I'm interested in understand how to derive the regularity result (with respect to HΓΆlder space (2,1) in space-time) in this PDE model:
https://arxiv.org/abs/1707.09622
But all articles that I find cite the Winkler's paper (https://www.tandfonline.com/doi/full/10.1080/03605302.2011.591865) direct or indirectly.
Winkler says "According to standard bootstrap arguments involving the regularity theories for parabolic equations and the Stokes semigroup [10], [8])"
[8] J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci. 26 (2016), 2071β2109
[10] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations 261 (2016), 2650β2669.
But I search in those references and I does not able to understeand how the results in those papers implies the regularity
Do you know some reference where develop this reasoning for beginers like me?
Thank you, and I'm sorry if this is a basic question, it's an interesting field!
Note this is in cows
Carretta MD, BarrΓa Y, Borquez K, et al. Ξ²-hydroxybutyrate and hydroxycarboxylic acid receptor 2 agonists activate the AKT, ERK and AMPK pathways, which are involved in bovine neutrophil chemotaxis. Sci Rep. 2020;10(1):12491. Published 2020 Jul 27. doi:10.1038/s41598-020-69500-2
https://doi.org/10.1038/s41598-020-69500-2
Abstract
Elevated plasma concentrations of the ketone body Ξ²-hydroxybutyrate (BHB), an endogenous agonist of the hydroxycarboxylic acid receptor 2 (HCA2), is associated with an increased incidence of inflammatory diseases during lactation in dairy cows. In the early stages of this pathology, an increase in neutrophil recruitment is observed; however, the role of BHB remains elusive. This study characterized the effect of BHB and synthetic agonists of the HCA2 receptor on bovine neutrophil chemotaxis and the signaling pathways involved in this process. We demonstrated that treatment with BHB concentrations between 1.2 and 10 mM and two full selective agonists of the HCA2 receptor, MK-1903 and nicotinic acid, increased bovine neutrophil chemotaxis. We also observed that BHB and HCA2 agonists induced calcium release and phosphorylation of AKT, ERK 1/2 and AMPKΞ±. To evaluate the role of these pathways in bovine neutrophil chemotaxis, we used the pharmacological inhibitors BAPTA-AM, pertussis toxin, U73122, LY294002, U0126 and compound C. Our results suggest that these pathways are required for HCA2 agonist-induced bovine neutrophil chemotaxis in non-physiological condition. Concentrations around 1.4 mM of BHB after calving may exert a chemoattractant effect that is key during the onset of the inflammatory process associated with metabolic disorders in dairy cows.
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