By Joaquim P. Marques de Sá
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Additional info for Applied Partial Differential Equations A Visual Approach
Meanwhile, this so called Keller–Segel model has become one of the most well analyzed systems of partial differential equations in mathematical biology, giving many insights into cell biology as well as into the analysis of nonlinear partial differential equations. The main unknowns of the Keller–Segel system are the nonnegative cell density r = r(x, t) and the chemical concentration S = S(x, t), where x denotes the one, two or three dimensional space variable and t > 0 the time variable. 1) where D0 is the positive cell diffusivity and c the positive chemotactic sensitivity.
Ridge of a Dune in Sossus Vlei, Namibia 3 Granular Material Flows 45 3 Granular Material Flows 46 Fig. 5. Wind ripples in Sossus Vlei, Namibia The obtained ﬂow equations have the form of a compressible Euler system with a temperature relaxation term. More precisely, the temperature T(x, t) relaxes according to the so-called Haff ’s law. This implies that – in the case of vanishing bulk velocity and time-independent position density – the temperature relaxes to 0 with the algebraic rate t−2 . For details and for a collection of references on granular ﬂows we refer to  and .
2 dimensions, where the spatial ground ﬂuctuations play the role of an external force ﬁeld. Comments on the Figs. e. grad Z has to be small. Clearly, this restricts the applicability of the model, in particular its use for waterfall modelling. Recently, an extension of the Saint–Venant system was presented in , which eliminates all assumptions on the bottom topography. There the curvature of the river bottom is taken into account explicitely in the derivation from the hydrostatic Euler system (assuming a small ﬂuid velocity in orthogonal direction to the ﬂuid bottom).
Applied Partial Differential Equations A Visual Approach by Joaquim P. Marques de Sá