Partial differential equations arise as key mathematical models in almost all branches of science, e.g. physics, chemistry, biology, geology, medicine and economics. Furthermore, such models are used to simulate industrial processes ranging from oil recovery to materials science at the nano scale. A unique feature of the research group in partial differential equations at CMA is the strong interrelation between analysis of the continuous models and the development of numerical methods. The research activity within the group can basically be divided into two subfields, (i) analysis of nonlinear hyperbolic conservation laws and related problems with applications to fluid mechanics and Hamilton Jacobi equations, and (ii) the construction and analysis of compatible discretization techniques. According to the recent evaluation of Norwegian mathematics, parts of this activity are at the forefront of current international development. In addition to the mathematical research, a number of more applied projects have been carried out during the CMA period in areas like oil recovery, astrophysics, computational quantum mechanics and so on. Furthermore, more recent joint projects with the Faculty of Medicine have been initiated.
The PDE research will benefit from more cooperation with other theoretical groups at MI. The properties of many PDE problems, and their discretization, strongly reflect geometrical, algebraic, and topological structures leading to research problems at the interface between PDEs and other branches of mathematics. Therefore, joint research activities across traditional discipline boundaries will be necessary in order to continue to be at the international forefront in theoretical mathematical research.
Also, computational methods should be integrated into any modern research activity in partial differential equations. Therefore, the strong link to the activity in computational mathematics will be maintained, and if possible, the connections to the activity at Simula Research on software development will be strengthened. The group should also make an effort to strengthen its applied profile. Many Norwegian research groups both within and outside the UiO could potentially benefit from more cooperation with the PDE group. In general, PDE research motivated from the life sciences is seen as an area of growing importance, with close ties to stochastic analysis and statistics, and of course to the life sciences themselves. There is also a large potential for closer cooperation between the PDE group and the fluid mechanics group at MI. Traditionally, the research activity in fluid mechanics at UiO has been more engineering-oriented. However, the two groups have a common interest in computational methods for fluid problems, and the research in both groups would benefit if more joint activities took place.