Details
| Original language | English |
|---|---|
| Pages (from-to) | 35-64 |
| Number of pages | 30 |
| Journal | Jahresbericht der Deutschen Mathematiker-Vereinigung |
| Volume | 122 |
| Issue number | 1 |
| Publication status | Published - Mar 2020 |
| Externally published | Yes |
Abstract
PDE systems describing chemotaxis, the directed motion of organisms in response to a chemical signal, contain a cross-diffusive term which in many cases causes the unavailability of strong regularity information. An important part of their mathematical analysis is thus concerned with their behavior in situations where solutions are known to blow-up or where singularities cannot be excluded a priori. In this note we review some results, as well as some underlying fundamental analytical ideas, from the context of rigorous blow-up detection, and discuss some approaches addressing the design of solution theories which are able to adequately cope with the possible destabilizing effects of chemotactic cross-diffusion.
Keywords
- Blow-up, Chemotaxis systems, Generalized solutions
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 122, No. 1, 03.2020, p. 35-64.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Facing Low Regularity in Chemotaxis Systems
AU - Lankeit, Johannes
AU - Winkler, Michael
N1 - Funding Information: The second author acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (No. 411007140, GZ: WI 3707/5-1). Publisher Copyright: © 2019, Deutsche Mathematiker-Vereinigung and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature.
PY - 2020/3
Y1 - 2020/3
N2 - PDE systems describing chemotaxis, the directed motion of organisms in response to a chemical signal, contain a cross-diffusive term which in many cases causes the unavailability of strong regularity information. An important part of their mathematical analysis is thus concerned with their behavior in situations where solutions are known to blow-up or where singularities cannot be excluded a priori. In this note we review some results, as well as some underlying fundamental analytical ideas, from the context of rigorous blow-up detection, and discuss some approaches addressing the design of solution theories which are able to adequately cope with the possible destabilizing effects of chemotactic cross-diffusion.
AB - PDE systems describing chemotaxis, the directed motion of organisms in response to a chemical signal, contain a cross-diffusive term which in many cases causes the unavailability of strong regularity information. An important part of their mathematical analysis is thus concerned with their behavior in situations where solutions are known to blow-up or where singularities cannot be excluded a priori. In this note we review some results, as well as some underlying fundamental analytical ideas, from the context of rigorous blow-up detection, and discuss some approaches addressing the design of solution theories which are able to adequately cope with the possible destabilizing effects of chemotactic cross-diffusion.
KW - Blow-up
KW - Chemotaxis systems
KW - Generalized solutions
UR - http://www.scopus.com/inward/record.url?scp=85116142595&partnerID=8YFLogxK
U2 - 10.1365/s13291-019-00210-z
DO - 10.1365/s13291-019-00210-z
M3 - Article
AN - SCOPUS:85116142595
VL - 122
SP - 35
EP - 64
JO - Jahresbericht der Deutschen Mathematiker-Vereinigung
JF - Jahresbericht der Deutschen Mathematiker-Vereinigung
SN - 0012-0456
IS - 1
ER -