Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model

In this paper we first study partial regularity of weak solutions to the initial boundary value problem for the system $-\mbox{div}\left[(I+\mathbf{m}\otimes \mathbf{m})\nabla p\right]=S(x),\ \ \partial_t\mathbf{m}-D^2\Delta \mathbf{m}-E^2(\mathbf{m}\cdot\nabla p)\nabla p+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m}=0$, where $S(x)$ is a given function and $D, E, \gamma$ are given numbers. This problem has been proposed as a PDE model for biological transportation networks. Mathematically, it seems to have a connection to a conjecture by De Giorgi \cite{DE}. Then we investigate the life-span of classical solutions. Our results show that local existence of a classical solution can always be obtained and the life-span of such a solution can be extended as far away as one wishes as long as the term $\|{\bf m}(x,0)\|_{\infty, \Omega}+\|S(x)\|_{\frac{2N}{3}, \Omega}$ is made suitably small, where $N$ is the space dimension and $\|\cdot\|_{q,\Omega}$ denotes the norm in $L^q(\Omega)$.


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 solutions of the system m(x, 0) = m 0 (x), x ∈ Ω (1.5) for given function S(x) and physical parameters D, E, γ with properties: (H1) S(x) ∈ L N (Ω); and (H2) D, E ∈ (0, ∞), γ ∈ ( 1 2 , ∞). This problem arises in the study of network formulation and transportation networks [8,9,1]. Examples of such networks one has in mind are the angiogenesis of blood vessels, leaf venation, and creation of neural pathways in nervous systems. The development of mathematical models to describe them has attracted a lot of attention. Our problem here was first proposed by Hu and Cai [9]. In this case the scalar function p = p(x, t) is the pressure due to Darcy's law, while the vector-valued function m = (m 1 (x, t), · · · , m N (x, t)) 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 [8].
In general nonlinear problems do not possess classical solutions. A suitable notion of a weak solution must be obtained for (1.1)- (1.5). It turns out [5] that we can introduce the following definition. It has been established in [5] that (1.1) -(1.5) has a weak solution, provided that, in addition to assuming S(x) ∈ L 2 (Ω) and (H2), we also have (H3) m 0 ∈ W 1,2 0 (Ω) ∩ L 2γ (Ω) Additional results concerning modeling, numerical simulations, the corresponding stationary equations, and the one-dimensional problem are obtained in [1,6]. However, the general regularity theory remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities in high space dimensions. Recently, Jian-Guo Liu and the author [12] obtained a partial regularity theorem for (1.1)-(1.5). It states that the parabolic Hausdorff dimension of the set of singular points can not exceed N , provided that N ≤ 3. In [17], the author shows that a weak solution to the 2-dimensional stationary problem must be a classical one. There are two very interesting mathematical features associated with the system. The first one is the elliptic coefficient matrix The currently available regularity theory requires that the largest eigenvalue λ l of A and the smallest one λ s be suitably "balanced". A typical example of such assumptions is that λ l ≤ cλ s and λ s is an A 2 -weight [7]. Here and in what follows the letter c denotes a generic positive number. Thus if m is not locally bounded a priori, our situation lies outside the scope of the existing regularity theory. The second one is the tri-linear term (m · ∇p)∇p in (1.2), which actually represents a cubic nonlinearity. We have not been able to find much research work done on this type of nonlinearities. In this paper we study the existence of a weak solution that possesses the additional property (D4) m ∞,Ω T < ∞ and sup 0≤t≤T ∇p q,Ω < ∞ for each q > 1. We would like to remark that if N = 2 then the two conditions in (D4) are equivalent (see Lemma 2.7 below).
, then a weak solution to (1.1)-(1.5) with the additional property (D4) is also a classical one.
The proof of this proposition will be presented at the end of Section 2. (H5) ∂Ω is C 1 . Then there is a positive number T determined by the given data such that (1.1)-(1.5) has a weak solution (m, p) with the property (D4) on Ω T . The next theorem reveals how the life-span of a classical solution depends on the size of given data. Theorem 1.3. Let the assumptions of Theorem 1.2 be satisfied. For each T > 0 there is a positive number δ = δ(T ) such that (1.1)-(1.5) has a weak solution on Ω T with the property (D4) whenever S(x) 2N 3 ,Ω + m 0 ∞,Ω ≤ δ. We believe that the fact that the number δ in the theorem has to depend on T is related to the time-independence of the source term S(x). We speculate that if the source term S is a function of both time and space and S q,Ω×[0,∞) is suitably small for some q > 1 we may be able to prove the existence of a classical solution on Ω × [0, ∞). Even though our results are not surprising, they should be viewed in terms of the new mathematical features mentioned earlier.
We would like to remark that nonlinearities play a rather peculiar role in blow-up of solutions. There are cases in which solutions blow up no matter how small the given data are [3].
The rest of the paper is organized as follows: In section 2 we collect some preparatory lemmas. Here we take or refine some relevant classical results. Section 3 is devoted to the proof of Theorems 1.2 and 1.3. A successive approximation scheme is employed for the second theorem. The key here is that one must show that the entire approximate sequence converges in a suitable sense. In the last section we consider the local boundedness estimates for p. This is motivated by the fact that in elliptic theory local boundedness estimates often result in local Hölder continuity. At least in the case where N = 2 we can infer from an argument in [18] that p ∈ L ∞ (0, T ; C α (Ω)) leads to Hölder continuity of m.

Preliminaries
In this section we prepare some background results. Some of them are well-known and some of them are a refinement of known results so that they fit our purpose.
Our first result is an elementary inequality whose proof is contained in ( [14], p. 146-148).
Lemma 2.1. Let x, y be any two vectors in R N . Then: (i) For γ ≥ 1, The next two lemmas deal with sequences of nonnegative numbers which satisfy certain recursive inequalities.
This lemma can easily be established via induction.

Lemma 2.4.
Let Ω be a bounded domain in R N with Lipschitz boundary ∂Ω. Assume that w is a weak solution of the initial boundary value problem where w 0 is Hölder continuous on Ω and |g| 2 , g 0 ∈ L q (Ω T ) for some q > 1 + N 2 . Then w is Hölder continuous on Ω T . That is, there is a number β ∈ (0, 1) such that This result is well-known, and it can be found, for example, in [11]. Next, we cite a result from ( [15], p.82).
Lemma 2.5. Let (H5) hold and assume (L1) A(x) is an N ×N matrix whose entries are continuous functions on Ω, satisfying the uniform ellipticity condition If u is a weak solution to the boundary value problem then for each q > 1 there is a positive c = c(N, q, Ω) with the property ∇u q,Ω ≤ c g q,Ω + g 0 Nq N+q ,Ω . We can easily infer from the preceding two lemmas that (D4) can be replaced by (D4) ′ m ∞,Ω T < ∞ and there is a q > 1 + N 2 such that sup 0≤t≤T ∇p 2q,Ω < ∞. Lemma 2.6. Let w be a weak solution of the initial boundary value problem Then there is a positive number c = c(N ) such that Proof. Denote by w i (resp. g i ) the i-th component of w (resp. g). Then we have Without loss of generality, we assume that Then define Without loss of generality, assume N > 2. Use (w i − k n ) + as a test function in (2.1) to derive, with the aid of Poincaré's inequality, that Use Poincaré's inequality again to obtain We can easily show that This puts us in a position to apply Lemma 2.2. Upon doing so, we arrive at Here we have made use of the fact that This completes the proof. Proof. Suppose that m ∞,Ω T < ∞. Then Equation (1.1) is uniformly elliptic. A result in [13] asserts that there is a q > 2 such that ,Ω . This together with an argument in [18] [also see ( [11], p.182)] implies that m is Hölder continuous on Ω T . Thus Lemma 2.5 becomes applicable to (1.1). This yields the desired result. Now assume that sup 0≤t≤T ∇p q,Ω < ∞ for each q > 1. Fix τ ∈ (0, T ]. By Lemma 2.6, there is a positive number c = c(N ) such that where c is independent of τ . Hence we can choose τ so that (2.8) the coefficient of m ∞,Ω×(0,τ ) on the right-hand side of (2.7) ≡ cτ This immediately gives m ∞,Ω×(0,τ ) < ∞. Obviously, [0, T ] can be divided into a finite number of subintervals with each one of them having length less than τ . Apply the preceding argument successively to each one of the subintervals, starting with [0, τ ]. The desired result follows.
