Cel
The mathematical theory of waves in media will be developed further on the basis of conservation laws through the achievement of the following objectives:
to investigate nonclassical shocks when diffusion-dispersion approximations are used for conservation laws;
to estimate the order of approximation and thickness of boundary and shock layers in parabolic approximations of conservation laws;
to investigate the stability of solutions with shocks for the MHD and semiconductor models;
to develop generalizations of the theory of characteristics and shock fronts for integrodifferential equations of mechanics of continua;
to derive and study hydrodynamic models for charge transport in semiconductors;
to simulate numerically electron flows in a silicon diode within the framework of the ballistic diode problem and evaluate the reliability of the respective hydrodynamic models;
to develop numerical methods for the problems with sharp fronts (central schemes, implicit-explicit schemes).
The research activity splits into six tasks, which closely connected and constructively interact with each other.
Task 1 concerns the study of nonclassical shocks which occur when a conservation law is considered as a zero diffusion-dispersion limit. Such shocks do not satisfy the classical Lax condition and a kinetic relation should be involved to further constrain the entropy dissipation. A nonclassical Riemann problem will be solved for a scalar conservation law and for systems of conservation laws, and the Cauchy problem will be studied in a class of nonclassical entropy solutions;
Task 2 is devoted to a problem of boundary and shock layers in conservation laws when small parabolic perturbations are involved. Concepts of thickness of boundary layer and shock layer will be introduced for nonlinear scalar conservation laws. A theory of viscosity limit will be constructed via strong estimates of solutions to parabolic perturbation equations outside the boundary and shock layers;
Task 3 consists of the study of stability of shock waves in conservation laws with applications in MHD. The stability problem reduces to an initial boundary-value problem with data on the shock front, and the well posedness of this problem will be investigated. To this end, the authors will develop further their method of dissipative energy integrals coupling it with Godunov's symmetrization approach. Particular attention will be paid to the relativistic MHD;
The aim of Task 4 is to develop a theory of perturbation and shock propagation for integrodifferential systems, which differ from hyperbolic systems because their continuous spectrum of characteristic velocities is not empty. Such integrodifferential systems find many applications in the theory of surface waves, the kinetic theory of bubbly flow and the plasma flow theory;
Task 5 is interdisciplinary to some extent. It concerns mathematical modelling the charge-carrier transport in semiconductors. The propagation of strong discontinuities will be studied for these models analytically and numerically. Particular attention will be paid to the influence of relaxation type terms;
Task 6 consists in developing accurate and efficient methods for the numerical solution of balance laws. It is supposed to develop second and third order schemes for parabolic systems with relaxation and for infinitely dimensional hyperbolic systems with applications to the shallow water model and to the Russo-Smereka model of bubbly flows.
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95125 Catania
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