scholarly journals Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process

2019 ◽  
Vol 78 (6) ◽  
pp. 1681-1711 ◽  
Author(s):  
Ramón G. Plaza
Keyword(s):  
Jump Process ◽  
Cross Diffusion ◽  
Velocity Jump Process ◽  
Velocity Jump ◽  
Parabolic Limit

scholarly journals Spreading in kinetic reaction–transport equations in higher velocity dimensions

2018 ◽  
Vol 30 (2) ◽  
pp. 219-247
Author(s):  
EMERIC BOUIN ◽  
NILS CAILLERIE
Keyword(s):  
Large Scale ◽  
Jump Process ◽  
Travelling Wave ◽  
Transport Equations ◽  
Kinetic Reaction ◽  
Velocity Jump Process ◽  
Minimal Speed ◽  
Crucial Difference ◽  
Velocity Jump ◽  
Reaction Transport

In this paper, we extend and complement previous works about propagation in kinetic reaction–transport equations. The model we study describes particles moving according to a velocity-jump process, and proliferating according to a reaction term of monostable type. We focus on the case of bounded velocities, having dimension higher than one. We extend previous results obtained by the first author with Calvez and Nadin in dimension one. We study the large time/large-scale hyperbolic limit via an Hamilton–Jacobi framework together with the half-relaxed limits method. We deduce spreading results and the existence of travelling wave solutions. A crucial difference with the mono-dimensional case is the resolution of the spectral problem at the edge of the front, that yields potential singular velocity distributions. As a consequence, the minimal speed of propagation may not be determined by a first-order condition.

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