Global well-posedness for the Keller--Segel--Navier--Stokes system with nonlinear boundary conditions

TL;DR

This paper proves the global existence and uniqueness of strong solutions for a Keller--Segel--Navier--Stokes system with nonlinear boundary conditions, extending previous results by handling complex boundary behaviors and establishing new maximal regularity results.

Contribution

It introduces a direct approach to construct solutions satisfying nonlinear boundary conditions and develops a new maximal regularity theorem for heat equations with inhomogeneous Neumann boundary conditions.

Findings

Established global strong solutions under small data assumptions.

Proved asymptotic stability of solutions.

Developed a new maximal regularity theorem for heat equations with nonlinear boundary conditions.

Abstract

In this paper, we consider the Keller--Segel--Navier--Stokes system with nonlinear boundary conditions in a bounded smooth (and not necessarily convex) domain $\Omega \subset \mathbb{R}^N$, $N \ge 2$, where the chemotactic sensitivity $S$ is assumed to have values in $\mathbb{R}^{N \times N}$ which accounts for rotational fluxes. In contrast to the case where $S$ is a scalar-valued function (or $S$ is the identity matrix), in our system, the normal derivative for the density $n$ of the cell is given as the product of the unknown functions, i.e., the function $n$ satisfies the nonlinear boundary condition. We show the existence and uniqueness of global strong solutions to the system under the smallness assumptions of given data, where the Lipschitz continuity of the solution mapping and the asymptotic stability of the solution are also shown. The proof is based on maximal regularity results for the linear heat equation and the Stokes system, where we establish a new maximal regularity theorem for some linear heat equation with an inhomogeneous Neumann boundary condition. Since we develop a direct approach to construct the solutions (i.e., without considering limiting procedure in certain regularized problem with homogeneous linear boundary conditions), our solutions indeed satisfy the boundary conditions, which was not addressed clearly in the previous contributions by Cao and Lankeit (2016) as well as Yu, Wang, and Zheng (2018). Under suitable regularity conditions on given data, the solution may be understood in a classical sense and, as a by-product, the non-negativity result is proved via the maximum principle of a new type.