ) λ k , (4.4) n k (y, τ ) = m(y k + r k y, τ k + r 2 k τ ), (4.5) We can easily verify Here and in what follows we suppress the dependence of ψ k , w k , n k on (y, τ ) for simplicity of notation. Our plan is to show that the lim sup of the left-hand side of the above inequality as k → ∞ can be made smaller than 1 2 if we adjust δ to be small enough, and thus the desired contradiction follows.
We estimate the last term in (4.9) as follows The last step is due to (1.9). We are ready to let k go to infinity in (4.9), thereby obtaining Remember that a is a constant vector. By the classical regularity theory for linear elliptic equations, there exist c > 0, α ∈ (0, 1) determined only by M and N with the property for all δ ≤ 1 4 . Subsequently, for all 0 < δ ≤ 1 4 . It is also well-known (see, e.g., Claim 1 in [17]) that there exist c > 0, α ∈ (0, 1) determined only by N, D such that for all 0 < δ ≤ 1 4 . If we could pass to the limit in (4.7), this would result in the desired contradiction. What prevents us from doing so is the lack of compactness of the sequence {ψ k } in the t-variable. To circumvent this problem, we fix a suitably small number 1 16 ≥ δ 0 > 0 and consider the decomposition We will show that {φ k } is precompact in L ∞ (− 1 2 δ 2 0 , 1 2 δ 2 0 ; L 2 (B δ 0 (0))), and this is enough for our purpose in spite of the fact that {η k } may not be precompact in the preceding function space. To see this, we first infer from (3.4) that Using η k − ψ k as a test function in (4.26) yields Note that (4.26) is an uniformly elliptic equation with constant coefficients. The classical regularity theory asserts that there exist c > 0, α ∈ (0, 1) depending only on M, N such that for all δ ≤ 1 2 δ 0 . Now we turn our attention to the sequence {φ k }. We wish to show ). This is where the subtlety of our analysis lies. We observe from (4.30) that In view of (4.31), {φ k } also satisfies the above estimate. By the interpolation inequality ( [6], p.146) it is sufficient for us to show |φ k (y, τ )|dy → 0 as k → ∞.
Note that the elliptic coefficients in (4.28) are constants. This puts us in a position to invoke the classical W 1,s estimate for φ k . That is, for each s ∈ (1, ∞) there is a positive number c with the property (4.37) ∇φ k s ≤ cλ k (n k · ∇ψ k )w k s + cλ k ((w k · ∇ψ k )m z k ,r k s + c F k sN s+N .
Remember that (4.37) does not hold for s = 1. To find a s > 1, we will show that there is a β > 0 such that Obviously, this will imply that for some s > 1. Consequently, the right-hand side of (4.37) goes to 0 as k → ∞. To establish (4.38), we will develop a suitable local version of (C2) in Proposition 2.1. This effort is complicated by the fact that a local version of (C1) in the proposition is not available. The remaining part of this section will be dedicated to the proof of (4.38), which will be divided into two claims.
Subsequently, we have We estimate from (4.12) and the Sobolev Embedding Theorem that Our assumption on the space dimension N implies By virtue of (4.30), we obtain The last term in (4.43) can be estimated as follows , δ > 0. (4.50) Substituting this and (4.45) into (4.43) and choosing δ suitably small in the resulting inequality yield . (0), and thereby obtain the claim.
Proof. Let ξ be given as in (4.41)-(4.42) and β > 0. We may assume that w k ∈ L ∞ (Ω T ) for each k. (Otherwise, we use the cut-off function in (2.13).) Then the function v β k w k ξ 2 is a legitimate test function for (4.10). Upon using it, we derive Now we analyze the last term in (4.53) to obtain To estimate the last two terms in the above inequality, we use ψ k v β k ξ 2 as a test function in (4.9) to obtain Observe that Here we have used the fact that v k ≥ K 2 and β < 1. The fifth integral in (4.57) can be estimated as follows.
Integrate (4.63) with respect to τ , choose ξ suitably, and remember Claim 1 to yield the desired result.
This finishes the proof of Proposition 1.2.