Before we conclude this section, we offer the proof of Proposition 1.1.
Proof of Proposition 1.1. We will only give an outline of the proof, leaving many well-known technical details out. Assume (D4) and (H2). By the Calderon-Zygmund inequality for parabolic equations [11], we have ∂ t m, ∆m ∈ (L q (Ω T )) N for each q > 1.
Differentiate both sides of (1.2) with respect to each one of the space variables and apply a local version of Lemma 2.4 to the resulting equations in a suitable way to conclude that ∇m is Hölder continuous on Ω T . As a result, the classical Schauder estimates ( [4], p.107) become applicable to (1.1). Upon applying, we yield that p ∈ L ∞ (0, T ; C 2,α (Ω)) for some α ∈ (0, 1). Differentiate (1.1) with respect to t to obtain This puts us in a position to use Lemma 2.5, from which follows ∇∂ t p ∈∈ (L q (Ω T )) N for each q > 1.
Differentiate both sides of (1.2) with respect to t and apply the Calderon-Zygmund inequality to the resulting equation to obtain ∂ 2 m ∂t 2 , ∆∂ t m ∈ (L q (Ω T )) N for each q > 1. Now the right-hand side of (2.9) is Hölder continuous in the space variables, and hence we can apply the Schauder estimates to it to get ∂ t p ∈ L q (0, T ; C 2,α (Ω)) for each q > 1.
This implies that ∇p is Hölder continuous on Ω T . On the other hand, owing to (H2), the term |m| 2(γ−1) m is also Hölder continuous. We can conclude from the parabolic Schauder estimates [10] that m is a classical solution of (1.2).

Proof of Theorems 1.2 and 1.3
We first establish the local existence of a smooth solution.
Proof of Theorem 1.2. For each ε > 0 let where m i (resp. m 0i ), i = 1, · · · , N , is the i th component of the vector m (resp. m 0 ) and the function θ ε (s) is defined by Obviously, we have w(x, 0) = m 0 (x) on Ω, (3.6) where p solves (3.3) coupled with (1.3). The latter problem has a unique solution if ε is sufficiently small. To see this, first observe that the elliptic coefficients on the left-hand side of (3.3) are continuous. Therefore, we are in a position to apply Lemma 2.5, from whence follows that for each q > 1 there is a positive number c determined only by q, m 0 , N , and Ω such that ∇p q,Ω ≤ c g ε ⊗ m 0 ∇p q,Ω + c g ε ⊗ g ε ∇p q,Ω + c S(x) Nq N+q,Ω ≤ c(ε + ε 2 ) ∇p q,Ω + c. Now fix a q > 2(1 + N 2 ). We have (3.8) ∇p q ≤ c if we choose ε so that the coefficient c(ε + ε 2 ) in (3.7) is strictly less than 1. From here on we assume that this is the case. Subsequently, Lemma 2.4 becomes applicable to (3.4). Upon using it, we obtain that w is Hölder continuous on Ω T . Therefore, we can claim that B is well-defined, continuous, and precompact. It remains to be seen that there is a positive number c such that This equation is equivalent to the following problem We still have (3.8). As a result, the right-hand side of (3.11) is bounded in L Suppose that w k−1 , p k−1 , k = 1, 2, · · · , are known. We define p k to be the unique solution of the boundary value problem (3.17) p k = 0 on Σ T , (3.18) while w k solves the problem The unigueness of a solution to the preceding problem can easily be inferred from Lemma 2.1. Obviously, if {(w k−1 , p k−1 )} satisfies (D4), so does {(w k , p k )}. The sequence {(w k , p k )} is welldefined. It follows from Lemma 2.6 that there is a