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Free energy and self-interacting particles. (English) Zbl 1082.35006

Progress in Nonlinear Differential Equations and their Applications 62. Boston, MA: Birkhäuser (ISBN 0-8176-4302-8/hbk). xiii, 366 p. (2005).
The mathematical analysis of the monograph under review is limited to a very special system of parabolic-elliptic PDE, but having numerous applications in mathematical biology (chemostatic feature of cellular lime or capillary formation of blood vessels in angeogenesis description), statistical mechanics and chemical kinetics (description of the motion of a mean field of many particles, interacting under the gravitational inner force or the chemical reaction). Therefore in the latter context such system is affiliated with a hierarchy of equations Langevin, Fokker-Planck, Liouville-Gel’fand, and the gradient flow, the mathematical principle of which is referred to as the quantized blow up mechanism. The blow up solution of the system develops delta-function singularities with the quantized mass.
A typical example of such systems is the chemotaxis model
\[ \left. \begin{aligned} u_t&=(\nabla, \nabla u - u\nabla v)\\ 0&=\Delta v-av+u \end{aligned}\right\}\;\text{ in }\Omega\times(0, T),\qquad \begin{aligned} \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0&\quad\text{on } \partial\Omega\times(0,T),\\ u|_{t=0}= u_0(x)&\quad\text{in }\Omega \end{aligned} \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary \(\partial\Omega\), \(a>0\) is a constant, \(\nu\) is the outer unit normal to \(\partial\Omega\), where \(u(x,t)\) and \(v(x,t)\) are the density of cellular slim molds and the concentration of chemical substances secreted by themselves.
The main results of the monograph, the theorems 1.1 and 1.2, important in the indicated applications, are formulated and proved in Ch. 15 using the boundary behavior of the Green function and the generation of the weak solution, established in Chs. 5 and 13. Theorem 1.2 asserts that if the solution of the system blows up in finite time then it develops delta-function singularities with the quantized mass (collapses) as the measure-theoretical singular part. If the collapse has an envelope, i.e. the region containing the whole blow up mechanism in space-time, then mass and entropy are exchanged at the wedge of this envelope.
Theorem 1.1. as a preparatory result of the author’s previous work, the formation of collapses and the estimation of their masses from below are presented with proofs in Ch. 11. Chs. 3–5 contain the classical theory for the system. Chs. 6–10 are devoted to the stationary problem, whereas Chs. 6–8 contain a survey and more or less known results by using new arguments (symmetrization), motivated by the study of nonstationary problems. Ch. 9 describes the effects of these unstable stationary solutions on the local dynamics. Ch. 10 is the application of this study to the local dynamics. In Ch. 16 the abstract theory of dual variation is developed and the results of Ch. 6 are extended with the aid of convex analysis. It is shown that the stationary problem has two equivalent variational formulations, where the cost functionals are associated with the particle density and the field distribution. These two functionals have the duality through the Legendre transform and are combined with a functional called the Lagrange function. Moreover, a stability result for the stationary solution is given.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
35Q72 Other PDE from mechanics (MSC2000)
35Q40 PDEs in connection with quantum mechanics
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
92C37 Cell biology
80A30 Chemical kinetics in thermodynamics and heat transfer
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