Vanishing capillarity on smooth manifolds

The topic of this research project is a refined analysis of a family of differential equations modelling diverse natural and social phenomena such as flow in porous media, unidirectional propagation of nonlinear, disper- sive, long waves, or population dynamics problems. In particular, we have in mind evolution problems in the widest sense governed by scalar conservation laws and pseudo-parabolic equations constrained to non-flat surfaces. The project is divided in two parts. In the first one, we intend to model the phenomena via the theory of Riemannian manifolds. Compared to the recently obtained results in the Euclidean setting, this will require new techniques and substantial modifications and re-interpretations of classical concepts. We will have to develop techniques not only for considering vanishing capillarity limits on manifolds, but also methods for analyzing quasi-linear pseudo-parabolic equations on manifolds. The second part of the project is devoted to an even more realistic modelling by allowing stochastic elements to be integrated into the picture. Still more so than in the first part, new mathematical tools will have to be developed. In particular, variants of the so-called velocity averaging lemmas and of the theory of micro-local defect measures (H-measures) will have to be developed in the stochastic setting. Apart from progress in understanding the phenomena modelled by these differential equations, we expect this progress to lead to a promising synthesis of hitherto separate theories, in this case scalar conservation laws, differential geometry, and stochastics. We also note that the proposal is motivated by an experiment from the flow in porous media showing inade- quacy of the scalar conservation law modelling the phenomenon. This might lead to a new (non-Kruzhkov) stable semigroup of solutions to scalar conservation laws which would be a fully original result in the theory of scalar conservation laws with regular flux. The principal investigators responsible for the successful accomplishment of the first (non-stochastic) part of the project are Darko Mitrovic and Michael Kunzinger, while for the stochastic part these researchers will be joined by Kenneth Karlsen. The collaboration with Kenneth Karlsen will enable us to investigate much more general situations concerning stochastic scalar conservation laws (with the stochastic elements in the form of both the stochastic forcing and the stochastic flux). Moreover, with the help of Kenneth Karlsen, we will be able to develop a numerical methods for the equations considered on manifolds.

In the frame of the project, we were investigating partial differential equations (PDEs) which model various natural phenomena along non-flat surfaces with stochastic effects taken into account. We were able to prove existence of solution to such types of PDEs and thus to help confirming that the models were properly derived from physical laws and thus useful. In the course of the proofs, we have developed different techniques which can be used to a much wider class of equation. In particular, we would like to emphasize the velocity averaging results for transport equation with stochastic forcing. This appeared to be an unexpectedly challenging task which forced us to get deep into the stochastic analysis and to rephrase the problem in an appropriate probability space through the Skorokhod-Jakubowski theory. The result is highly non-trivial and it represents a substantial novelty in the field. We also needed to develop original methods from the theory of micro-local defect measures. Namely, in the frame of the stochastic analysis, the time variable involves several technical obstacles. The main one is regarded derivative of composition of function where one needs to use the Ito lemma instead of the standard formula. Moreover, the presence of the Wiener measure causes subtle measurability problems and it was not possible simply to adapt the approach from the deterministic setting. Finally, the fact that we are considering equations on manifolds (i.e. non-flat media) urged non-trivial re-derivation of results and properties known from the Euclidean case. We thus believe that the developed methods and achieved results represent essential contribution in the field of stochastic evolution equations (and in particular stochastic scalar conservation laws) on manifolds, but also in the standard Euclidean (flat) setting.