University of Zagreb
Nenad Antonić
Faculty of Natural Sciences

Oscillatory solutions of partial differential equations

Croatian Ministry of Science, Education and Sport (Jan 2007–Dec 2013)
Project leader: Nenad Antonić
Collaborators: Krešimir Burazin, Marko Erceg, Ivan Ivec, Martin Lazar, Marko Vrdoljak
Consultant: Luc Tartar

Many effects in natural sciences include multiple scales which lead to mathematical models described by (nonlinear) differential equations in heterogeneous media. Fundamental laws valid at the microscale are usually known, and it is important to understand which equations are valid at the macroscale. The existing mathematical models are mainly based on various notions of weak convergence. The issues known under the name of homogenisation are connected to the properties of mixtures of two or more materials. It is of interest to determine the precise bounds for qualitative physical parameters of the mixture (the effective coefficients), and whether the properties of mixtures obey the same type of physical law as the properties of components (which is not the case for memory effects). The second set of questions is related to the calculus of variations: nonconvex optimisation and microstructures, with intended description of nonlinear elastic and magnetic materials, as well as the related questions of optimal design. The technology has recently reached the maturity, so it is possible to construct nanodevices, where numerical modelling depends on the averaging by microstructure. The third aspect concerns hyperbolic conservation laws and admissibility conditions for shocks, as well as the propagation and interaction of singularities (oscillations). The microlocal analysis (H-measures) is being used as a tool here. It is possible to apply the mentioned methods for questions of turbulence in fluids and statistical mechanics. Practical computation (on computers) of coefficients in specific applications is of particular importance. As the domain of unknown functions is contained in multidimensional spaces, it is often needed to develop efficient numerical methods. The described research is a part of a worldwide trend; within the proposed project we shall join the development of new mathematical techniques and their wider applications in the modelling of physical processes.

Center of Excellence for Applications of Mathematics

DAAD (2003–)
Project leader (Universität Duisburg-Essen): Heiner Gonska
Coordinator for University of Zagreb: Nenad Antonić

Mathematical modelling of geophysical phenomena

Croatian Ministry of Science, Education and Sport (Mar 2008–)
Project leader: Marko Vrdoljak
Collaborators: Nenad Antonić, Tomi Haramina, Mladen Jurak, Martin Lazar

The goal of this project are applications of modern mathematical tools to the study of geophysical phenomena, which are often described by partial differential equations. In the transport of energy from the microscopic/mesoscopic to macroscopic scale in fluid dynamics, important role is being played by the turbulence effects. A study of an analytic model of thermohaline circulation with constant coefficients in the Adriatic gives an estimate on lateral turbulent friction coefficients, which could be significantly enhanced by the application of perturbative techniques to the model with variable coefficients. New mathematical results related to geophysical models for turbulent processes will be applied to particular localities in the Adriatic, and obtained results could be compared to available measurements. The mathematical model of double-diffusion process describing the mixing of cold fresh water with the warm sea will be studied; the same mechanism could be applied to the transport of pollutants in underground waters as well. Problems with the strong nonhomogeneity of medium, where the classical analytical and numerical procedures are not feasible, are of particular interest. Such problems are approached within the theory of homogenisation; by its application we arrive to the formulation of effective parameters via the solutions to auxiliary boundary value problems, providing a sound foundation for numerical treatment. In particular cases (e.g. iterated laminates) this method gives explicit formulas for effective parameters. We shall investigate the applicability of homogenisation in inverse problems of geophysical interest. The results on continuous dependence of solutions on the coefficients in the equation, given by that method, together with the optimal control theory, will be the basis for our approach to such problems. The homogenisation method is also crucial in inferring the good effective models for naturally nonhomogeneous porous media. The numerical treatment of models for transport of mass and energy within certain geophysical system (singularly perturbed equations of convection-diffusion) will be primarily based on the methods of finite elements and finite volumes, with a particular technique of discretisation for the convective term in the equation.

Functional analysis methods in mathematical modelling

Billateral Croatian-Serbian project (Jan 2011–Dec 2012)
Project leaders: Jelena Aleksić, Marko Vrdoljak
Collaborators: Nenad Antonić, Krešimir Burazin, Nataša Krejić, Stevan Pilipović