Mathematical Models and Methods in Applied Sciences

In this paper the system of partial differential equations

We study the Vlasov–Poisson system with time periodic boundary conditions. For small data we prove existence of weak periodic solutions in any space dimension. In the one-dimensional case the result is stronger: we obtain existence of mild solution and uniqueness of this solution when the data are smooth. It is necessary to impose a nonvanishing condition for the incoming velocities in order to control the lifetime of particles in the domain.

In this paper, we prove the existence of weak solutions for mathematical models of miscible and immiscible flow through porous medium. An important difficulty comes from the modelization of the wells, which does not allow us to use classical variational formulations of the equations.

We analyze steady motions of incompressible second- and third-grade fluids in an aperture domain. Such flows exist due to a pressure drop and provided the flux through the hole is small. We prove that the asymptotic behavior of their velocity field far from the aperture is of order |x|^-2, which is the optimal result.

We prove a posteriori error estimates for a finite element method for steady-state, energy dependent, Fokker–Planck and Fermi pencil beam equations in two space dimensions and with a forward-peaked scattering (i.e. with velocities varying within the right unit semi-circle). Our estimates are based on a transversal symmetry assumption, together with a strong stability estimate for an associated dual problem combined with the Galerkin orthogonality of the finite element method.

Differential inequalities play a vital role in the study of ordinary and partial differential equations. In this paper we make use of first-order differential inequalities to investigate the decay of solutions to two ill-posed elliptic problems in a semi-infinite cylindrical domain.

We present the numerical analysis of a new a posteriori finite element procedure that provides rigorous, constant-free, lower and upper bounds for linear functional outputs of coercive second-order partial differential equations. We prove that, for a linear finite element space, the bounds converge to the exact result at the optimal rate: for sufficiently regular data and output functionals, where H is the diameter of the mesh which provides the solution and adjoint required by our technique.