On the vanishing viscosity limit of a chemotaxis model

A vanishing viscosity problem for the Patlak-Keller-Segel model is studied in this paper. This is a parabolic-parabolic system in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} , with a vanishing viscosity \begin{document}$ \varepsilon\to 0 $\end{document} . We show that if the initial value lies in \begin{document}$ W^{1, p} $\end{document} with \begin{document}$ p>\max\{2, n\} $\end{document} , then there exists a unique solution \begin{document}$ (u_\varepsilon, v_\varepsilon) $\end{document} with its lifespan independent of \begin{document}$ \varepsilon $\end{document} . Furthermore, as \begin{document}$ \varepsilon\rightarrow 0 $\end{document} , \begin{document}$ (u_\varepsilon, v_\varepsilon) $\end{document} converges to the solution \begin{document}$ (u, v) $\end{document} of the limiting system in a suitable sense.

1. Introduction. In this paper we study the parabolic-parabolic system with a small parameter ε > 0: in Ω T := Ω × (0, T ), v t − ε v = g (u, v) in Ω T := Ω × (0, T ). (1.1) Here Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, u = u(x, t) and v = v(x, t) are unknown functions of (x, t) ∈ Ω × (0, T ), the functions f ∈ C 2 (R), g ∈ C 2 (R 2 ). Throughout this paper a homogeneous Neumann boundary condition will be coupled with (1.1), ∂u ∂ν = ∂v ∂ν = 0 on ∂Ω × (0, T ). (1. 2) The initial value is given by 3) The case f (y) = y and g(x, y) = x is a simplified Patlak-Keller-Segel (PKS) model for chemotaxis [7]. This model is used to describe oriented movement of cell populations, guided by a chemical gradient which is produced by the cells themselves. In the biological interpretation, u(x, t) is a (scalar) particle density, v(x, t) denotes the chemical concentration, ε is the diffusion constant for the signal while the diffusion constant of the species has been scaled to 1.

HUA CHEN, JIAN-MENG LI AND KELEI WANG
When the diffusion of the chemical substance is so small that it is negligible, i.e. ε → 0, we can formally take ε = 0 in (1.1), which leads to a parabolic-ODE system: in Ω T , v t = g (u, v) in Ω T (1.4) with the no-flux boundary condition (1.5) Such a system has been proposed by Othmer and Stevens [13] to model the case where there is no diffusion for the chemicals and chemicals can modify the local environment for succeeding passages. The PKS model has been considered by many people, see for instance the monograph [16] and references therein. Being a parabolic system, the local well-posedness of (1.1) can be proved by standard methods, see [16,Chapter 3], while it is apparently not the case for (1.4). Hence the local well-posedness problem for (1.4) has attracted a lot of interest, see [4,9,20,21]. Moreover, a similar model was studied in [18] for the one dimensional case and in [11] for the multidimensional cases. In [22], global existence and uniqueness of small smooth solutions are established for the model on bounded domains subject to no-flux boundary condition. In [2,3], the authors studied the following model on bounded domains (1. 6) It was shown in [3] that, under smallness assumptions on the initial data, there exist global weak solutions to (1.6) and the solution converges to a constant state as time goes to infinity. Finally, the problem to identify (1.4) (when f (v) = log v) as the vanishing viscosity limit of (1.1) (as ε → 0) was also studied in [15,19], by using a Cole-Hopf transformation and then considering a hyperbolic system. In this paper, we verify rigourously the convergence from (1.1) to (1.4). More precisely we prove (1) uniform in ε well-posedness for system (1.1) in W 1,p (Ω), if p > max{2, n}; (2) well-posedness for system (1.4) in W 1,p (Ω), if p > max{2, n}; (3) convergence of solutions to (1.1) to the one of (1.4).
For k ∈ (0, 1), as in [5], a function u ∈ W k,p (Ω) if When working with W 1,p solutions, the equations (1.1)-(1.3) are understood in weak sense, i.e. for any η ∈ C ∞ (Ω T ) which vanishes on Ω × {T }, we have The existence part will be proven by Banach's fixed point theorem. However, different from the treatment in many other problems, the contraction property relies on an L ∞ estimate for the parabolic operator ∂ t −∆+div(b·), where b is a vector field in L ∞ (0, T 0 ; L p (Ω)). This L ∞ estimate allows us to apply standard W 2,p estimate to the first equation in (1.1), and then define a map from L p (0, T 0 ; W 1,p (Ω)) into itself. We will use Moser iteration to prove this L ∞ estimate. This is the reason we need p > n in the above theorem, see Ladyzhenskaia, Solonnikov and Ural'ceva [8,Theorem 10.1]. Further use of Moser iteration and heat kernel representations then give uniform Hölder continuity of u ε and v ε . Finally, we stress that although this L ∞ estimate seems only to be a technical point, (1.1) could not be well-posed in W 1,n when n ≥ 2.
The second result is about (1.4).
Theorem 1.2. Suppose the initial value satisfies There exists a unique local solution (u, v) of (1.4), which satisfies with T 0 being the same as in Theorem 1.1. Furthermore, (u, v) enjoys the following regularity.
The proof of this theorem is similar to the one for Theorem 1.1. Here the equations (1.4)-(1.5) are also understood in weak sense, i.e. for any η ∈ C ∞ (Ω T ) which vanishes on Ω × {T }, we have (1.8) For the limiting system (1.4), we also prove Theorem 1.3. If the initial value satisfies u 0 ∈ W 2,p (Ω), v 0 ∈ W 2,p (Ω), for some p > max{2, n}.
(ii) If the initial value is taken as Then there exists a constant C such that Notations.
• The Laplcian ∆, when viewed as an operator between two spaces defined on Ω, is always coupled with the homogenous Neumann boundary condition. • We denote by · p , · k,p the norms in L p (Ω), W k,p (Ω), respectively. • Throughout this paper, C(t) denotes a constant satisfying lim t→0 C(t) = 0.
It may be different in different places.
The remaining part of this paper is organized as follows. In Section 2 we provide several technical tools, in particular, an L ∞ estimate which is crucial for our proof. In Section 3 we construct solutions (u ε , v ε ) to (1.1). This is the existence part of Theorem 1.1. Further properties of (u ε , v ε ) are established in Section 4. Section 5 and Section 6 are devoted to the proof of existence part in Theorem 1.2 and Theorem 1.3 respectively. Finally, we study the convergence of (u ε , v ε ) in Section 7.
2. Preliminary. In this section, we present some tools which will be used in the proof of Theorem 1.1.
where C depends on n, p, t and it is increasing in t.

Remark 1.
It is not expected that the constant C in these two propositions goes to 0 as T → 0. Therefore using only these two propositions are not sufficient to construct contraction maps and solutions to (1.1).
The following is an L ∞ estimate, which will be crucial for our following proofs.
Proof. We use Moser iteration to prove this L ∞ estimate.
Step 1. Multiplying the first equation of (2.5) by w, we obtain (2.6) Using Young inequality and Hölder inequality, we get For n > 2, we make use of interpolation inequality which holds for θ = 1−n/p. By Sobolev embedding theorem, there exists a constant C S such that . The treatment is similar to the n > 2 case, so we will assume n > 2.
The second term in the right-hand side of (2.7) is estimated from above by Concerning II, by Cauchy inequality and Hölder inequality, we get Putting these back into (2.6), we get By Gronwall's inequality, we have where C depends on Ω, a L ∞ (0,T ;L p (Ω)) , p, n, t and is increasing in t. Therefore w ∈ L ∞ (0, T ; L 2 (Ω)) and sup Step 2. For any odd number q ≥ 3, multiplying the first equation of (2.5) by w q , we get Using Young's inequality, we have Now, for n > 2, we make use of Gagliardo-Nirenberg-Sobolev inequality in the form of 10) where 1/r = 2n/(s(n + 2)) − (n − 2)/(n + 2), C GN S ≥ 1 is a constant independent of r ∈ (s, r 0 ] if r 0 > s is prescribed in advance and s ∈ [1, 2n n−2 ). For n ≤ 2, we have h s ≤ C GN S ∇h 1−1/s 2 h 1/s 1 + h 1 , for any s ≥ 1. The treatment is similar to the n > 2 case, so we will assume n > 2.
We apply (2.10) with h(t) = w(t) (q+1)/2 and s = 2p p−2 , which gives Without loss of generality, we can assume h(t) 1 > 1. Next, we apply (2.10) with Here we have used h s ≤ C(Ω) h s . Thus for any t ∈ [0, T ), by (2.9), we get where C depends on C GN S , n and p. From this differential inequality, we deduce that for any odd number q ≥ 3, Take q + 1 = 2 k+1 and denote Next, we need some estimates on g(u, v) 1,p and f (v) 1,p . Since ∇(f (y) − f (0)) = ∇f (y), without loss of generality, we may assume f (0) = 0. Denoting K = max{D α g(0, 0), f (0)}, for |α| ≤ 1 and take L to be the maximal of the Lipschitz constants for g, ∇g, f and f , we have

11)
where C depends on n, Ω, K and L.
Proof. For any u, u , v, v ∈ W 1,p (Ω), by the Lipschitz continuity of f and g, we get It follows from (2.12) that Next for j = 1, · · ·, n, by Sobolev embedding theorem and Young's inequality, where C is a constant determined by Ω, K and L. These relations are summarized as (2.11).
To prove Theorem 1.1, we also need a generalized Gronwall's inequality.
Lemma 2.4. Let η(t) be nonnegative, absolutely continues function on [0, T ], which satisfies for a.e. t ∈ (0, T ) the differential inequality where m > 1 is a positive integer and a(t), b(t), k(t) are positive, continuous functions on [0, T ]. Then for 0 ≤ t ≤ α m , Proof. Let Hence Integrating this inequality in (0, t), we get Then for t ∈ [0, α m ], we have 3. Proof of Theorem 1.1. Now we are ready to prove Theorem 1.1, Proof. We will apply Banach's fixed point theorem on L p (0, T ; W 1,p (Ω)). The proof is divided into three steps.
Step 1. Given a function z ε ∈ L p (0, T ; W 1,p (Ω)), we solve the problem with homogeneous Neumann boundary. By the semigroup theory, v ε is given by Applying Lemma 2.1 and Lemma 2.
Thus we can apply Lemma 2.2 and (3.2) to (3.5), which gives a constant C depending on v 0 1,p , u 0 1,p , M and T , such that Combining (3.6) and (3.7), we get (3.9) Combining (3.6) and (3.9) leads to where C is the coefficient of t in (3.9).
Step 3. By the previous two steps, we can define a map B : L p (0, T ; W 1,p (Ω)) → L p (0, T ; W 1,p (Ω)) as Bz ε = u ε . By (3.10), there exist M and T such that if We show when T is small enough, B is a contraction on B M . To this aim, consider two functions z 1 ε , z 2 ε ∈ B M and denote the images by u 1 ε = Bz 1 ε , u 2 ε = Bz 2 ε , respectively. The corresponding solutions of the equation (3.1) are denoted by v 1 ε and v 2 ε , respectively. By the semigroup theory, it is directly deduced that By (3.2) and (3.11), we find a constant χ, depending only on L, M , K and v 0 1,p such that , for i = 1, 2 and C is a constant determined by χ, p, K, M and L.
Applying Gronwall's inequality, we obtain (3.14) Direct calculation gives By Lemma 2.2 and (3.8), there exists a constant C such that Combining this estimate with Proposition 1 and (3.12), we deduce that where C depends on M , L, K and χ. Combining (3.14), (3.15) and (3.16), we get  .7)) However, for each ε fixed, standard parabolic regularity theory implies that u ε and v ε are smooth enough in Ω × (0, T 0 ). Hence these boundary conditions are satisfied in the classical sense. Then the Neumann boundary condition (1.2) is also satisfied by (u ε , v ε ).
4. Further properties of (u ε , v ε ). In the following, (u ε , v ε ) denotes the solution constructed in the previous section. Since for each ε > 0 fixed, (1.1) is a standard parabolic equation, (u ε , v ε ) is smooth in Ω × (0, T 0 ) and continuous up to t = 0. However, such regularity may not be uniform in ε.
In this section we establish some regularity for (u ε , v ε ), which is uniform as ε → 0. First, we prove a uniform Hölder continuity in t of u ε (t) with respect to W k,p (Ω) norms, for any k ∈ [0, 1).
Before proceeding to the proof, we recall a lemma about the mapping properties of the heat semigroup, which is taken from Taylor [17].
By Minkovski inequality, for any u ∈ L p (0, t; W k,p (Ω)), we have Hence by Lemma 4.1, we have (4.1) The first term in the right-hand side of (4.1) is estimated by applying Lemma 4.1, which gives where C is determined by k, L, M and χ. For the second term J, since u ε ∇f (v ε ) ∈ L ∞ (0, T ; L p (Ω)), we have where C is determined by k, L, M and χ. Combining (4.1), (4.2) and (4.3), we finish the proof.
Combining this proposition with Sobolev embedding theorem, we obtain the uniform Hölder continuity of u ε . Corollary 1. There exists an α ∈ (0, 1) depending on n and p such that u ε are uniformly bounded in C α,α/2 (Ω T0 ).
Note that interior Hölder continuity of u ε can also be proved by utilizing Moser iteration and Moser's Harnack inequality, see [8,Theorem 10.1].
Next, we show that v ε are uniformly Hölder continuous in t.
Since W 1,p (Ω) → C 1− n p (Ω), we have G(x, y, t)|y − x| 1− n p dy ≤ Ct Furthermore, by [1], we have the Gaussian bounds on heat kernel where c is a constant. Therefore where the constant C depends on n.
Next, since g(x, y) is a C 2 Lipschitz-continuous mapping, we have t 0 Ω G(x, y, ε(t − s))g(u ε (y, s), v ε (y, s))dyds where C is a constant determined by n, L, K, u 0 1,p and v 0 1,p . These relations are summarized as the conclusion.

