Mathematical Models and Methods in Applied Sciences

This work is concerned with a prototypical model for the spatio-temporal evolution of a forager–exploiter system, consisting of two species which simultaneously consume a common nutrient, and which interact through a taxis-type mechanism according to which individuals from the exploiter subpopulation move upward density gradients of the forager subgroup. Specifically, the model

We develop a rigorous framework for modeling the geometry equilibration of crystalline defects. We formulate the equilibration of crystal defects as a variational problem on a discrete energy space and establish qualitatively sharp far-field decay estimates for the equilibrium configuration. This work extends [V. Ehrlacher, C. Ortner and A. Shapeev, Analysis of boundary conditions for crystal defect atomistic simulations, Arch. Ration. Mech. Anal. 222 (2016) 1217–1268] by admitting infinite-range interaction which in particular includes some quantum chemistry based interatomic interactions.

In this work, we introduce a new system of partial differential equations as a simplified model for the evolution of thermally induced reversible martensitic transformations in low hysteresis alloys. The model is developed in the context of nonlinear continuum mechanics, where the developed theory is mostly static, and cannot capture the influence of dynamics on martensitic microstructures. First, we prove existence of weak solutions; second, we study the physically relevant limit when the interface energy density vanishes, and the elastic constants tend to infinity. The limit problem provides a framework for the moving mask approximation recently introduced by the author. In the last section, we study the limit equations in a one-dimensional setting. After closing the equations with a constitutive relation between the phase interface velocity and the temperature of the one-dimensional sample, the equations become a two-phase Stefan problem with a kinetic condition at the free boundary. Under some further assumptions, we show that the phase interface reaches the domain boundary in finite time.

Our results provide a first step to make the formal analysis rigorous in terms of $\frac{1}{c^2}$ proposed by Chandrasekhar [S. Chandrasekhar, The post-Newtonian equations of hydrodynamics in general relativity, Astrophys. J. 142 (1965) 1488–1512; S. Chandrasekhar, post-Newtonian equations of hydrodynamics and the stability of gaseous masses in general relativity, Phys. Rev. Lett. 14 (1965) 241–244], motivated by the methods of Einstein, Infeld and Hoffmann, see Thorne [K. S. Thorne, The general-relativistic theory of stellar structure and dynamics, in Proc. Int. School of Physics “Enrico Fermi,” Course XXXV, at Varenna, Italy, July 12–24, 1965, ed. L. Gratton (Academic Press, 1966), pp. 166–280]. We consider the non-relativistic limit for the local smooth solutions to the free boundary value problem of the cylindrically symmetric relativistic Euler equations when the mass energy density includes the vacuum states at the free boundary. For large enough (rescaled) speed of light $c$ and suitably small time $T,$ we obtain uniform, with respect to $c,$ “a priori” estimates for the local smooth solutions. Moreover, the smooth solutions of the cylindrically symmetric relativistic Euler equations converge to the solutions of the classical compressible Euler equation at the rate of order $\frac{1}{c^2}$.