The project is split into three work packages:
A: Singularities of solutions (Ehrnström, Grunert, Holden)
Concerned with solutions of nonlocal, nonlinear, qualitatively hyperbolic equations with very weak dispersion, in particular approximations to the Euler equations such as the Burgers–Hilbert, the Burgers–Poisson, the Whitham equations and the Camassa–Holm equation.
B: Nonlocal partial differential equations (Ehrnström, Holden, Jakobsen, Lindqvist)
Focuses on the existence, uniqueness and properties of solutions of a general family of nonlinear and nonlocal partial differential equations. A specific emphasis is put on three classes of problems:
- the Euler equations for steady free-surface water waves
- degenerate parabolic convection-diffusion equations
- elliptic and parabolic fractional \(p\)-Laplace equations
Common to all problems is the challenging combination of singular solutions and nonlinear and nonlocal effects.
C: Uncertainty and partial differential equations (Fjordholm, Holden, Jakobsen)
Focuses on the interplay between uncertainty quantification, stochastics, and partial differential equations. One part is devoted to hyperbolic conservation laws with uncertain or stochastic data, and the other part to boundary value problems for nonlocal partial differential equations that are associated with reflecting jump processes. We will investigate the existence, uniqueness numerical approximation and characterization of the solutions of these new problems.