Proof of Theorem 1.2: Well-posedness.
Proof. We apply Banach's fixed point theorem on the Banach space L p (0, T ; W 1,p (Ω)). The proof is divided into three steps.
Step 1. Given a function z ∈ L p (0, T ; W 1,p (Ω)), we solve with the initial value (1.3). This is equivalent to the integral equation Furthermore, it is easy to deduce that v ∈ C(0, T 0 ; W 1,p (Ω)).
With v defined as in (5.1), we solve the equation for u. Similar to (3.10), we get where C is the same as (3.10) with z ε L p (0,T ;W 1,p (Ω)) replaced by z L p (0,T ;W 1,p (Ω)) .
where χ is the constant in (3.12). Similar to (3.14), we have It follows that where the constant C is the same as in (3.17). Hence the mapping T is a contraction on B M with the same T 0 as in Theorem 1.1. Applying Banach's fixed point theorem, we get the unique fixed point of T , which is the unique weak solution (u, v) of (1.4) in [0, T 0 ]. 6. Proof of Theorem 1.3. In order to prove Theorem 1.3, we need the following lemma. Denoting K 1 = max{f (0), f (0)}, we have where C is a constant dependent on v 2,p , v 2,p , K 1 and L.
Proof. Since the Lipschitz constant of f is L, Applying Sobolev embedding theorem and Young inequality, for i = 1, · · ·, n, we have where C depends on L, v 1,p , v 1,p and K 1 . Next, for i, j = 1, · · ·, n, we have where C depends on L, v 1,p , v 1,p and K 1 . These relations are summarized as the conclusion.
Proof. We apply Banach's fixed point theorem on the Banach space L p (0, T ; W 2,p (Ω)). The proof is divided into three steps.

(6.2)
Step 2. Withṽ defined as in (6.1), we study the equation forũ. Take the decom- in Ω and κ 2 is the solution of in Ω.
By the semigroup theory, we have On the other hand, applying Proposition 2, Sobolev embedding theorem and Lemma 6.1 to (6.3), we get where C depends Ω, p, n and χ. Combining this with (6.2) leads to Step 3. By the previous two steps, we can define a map H : L p (0, T ; W 2,p (Ω)) → L p (0, T ; W 2,p (Ω)) as Hτ =ũ. Choose M as in (3.9). By (6.4), if τ L p (0,T ;W 2,p (Ω)) ≤ M and T is small enough, it holds that We show when T is small enough, the mapping H is a contraction on B M . To this aim, consider two functions τ 1 , τ 2 ∈ B M and denote the images by u 1 = Hτ 1 , u 2 = Hτ 2 , respectively. The corresponding solutions of the equation (6.1) are denoted by v 1 and v 2 , respectively. By (6.2) and (6.5), we find a constant λ, depending on M and v 0 W 2,p (Ω) , such that sup t∈[0,T ] v i (t) 2,p ≤ λ, i = 1, 2. (6.6) By the semigroup theory, Lemma 6.2 and (3.14), if T is small enough, we deduce that, for all t ∈ [0, T ], where C depends on L, M , χ, p and λ. Applying Gronwall's inequality, we get Next we come to the estimate of u 1 − u 2 , which satisfies Take the decomposition u 1 − u 2 as u 1 − u 2 = κ 1 + κ 2 , where κ 1 is the solution of in Ω × (0, T ), By (6.6), Lemma 6.1 and Sobolev embedding theorem, we have where C is a constant dependent of M , K 1 , λ, L and χ.
The function κ 2 is the solution of in Ω × (0, T ), It follows from Lemma 6.1 that where C depends on M . Combining (6.7), (6.8) and (6.9), we get 7. Proof of Theorem 1.4. In this section, we study how the solution (u ε , v ε ) of system (1.1) converges to the solution (u, v) of the limiting system (1.4), as ε → 0.
With these convergence in hand, passing to the limit in (1.7) leads to (1.8), that is, (ū,v) is a solution of (1.4) in the weak sense. In particular, since g(ū,v) ∈ C α,α (Ω T0 ), the second equation in (1.4) implies that the distributional derivativev t is a C α,α (Ω T0 ) function and this equation can be understood in the classical sense.
Since (ū,v) enjoys the same regularity with (u, v), by the uniqueness of solutions to (1.4), we deduce that (ū,v) ≡ (u, v) in Ω T0 . In particular, the limit of (u ε , v ε ) is independent of the choice of subsequences and we get the full convergence of (u ε , v ε ) to (u, v) as ε → 0.
Finally, by these convergence, (u, v) inherits the regularity of (u ε , v ε ). Furthermore, since ∂ t v ∈ L ∞ (0, T 0 ; L ∞ (Ω)), we get the Lipschitz regularity in t for v. This gives the regularity of (u, v) in Theorem 1.2.