Hölder continuity of Keller–Segel equations of porous medium type coupled to fluid equations

Abstract

We consider a coupled system consisting of a degenerate porous medium type of Keller-Segel system and Stokes system modeling the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global existence of weak solutions and Hölder continuous solutions in dimension three, under the assumption that the power of degeneracy is above a certain number depending on given parameter values. To show Hölder continuity of weak solutions, we consider a single degenerate porous medium equation with lower order terms, and via a unified method of proof, we obtain Hölder regularity, which is of independent interest.

Regularity Properties of Degenerate Diffusion Equations with Drifts

“…Below we state two theorems that summarizes our main results. Regarding (c), the only relevant result for (1.1) that we are aware of is from [12], where integrability conditions are assumed on both V and ∇V . Let us also very briefly mention some results for the linear case m = 1 where the threshold L ∞ t L d x remains the same.…”

Uniqueness of solutions for Keller–Segel system of porous medium type coupled to fluid equations

“…We note that Hölder regularities in [7] are enough to control J (t). Hence, we derive the following a prior estimate by Gronwall's inequality Indeed, when we estimate the right-hand side of (3.14), we need to perform the integration by parts to move one derivative in ∆ζ to the other terms in II.…”

“…Hence, we derive the following a prior estimate by Gronwall's inequality Indeed, when we estimate the right-hand side of (3.14), we need to perform the integration by parts to move one derivative in ∆ζ to the other terms in II. If q > 1, A 1+α appears in II and it requires that η ∈ W 1,∞ (R 3 T ), which is beyond the regularity in [7]. Developing new methods dealing with (1.1) for the case q > 1 and possibly other equations, having solutions but no uniqueness, will be some of our next study.…”

“…Since the system (1.2) satisfies the assumptions in [7,Theorem 1.10], it is straightforward that there exist Hölder continuous weak solutions when α > 1 8 and initial data are sufficiently regular. In this case, we can show that such solutions become unique.…”

Existence of weak solutions for Porous medium equation with a divergence type of drift term

“…There are works showing counterexamples of lack of regularities under certain conditions on the drift, for example, [43,46,52] for (fractional) linear diffusion and [27,33] for porous medium equations. From the perspective of critical conditions on V providing continuity of (1.1) (for example, [14,27,33]), we recall…”