Maximal regularity and a singular limit problem for the Patlak-Keller-Segel system in the scaling critical space involving \textit{BMO}
The authors study the singular limit of the Cauchy problem for the parabolic-parabolic Patlak-Keller-Segel system in \(\mathbb{R}^n\), \(n \geq 2\), \[ \begin{cases} \partial_tu_{\tau} - \Delta u_{\tau} + \nabla \cdot (u_{\tau}\nabla \psi_{\tau}) = 0, & t>0, \ x \in \mathbb{R}^n,\\ \frac{1}{\tau} \partial_t \psi_{\tau} - \Delta \psi_{\tau} + \lambda \psi_{\tau} = u_{\tau}, & t>0, \ x \in \mathbb{R}^n,\\ u_{\tau}(0,x) = u_0, \psi_{\tau}(0,x) = \psi_0(x), & t=0, \ x \in \mathbb{R}^n, \end{cases}\tag{1} \] as the relaxation time \(\tau \rightarrow \infty\). Here in the above equation \(u_{\tau}, \psi_{\tau}\) denotes the density of molds and the concentration of the chemical substances respectively and \(\lambda \geq 0\). Formally, the singular limit \(u = \lim_{\tau \rightarrow \infty} u_{\tau}, \psi = \lim_{\tau \rightarrow \infty} \psi_{\tau}\), if exists, would expected to satisfy the parabolic-elliptic Patlak-Keller-Segel system \[ \begin{cases} \partial_t u - \Delta u + \nabla \cdot (u \nabla \psi) = 0, & t >0, \ x \in \mathbb{R}^n \\ -\Delta \psi + \lambda \psi = u, & t>0, \ x \in \mathbb{R}^n \\ u(0,x) = u_0(x), & t =0, \ x \in \mathbb{R}^n. \end{cases}\tag{2} \] One of the main obstruction in the singular limit problem is, in general, the solution space for \(\psi_{\tau}\) and \(\psi\) are different from each other. The authors rigorously justified the above formal claim in the scaling invariant critical spaces of Bochner-Lebesgue type: \begin{align*} (u_0, \psi_0) \in \Big(L^{\frac{n}{2}}(\mathbb{R}^n) \cap B_{q,\theta}^{-\frac{2}{\theta}}(\mathbb{R}^n) \Big) \times \Big(VMO(\mathbb{R}^n) \cap B_{r,\theta}^{1-\frac{2}{\theta}}(\mathbb{R}^n)\Big) \end{align*} for large data time local and \begin{align*} (u_0, \psi_0) \in \Big(L^{\frac{n}{2}}(\mathbb{R}^n) \cap \dot{B}_{q,\theta}^{-\frac{2}{\theta}}(\mathbb{R}^n) \Big) \times \Big(BMO(\mathbb{R}^n) \cap \dot{B}_{r,\theta}^{1-\frac{2}{\theta}}(\mathbb{R}^n)\Big) \ \end{align*} for small data time global, where ``\(B\)'' stands for the non-homogeneous Besov space, and ``\(BMO\)'' (respectively, ``\(VMO\)'') denotes the space of bounded mean oscillation (respectively, vanishing mean oscillation) and the dot denotes the corresponding homogeneous spaces. For dimension \(2,\) further integrability assumption on the initial data needed. Here the exponents \(q,\theta\) and \(r\) satisfy scale invariant (Serrin) admissibility criterion \[ \begin{cases} \frac{2}{\theta} + \frac{n}{q} = 2, \ \ \frac{n}{2} < q < \theta, \ \ \theta \geq 2, \\ \frac{2}{\theta} + \frac{n}{r} = 1, \ \ n < r < \theta, \ \ \frac{rq}{r + q} \geq 1. \end{cases} \] The choice of this class is, in some sense, the optimal where the existence of large data local solutions and small data global solutions for the Cauchy problem (1) can be proven, and the singular limit for \(\psi_{\tau}\) can be justified using the maximal regularity properties of the heat semigroup. The result of this article improves the earlier works of the same authors [J. Differ. Equations 269, No. 10, 8959--8997 (2020; Zbl 1443.35070); J. Evol. Equ. 20, No. 2, 421--457 (2020; Zbl 1444.35096)], and the main difference is the solution here is considered in the end-point critical function class \(VMO(\mathbb{R}^n)\) for time local solution and \(BMO(\mathbb{R}^n)\) for small solution. This nicely written article has been divided into several sections. After a very descriptive introduction section, necessary basic estimates have been listed. A brief section has been devoted on the study of maximal regularity of the heat semigroup, and then the authors demonstrated the existence of local solutions and small data global solutions in the above mentioned function spaces followed by the proof of singular limit problem and detection of the initial layer. The article contains a broad appendix section where several necessary results have been recalled and improved in this particular setting making this article as self-contained as possible.
scientific article; zbMATH DE number 7451785
scientific article