Hiperbolic conservation laws 1000-1S13HPZ

We will consider the particular type of partial differential equations, namely hyperbolic equations. These equations are usually not the subject of the basic course in "Partial differential equations", so we do not require previous participating in this course. We will provide the examples of phenomena described by hyperbolic equations originating in gas dynamic, granular avalanche flows, traffic flow on highways, structured population models. We will consider the problems of existence , uniqueness and asymptotic behaviour of solutions. The seminar last for two semesters what will allow to discuss modern and advanced techniques and on the other side to provide detailed explanations of analytical details understable for all participants. We will concentrate on the equations which do not posess global in time classical solutions (i.e. enough regular). Thus we will be interested in solutions of different sense. The first part of the lecture concerns the so-called weak entropy solutions and renormalized solutions. Next, we will present the compensated compactness method and consequently the notion of measure-valued solutions.