positive number c = c(N, Ω) with On the other hand, we can deduce from Lemma 2.5 that there is a positive number c = (N, Ω) such that Adding (3.23) to (3.22), we derive Observe from (3.15)-(3.16) that ∇p 0 2N,Ω ≤ c S(x) 2N 3 ,Ω . In view of Lemma 2.3, if We must show that the whole sequence {w k , p k } converges in a suitable sense. To this end, we conclude from (3.19) that Use w k − w k−1 as a test function in (3.26) and keep the above inequality and (3.25) in mind to derive 1 2 in Ω T , k = 2, 3, · · · . (3.30) Upon using p k − p k−1 as a test function in the above equation, we arrive at We represent Plug the preceding estimates into (3.31) to derive Add (3.35) to (3.29) and integrate the resulting equation over (0, T ) to yield Hence if (3.39) cC 4 0 < 1, then the two series's ∇w 0 + ∇w 1 − ∇w 0 + · · · +∇w k − ∇w k−1 + · · · and (3.40) ∇p 0 + ∇p 1 − ∇p 0 + · · · +∇p k − ∇p k−1 + · · · (3.41) converge in L 2 (0, T ; W 1,2 (Ω) N ) and L 2 (0, T ; (W 1,2 (Ω)), respectively. It immediately follows that the two sequences {w k } and {p k } also converge in L 2 (0, T ; W 1,2 (Ω) N ) and L 2 (0, T ; W 1,2 (Ω)), respectively. We can also deduce from (3.25) and Lemma 2.4 that {w k } is uniformly convergent on Ω T . We can let k → ∞ in (3.17) and (3.19). Note from (3.25) that (3.39) is valid if we make the term m 0 ∞,Ω + S(x) 2N 3 ,Ω suitably small. The proof is complete.

Local boundedness estimates for p
It is known from [12] that p ∈ L ∞ (Ω T ) no matter what the space dimension is. Local boundedness estimates for p turn out to be a much more delicate issue. As we know from the regularity theory for linear elliptic equations [4], local boundedness estimates often lead to local Hölder continuity. A result in [17] asserts that p ∈ L ∞ (0, T ; C loc (Ω)) if the space dimension N is 2. If we could improve this result to p ∈ L ∞ (0, T ; C α (Ω)) for some α ∈ (0, 1) then m would be Hölder continuous on Ω T [18]. In this section, we shall consider the case where N = 3. Proposition 4.1. Assume that S(x) ∈ L q (Ω) with q > 3 2 and N = 3. Let (m, p) be a weak solution of (1.1)- (1.5). Then for each y in Ω, each 0 < r < 1 2 dist(y, ∂Ω), and each 1 > α > 0 there is a positive number c such that and As a result, we can apply Proposition 2.1 in [12] to the above problem. This yields (4.6) sup To estimate u, we set where k is a positive number to be determined. Then choose a sequence of smooth functions η j so that η j = 1 on B r j+1 (y), (4.9) η j = 0 outside B r j (y), (4.10) j as a test function in (4.2) to obtain Fix 2 > s > 1. For each j write B j = {x ∈ B r j (y) : u(x) ≥ k j }, (4.14) With the aid of Sobolev's inequality, we estimate Br j (y) . Now recall that 2r ≤ dist(y, ∂Ω). Pick a smooth cutoff function ξ such that ξ = 1 on B r (y), ξ = 0 outside B 2r (y), 0 ≤ ξ ≤ 1 on B 2r (y), |∇ξ| ≤ c r on B 2r (y).
Subsequently, p is continuous at y.
The proof is similar to that of Lemma 2.2 in [16]. We shall omit it here.
As we can see from the preceding analysis, there is nothing for us to cancel out the troubling term − B 2r (y) (|m| + 2) 6 dx α+2 3(α+1) in (4.1) because the weight only appears on the right-hand side in (1.6). This also explains why we can only establish a partial regularity theorem in the large-datum case [12].