Radial boundary layers for the singular Keller-Segel model

Chemotaxis describes the oriented movement of species stimulated by uneven distribution of a chemical substance in the environment. It is a significant mechanism accounting for abundant biological process/phenomenon, such as aggregation of bacteria [34, 44], slime mould formation [15], fish pigmentation [35], tumor angiogenesis [4, 7, 6], primitive streak formation [36], blood vessel formation [12], wound healing [39]. Mathematical models of chemotaxis were first proposed by Keller and Segel in their seminal works [19, 20, 21]. In this paper, we are concerned with the following chemotaxis model:

where $\Omega$ is a domain in $\mathbb{R}^{n}$ with smooth boundary. System (1.3) was first advocated in [21] to describe the traveling band propagation of bacterial chemotaxis observed in the experiment of Adler [1, 2]. It later appeared in the work by Levine et al [23] to model the initiation of tumor angiogenesis, where $u(x,t)$ represents the density of vascular endothelial cells and $c(x,t)$ denotes the concentration of signaling molecules vascular endothelial growth factor (VEGF). The parameters $D>0$, $\varepsilon\geq 0$ are diffusion coefficients of the endothelial cells and the chemical VEGF respectively, $\chi>0$ is the chemotactic coefficient measuring the intensity of chemotaxis and $\mu\geq 0$ is the chemical consumption rate by cells. In particular it was pointed out in [23] that the chemical diffusion process is far less important comparing to its interaction with endothelial cells and thus the diffusion coefficient $\varepsilon$ could be small or negligible. Despite of its biological significance, (1.3) is difficult to study mathematically due to the singularity of $\ln c$ at $c=0$. The well-known way to overcome this singularity was applying the following Cole-Hopf transformation (cf. [22, 31]):

to transform (1.3) into a system of conservation laws:

The transformed system (1.5) attracts extensive attentions and numerous interesting results have been developed. We briefly recall these results by the dimension of spaces. In the one dimensional case, the global well-posedness along with large time behavior of solutions was investigated when $\Omega=\mathbb{R}$ in [13, 25] with $\varepsilon=0$ and in [37, 33] with $\varepsilon>0$. When $\Omega=(0,1)$, authors in [54, 28] obtained the unique global solution under Neumann-Dirichlet boundary conditions for $\varepsilon=0$, and the result was later extended to the case $\varepsilon>0$ in [49, 43]. The problem with $\varepsilon\geq 0$ is also globally well-posed [26] with Dirichlet-Dirichlet boundary conditions. Furthermore, the existence and stability of traveling wave solutions were studied in [18, 29, 30, 31, 32, 27, 3]. However to the best of our knowledge, except when it is associated with radially symmetric initial data, the known well-posedness results of problem (1.5) in the multi-dimension are merely confined to local large and global small solutions, cf. [24, 14, 8, 38, 45, 48] for details when $\Omega=\mathbb{R}^{n}$ ($n\geq 2$) and [28, 41] when $\Omega\subset\mathbb{R}^{n}$ ($n\geq 2$) is bounded. If the initial data are radially symmetric and $\Omega=B_{R}(0):=\{x=(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}\,|\,\,|x|<R\}$ ($n\geq 2$), Winkler [50] recently proved that (1.3) with $\varepsilon>0$ subject to Neumann boundary conditions admits a global generalized (weak) solution which is radially symmetric and smooth away from the origin $x=0$.

In addition to the above well-posedness results, the asymptotic behavior of solutions as $\varepsilon\rightarrow 0$ is a particularly relevant issue (mentioned in [23] that $\varepsilon$ could be small/negligible) and has been studied in several circumstances for equation (1.5) and thus for the original equation (1.3) via transformation (1.4). For illustration, denote by $(u^{\varepsilon},\vec{v}^{\,\varepsilon})$ and $(u^{0},\vec{v}^{\,0})$ the solutions of (1.5) with $\varepsilon>0$ and $\varepsilon=0$ respectively (when $n=1$ we shall use the notation $v$ instead of $\vec{v}$ since it is a scalar). First in unbounded domains, it has been shown that both traveling wave solutions (cf. [47]) in ${\mathbb{R}}$ and the global small-data solution of the Cauchy problem (cf. [48, 38]) in ${\mathbb{R}}^{n}(n=2,3)$ are uniformly convergent in $\varepsilon$, namely $(u^{\varepsilon},\vec{v}^{\,\varepsilon})$ converge to $(u^{0},\vec{v}^{\,0})$ in $L^{\infty}$-norm as $\varepsilon\to 0$. With $\Omega=(0,1)$, the solutions still converge (cf. [49]) as $\varepsilon\rightarrow 0$ when (1.5) is endowed with the following mixed homogeneous Neumann-Dirichlet boundary conditions

However if both of $u$ and $v$ are subject to the Dirichlet boundary conditions, one can not preassign the boundary value for $v^{0}$ when $\varepsilon=0$ since it is intrinsically determined by the second equation of (1.5) as $v^{0}|_{x=0,1}=v_{0}|_{x=0,1}+\int_{0}^{t}u_{x}^{0}|_{x=0,1}\,d\tau$. Thus the plausible Dirichlet boundary conditions should be prescribed as:

where $\bar{u}\geq 0,$ $\bar{v}\in\mathbb{R}$ are constants. In this case, if the boundary values of $v$ with $\varepsilon>0$ and $\varepsilon=0$ do not match, then the solution component $v$ would diverge near the end points $x=0,1$ as $\varepsilon\rightarrow 0$ and this phenomenon is termed as the boundary layer effect, which has been an important topic in the fluid mechanics [42] when investigating the inviscid limit of the Navier-Stokes equations near a boundary and has attracted extensive studies (cf. [9, 10, 11, 17, 46, 52, 53]) since the pioneering work [40] by Prandtl in 1904. In particular, this boundary layer effect for problem (1.5)-(1.6) has been recently numerically verified in [26] and rigorously proved in [16].

