A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data

Abstract

The chemotaxis systemis considered in a bounded domain Ω ⊂ R n with smooth boundary, where χ > 0.An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter χ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis thatfor all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that u ∈ L 1 loc (Ω × [0, ∞)).

A Keller‐Segel‐fluid system with singular sensitivity: Generalized solutions

“…While for small χ , the proofs of global existence of classical solutions (see Winkler) and even boundedness, rely on an ODI for for some p > 1 and suitable r , the decisive estimates for the construction of generalized solutions for larger χ in Lankeit and Winkler are based on a similar observation concerning for p below 1.…”

“…It turned out that corresponding estimates allow for a proof of a supersolution property involving the compound quantity n p c − r with p , − r ∈ (0,1), which if combined with a more common notion of weak solubility for the second equation and with the condition that the mass be nonincreasing (as a faint subsolution requirement) serves to yield a solution concept which is compatible with the usual concept but can cope with much less regularity information, and has successfully been employed in systems where the existence of global solutions of any kind had been unknown (Lankeit and Winkler and, in a parabolic‐elliptic setting, Black). Up to now, however, the treatments of this approach do not extend to any fluid‐coupled systems.…”

“…If we want to control for some ρ > 1 (which is crucial not only for some of the convergence results in Lemma but also for obtaining the minimal regularity we desire for our solutions if they are meant to exclude blow‐up into a persistent Dirac‐type singularity), as in Lemma , the restriction will force us (cf ) to pose a stronger condition on χ in the three‐dimensional case than was needed in the fluid‐free setting in Lankeit and Winkler …”

Singular sensitivity in a Keller–Segel-fluid system

“…As in related situations (see [38,9], but also [22]), the key for establishing estimates significantly going beyond those of Lemma 2.2 lies in the following:…”

“…There is still a range of values for χ where it is unknown whether blow-up can occur. Attempts at gaining insight here include the consideration of system variants where either component is assumed to diffuse slowly (though not infinitely slowly) if compared to the other ( [10] and [11]) and the constructions of solutions within a weaker framework ( [38,30,22]) that at least cannot undergo blow-up in form of a persistent delta-type singularity.…”