Traces of solutions to evolution equations

In the framework of this project, we are going to investigate boundary properties of solutions to equations describing a wide range of natural phenomena. As the name of the equations suggests, they are modelling processes that evolve with time, which includes e.g. the dynamics of oil/gas in a porous medium, traffic flow, different predator-prey models, etc. Roughly speaking, the mathematical research of such equations goes in two directions. The first one is the development of numerical methods which provide explicit approximate solutions to the equations. Such approximations are then used in industry to simplify and improve decision making processes, to predict future assets (financial or natural, such as oil or gas), to make engineering projects cheaper, etc. On the other hand, it is of no less importance to investigate intrinsic properties of solutions to the equations since, for instance, this can prove or disprove the correctness of the modelling equation. More precisely, if we are able to prove existence and uniqueness of solutions then we can say that the model is realistic. Another expected property of a realistic model is existence of traces for solutions of the modelling equations. Namely, since we are dealing with evolution processes, it is natural to assume initial data, which then evolve according to appropriate physical/social/biological laws. This is exactly the area of investigation of the current project. We plan to prove that physically relevant solutions to various evolution equations admit strong traces at time t=0, i.e., that they naturally arise from certain initial situation. This shows that solutions to the considered equations indeed describe the corresponding evolution process. Another implication of our research is the development of new mathematical tools which will, due to the substantial complexity of the problem, have an impact on various fields such as partial differential equations, functional and stochastic analysis, and differential geometry. Regarding the concrete results that we plant to obtain, we list them below. 1. Existence of strong traces of entropy solutions to scalar conservation laws with a regular flux explicitly depending on the space variable. 2. Existence of strong traces at t = 0 of solutions to degenerate parabolic equations. 3. Existence of strong traces at t = 0 for velocity averaged solution to diffusive transport equations. 4. Solution concept for the initial boundary value problem for degenerate parabolic equations. 5. Traces of entropy solutions to degenerate parabolic equations on manifolds with stochastic forcing. The mentioned issues are mainly long standing open problems that require substantially new methods and approaches in order to solve them. At the end of the project, we hope to have a clear vision what kind of properties are expected from solutions to the quite general class of evolution equations under consideration.