Enlightened by these results, it is natural to expect that (1.5) in multi-dimension ($n\geq 2$) possesses boundary layer solutions as well when prescribing appropriate Dirichlet boundary conditions. In particular we aim to investigate this issue for its radial solutions in the present paper. To this end, we first rewrite (1.3) in its radially symmetric form by assuming that the solutions $(u,c)$ are radially symmetric, depending only on the radial variable $r=|x|$ and time variable $t$. In a domain bounded by two concentric sphere, i.e. $\Omega=\{x=(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}\,|\,\,\,0<a<|x|<b\}$, (1.3) reads as

where $D=\chi=\mu=1$ have been assumed without loss of generality. Similar as deriving (1.5) from (1.3), we apply the following Cole-Hopf type transformation

which turns (1.10) into

Similar to (1.6), the Dirichlet boundary conditions for (1.15) are prescribed as

In this paper, we shall investigate the asymptotic behavior of solutions to (1.15)-(1.16) as $\varepsilon\rightarrow 0$ for $n\geq 2$ (if $n=1$, it coincides with the one-dimensional model (1.5)-(1.6) which has been studied in [16] as aforementioned). In particular, the solution component $v$ is proved to have a boundary layer due to the mismatch of its boundary values as $\varepsilon\rightarrow 0$ (see Theorem 2.2).

To study the boundary layer effect, we first present the global well-posedness and regularity estimates for solutions of (1.15)-(1.16) with $\varepsilon=0$ in Theorem 2.1. By these estimates, we then state the main result on the convergence for $u$ and boundary layer formation by $v$ in Theorem 2.2. Finally, the result is converted to the original chemotaxis model (1.10) via (1.11). We begin with introducing some notations.

Notations. Without loss of generality, we assume $0\leq\varepsilon<1$ since the zero diffusion limit as $\varepsilon\rightarrow 0$ is our main concern. Throughout this paper, unless specified, we use $C$ to denote a generic positive constant which is independent of $\varepsilon$ and dependent on $T$. In contrast, $C_{0}$ denotes a generic constant independent of $\varepsilon$ and $T$. For simplicity, $L^{p}$ represents $L^{p}(a,b)$ with $1\leq p\leq\infty$, $H^{k}$ denotes $H^{k}(a,b)$ with $k\in\mathbb{N}$ and $\|\cdot\|$ stands for $\|\cdot\|_{L^{2}}$. Moreover, if $f(r,t)\in L^{p}(a,b)$ for fixed $t>0$, we use $\|f(t)\|_{L^{p}}$ to denote $\|f(\cdot,t)\|_{L^{p}}$.

The first result is on the global well-posedness of (1.15)-(1.16) with $\varepsilon=0$.

We proceed to recall the definition of boundary layers (BLs) following the convention of [10, 11].

Our main result is as follows.

By employing transformation (1.11), we next convert the above results for (1.15)-(1.16) to the pre-transformed chemotaxis model (1.10). The counterpart of the original model reads as follows:

At the end of this section, we briefly introduce the main ideas used in the paper. Although the system (1.15)-(1.16) with $n\geq 2$ is in a similar form to its counterpart with $n=1$ for which the vanishing diffusion limit has been studied in [16] based on a $\varepsilon$-independent estimate for solutions with $\varepsilon>0$, the methods used there can not be applied to study the present problem since when $n\geq 2$ the system (1.15)-(1.16) with $\varepsilon>0$ lacks an energy-like structure or a Lyapunov function to provide a preliminary estimate uniformly in $\varepsilon$. Moreover, one can not use the estimates derived in [50] for the present problem either since those estimates depend on $\varepsilon$. The difficulty in our analysis consists in deriving the $\varepsilon$-convergence estimates in (2.5) and (2.6) without any uniform-in-$\varepsilon$ priori bounds on solutions $(u^{\varepsilon},v^{\varepsilon})$. Inspired by the works [5, 51], this will be achieved in section 4 by regarding $(u^{\varepsilon},v^{\varepsilon})$ with small $\varepsilon>0$ as a perturbation of $(u^{0},v^{0})$ and then estimating their difference $(u^{\varepsilon}-u^{0},v^{\varepsilon}-v^{0})$ by the method of energy estimates and a new Gronwall’s type inequality (see Lemma 4.1) on ODEs. The proof of Theorem 2.1 is standard and will be given in section 3.

This section is to prove Theorem 2.1 based on the following lemmas where the a priori estimates on solution $(u^{0},v^{0})$ of (1.15)-(1.16) with $\varepsilon=0$ are derived by the energy method. We set off by rewriting (1.15)-(1.16) with $\varepsilon=0$ as follows:

We proceed to derive higher regularity properties for the solution $(\tilde{u},v^{0})$ of (3.12).