• A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

      Kavallaris, Nikos I.; Ricciardi, Tonia; Zecca, Gabriela; University of Chester; Universita` di Napoli Federico II (Cambridge University Press, 2017-10-09)
      We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted vs.\ repelled by a single chemical substance. The production vs.\ destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model we investigate the variational structures, in particular we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy-Littlewood-Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.
    • A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding

      Kavallaris, Nikos I.; Yan, Yubin; University of Chester (Cambridge University Press, 2015-10-02)
      In the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm.