Braukhoff, M., Huber, F., & Jüngel, A. (2023). Global martingale solutions for stochastic Shigesada–Kawasaki–Teramoto population models. Stochastics and Partial Differential Equations: Analysis and Computations. https://doi.org/10.1007/s40072-023-00289-7
Stochastics and Partial Differential Equations: Analysis and Computations
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ISSN:
2194-0401
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Datum (veröffentlicht):
2023
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Umfang:
51
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Verlag:
Springer
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Peer Reviewed:
Ja
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Keywords:
Cross diffusion; Entropy method; Martingale solutions; Multiplicative noise; Population dynamics; Tightness of laws
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Abstract:
The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada–Kawasaki–Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods for evolution equations. Instead, the existence proof is based on the entropy structure of the model, a novel regularization of the entropy variable, higher-order moment estimates, and fractional time regularity. The regularization technique is generic and is applied to the population system with self-diffusion in any space dimension and without self-diffusion in two space dimensions.
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Projekttitel:
Analysis von PDEs mit Kreuzdiffusion und stochastischen Termen: I 3401-N32 (FWF - Österr. Wissenschaftsfonds) Multikomponentensysteme mit unvollständiger Diffusion: P 33010-N (FWF - Österr. Wissenschaftsfonds) Doktoratskolleg "Dissipation and Dispersion in Nonlinear Partial Differential Equations": W1245-N25 (FWF - Österr. Wissenschaftsfonds) Emergente Netzwerkstrukturen und neuromorphische Anwendungen: 101018153 (European Commission)