Search

In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect to the Wasserstein metric. We show that systems where the population pressure is given by a function of the total population are critical with respect to cross-diffusion perturbations. This criticality is showcased by proving that adding an extra cross-diffusion term that breaks the symmetry of the population pressure in the system leads to completely different behaviours, namely segregation or mixing, depending on the sign of the perturbation. We show these results at the level of the minimisers of the associated free energy functionals. We also analyse certain implications for the gradient flow systems of the associated PDEs and present a numerical exploration of the time evolution of these phenomena. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence, uniqueness and asymptotic behaviour of a strong regular solution for a class of nonlinear equations of reaction–diffusion nonlocal type with moving boundaries: u t - a ∫ Ω t u ( x , t ) dx u xx = f ( x , t ) , ( x , t ) ∈ Q t , u ( α ( t ) , t ) = u ( β ( t ) , t ) = 0 , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω 0 = ] α ( 0 ) , β ( 0 ) [ , where Q t is a bounded non-cylindrical domain defined by Q t = ( x , t ) ∈ R 2 : α ( t ) < x < β ( t ) , for all 0 < t < T . Moreover, we study the properties of the solution and implement a numerical algorithm based on the Moving Finite Element Method (MFEM) with polynomial approximations of any degree, to solve this class of problems. Some numerical tests are investigated to evaluate the performance of our Matlab code based on the MFEM and illustrate the exponential decay of the solution. [ABSTRACT FROM AUTHOR]

In Ahmed et al. (2019) a space–time domain decomposition method was proposed for two-phase flow in a porous medium composed of two different rock types, so that the capillary pressure field is discontinuous at the interface between the rocks. For this nonlinear and degenerate parabolic problem, with nonlinear and discontinuous transmission conditions on the interface, the Optimized Schwarz waveform relaxation method (OSWR) with Robin or Ventcell transmission conditions was considered. A guaranteed and fully computable a posteriori error estimate was derived, which in particular took into account the domain decomposition error. In this paper we provide a mathematical and numerical analysis of this space–time domain decomposition method in the Robin case. Complete numerical approximation is achieved by a finite volume scheme in space and the lowest order discontinuous Galerkin method in time. We prove the existence of a weak solution of the two-phase flow subdomain problem with Robin boundary conditions by analyzing the convergence of the finite volume scheme. The domain decomposition algorithm is based on the solution of space–time nonlinear subdomain problems over the whole time interval, allowing for different time steps in different parts of the domain, adapted to the physical properties of each subdomain, and we show that such an algorithm is well-defined. Numerical experiments on three-dimensional problems with different rock types illustrate the performance of the domain decomposition method. • Space–time domain decomposition for degenerate diffusion equation is well posed. • Existence for local problem with nonlinear Robin boundary conditions. • Formulation as a space–time interface problem. [ABSTRACT FROM AUTHOR]