Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with subcritical sensitivity

Abstract

This paper deals with the Keller–Segel–Navier–Stokes system [Formula: see text] in a bounded domain [Formula: see text] with smooth boundary, where [Formula: see text] and [Formula: see text] are given functions. We shall develop a weak solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component [Formula: see text]. Under the assumption that there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] it is finally shown that for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined weak solution. In comparison to the result for the corresponding fluid-free system, it is easy to see that the restriction on [Formula: see text] here is optimal. This result extends previous studies on global solvability for this system in the two-dimensional domain and for the associated chemotaxis-Stokes system obtained on neglecting the nonlinear convective term in the fluid equation.

Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions

“…we obtain the following main results. For the linear diffusion case m = 1 Theorem 1.1 provides the existence of a global weak solution for α > 1 3 , extending the results of [16] and [10], which provided the existence of a global very weak solution for α > 1 3 and a global weak solution for α > 3 7 , respectively. If we merely prescribe m + 2α ≤ 5 3 , we have to weaken the solution concept in order to verify the existence of global solutions -which is due to the obtainable a priori information being so weak that we have to consider a sublinear functional of n for our testing methods.…”

“…As a first step in the construction of global solutions in either of the senses above we will first adapt the approaches undertaken in [21,16,2] to our setting in order to approximate the system (1.4) by problems in which the no-flux boundary condition of the first component reduces to a homogeneous Neumann boundary condition and which are solvable globally in time. With a family (ρ ε ) ε∈(0,1) ⊂ C ∞ 0 (Ω) of cut-off functions in Ω satisfying…”

“…Because of 4 5−6α ≤ 2 for α ≤ 1 2 , this now allows to pursue a similar reasoning as before, while making use of the fact that s ln s ≥ − 1 e for all s > 0. While the main idea of utilizing the latter spatio-temporal bound for ∇(n ε + ε) m+α−1 to establish timespace bounds for n ε + ε remains unchanged from the previous works [16,Lemma 4.2] and [2, Lemma 4.3], we have to treat the term more delicate in order to prepare sufficient information for the limiting procedure later on. In particular, there exist r ∈ (1, 2) and C > 0 such that hold for each ε ∈ (0, 1) and all t ≥ 0.…”

“…linear diffusion) and tensor-valued S(x, n, c) satisfying |S(x, n, c)| ≤ (1 + n) −α global weak solutions were shown to exist for α ≥ 3 7 ([10]) and global very weak solutions were established whenever α > 1 3 ([16]). In space dimension N = 2 the optimal condition α > 0 can even be reached with global bounded classical solutions ( [17]). If we simplify to Stokes-fluid (κ = 0 in (1.2)) instead of full Navier-Stokes-fluid, more regular solutions can also achieved in dimension N = 3, as indicated by the recent studies on bounded classical solutions in [24].…”

Global existence and time decay estimate of solutions to the Keller–Segel system

“…As an immediate byproduct, the authors also obtained the usual L p − L 2 (1 ≤ p ≤ 2) type of the decay rates without requiring that the L p norm of initial data is small. For more details, one can refer to previous studies [15][16][17][18][19][20][21][22][23][24][25] and the reference therein.…”