Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation

Hele-Shaw and Stokes flow with a source or sink : stability of spherical solutions

Qualitative aspects of mathematical models for the dynamics of liquids with a moving boundary are studied. These models describe for instance groundwater flow, extraction of oil, the growth of tumours and viscous sintering in the production of glass. Stability of radially symmetric solutions and decay properties of perturbations are studied for the case that in a single point fluid is injected or extracted. For the motion of the moving boundary a nonlinear non-local evolution equation is derived. The domain is rescaled in such a way that the spherical solution is represented by a stationary solution. Because of this rescaling, the evolution operator is time dependent. The nonlinear stability results are based on linearisation, energy estimates and the principle of linearised stability. The Hele-Shaw model is studied for several boundary conditions, describing various physical situations. In the case of zero pressure on the boundary, it is proved for the injection problem that balls around the injection point are asymptotically stable with respect to small star-shaped perturbations. If surface tension regularisation is included, then balls are stable even for the case of suction under additional assumptions on the initial geometry, suction speed and dimension. Moreover, perturbations turn out to decay algebraically fast. For two dimensional suction, the influence of surface tension dominates the influence of the sink for large time. As a consequence, no condition on the suction speed is necessary. In contrast to the two dimensional problem there is a bound on the suction speed for the 3D problem. In dimensions higher or equal to four the influence of the sink dominates the influence of surface tension. This leads to linear instability for the spherical solution for any suction speed. Making use of the autonomous character of the evolution equation, existence of nontrivial self-similarly vanishing solutions to the three dimensional suction problem with surface tension is proved. These solutions are found as bifurcation solutions from the trivial spherical solution. The suction speed plays the role of bifurcation parameter. Moreover, one branch of bifurcation solutions turns out to be stable with respect to a certain class of perturbations. For the closely related Stokes flow stability of the spherical solution in the case of injection has been proved for dimensions two and three. For the suction problem for these dimensions the spherical solution is linearly unstable