Mathematical Model

  • Vladimir DanilovEmail author
  • Roman Gaydukov
  • Vadim Kretov
Part of the Heat and Mass Transfer book series (HMT)


This chapter is a “mathematical” one. Here we collect the mathematical background related to the mathematical model of phase transition based on the phase field system introduced by G. Caginalp. Sections 3.1 and 3.2 of the chapter contain some preliminaries and considerations about mathematical models from the physical viewpoint. In Sect. 3.3, we give the results of asymptotic analysis applied to the phase field system. In Sect. 3.4, we discuss a new definition of the generalized solution to the phase field system which is stable under passing to the limiting Stefan–Gibbs–Thomson problem. Finally, in Sect. 3.5, we discuss an approach which is a combination of mathematical (asymptotic) investigation and numerical analysis.


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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Vladimir Danilov
    • 1
    Email author
  • Roman Gaydukov
    • 1
  • Vadim Kretov
    • 1
  1. 1.National Research University Higher School of EconomicsMoscowRussia

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