The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system

The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System

“…In order to successfully extract a uniform stabilization for n (κ) ε and u (κ) ε in certain L p spaces we will require the following auxiliary lemma for ODEs, which we have taken from [26,Lemma 4.3].…”

“…for large η > 0 independent of κ ∈ [−1, 1] to derive, after a some bootstrapping, κ-independent bounds in C 1 Ω × C 2 Ω × D(A α ) uniform in time. These bounds, when combined with decay properties of (Λ κ ), then become the driving force of the exponential stabilization featured in [26]. In stark contrast, in the current three-dimensional framework we cannot utilize a corresponding quasi-energy functional immediately, as for (Λ κ ) only the global existence of a weak solution obtained by a limiting procedure from approximating systems is known ( [30]).…”

“…Rigorous mathematical results appear to be mostly lacking. In fact, only recently it was shown in the two-dimensional setting that upon taking κ → 0 the global classical solution n (κ) , c (κ) , u (κ) of the chemotaxis-Navier-Stokes system convergences uniformly in time towards the global classical solution (n (0) , c (0) , u (0) of (Λ 0 ) in the sense that there exist C > 0 and µ > 0 such that whenever κ ∈ (−1, 1), n (κ) (·, t) − n (0) (·, t) L ∞ (Ω) + c (κ) (·, t) − c (0) (·, t) L ∞ (Ω) + u (κ) (·, t) − u (0) (·, t) L ∞ (Ω) ≤ C|κ|e −µt holds for all t > 0 ( [26]).…”

“…Mathematical challenges and the approach. In the two-dimensional setting investigated in [26], it is known that (Λ κ ) already admits a classical solution on Ω × (0, ∞), which in turn allows for testing procedures immediately targeting the quasi-energy functional Ω n (κ) ln n (κ)…”

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.…”