Geometric Partial Differential Equations: Surface and Bulk Processes

The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems. Mathematics Subject Classification (2010): 35-XX, 49-XX, 65-XX. Introduction by the Organisers Geometric partial differential equations are a flourishing research area at the interface between differential geometry and pde theory. These equations arise from a variety of problems, and describe a host of phenomena. One traditional and important source are minimization problems involving geometric objects like curves or surfaces. More recently, pdes on stationary or evolving surfaces possibly coupled to equations that hold in the domain bounded by the surface have emerged as a very active research field. While the the study of geometric partial differential equations leads to interesting and challenging problems both from the analytical and the numerical point of view, they also frequently arise in the modeling of physical interfaces or membranes. For this reason, understanding the mathematical theory behind these equations and constructing efficient algorithms to approximate solutions is of great importance in application areas such as materials science, biology, image processing and astrophysics. 2 Oberwolfach Report 55/2015 The workshop brought together about 50 experts from Europe and America covering a wide variety of aspects bridging analysis, numerical analysis, scientific computation and real-life applications. The scientific programme consisted of 26 plenary talks with the following topics emerging as the main directions of current research: Curvature energies: Pozzi considered the minimisation of the anisotropic Willmore energy via a natural formulation of the corresponding gradient flow, while Bartels suggested and analysed a numerical approach for the minimisation of a bilayer bending energy involving an isometry constraint. Röger introduced a ’mesoscale’ model for lipid bilayer membranes and considered a corresponding macroscopic limit. The talk by Lipowsky was concerned with the stability of certain membrane shapes in a spontaneous curvature model. In their two talks Gräser and Schmeiser discussed mathematical modelling of different aspects of the mutual interaction of cell membranes, membrane proteins and the cytoskeleton, while a hyperboliparabolic model for the polymerization of the filaments in the cytoskeleton was presented by Stevens. Simulations of the wrinkling behavior of thin elastic sheets were presented by Sander who used a Cosserat shell model and geodesic finite elements. PDEs on surfaces and evolving domains: Several talks were concerned with refinements and extensions of methods that have been developed recently. Olshanskii presented new ideas for mesh adaption on the basis of octree background meshes while Reusken introduced a higher order trace finite element method. Giesselmann’s talk focussed on hyperbolic conservation laws on surfaces with particular emphasis on the approximation of geometry. Lubich derived an error analysis for an advection–diffusion equation on an evolving hypersurface whose velocity is given through its coupling to a regularized mean curvature type evolution law. Optimal control problems for PDEs on surfaces and their numerical analysis were the subject of Hinze’s talk. Ranner presented an abstract framework for the analysis of finite element discretization for PDEs in evolving domains. Coupled surface/bulk problems and applications: Several presentations showed how sophisticated numerical methods for surface PDEs can be incorporated into the simulation of complicated coupled surface/bulk problems. Garzon used the level set method in order to treat a fluid/interface problem arising in electrohydrodynamics, while Stricker considered the motion of liquid droplets driven by Marangoni flow. Optimal control of multiphase fluids and droplets described by a phase-field model was presented by Hintermüller. An algorithm for solving coupled surface/bulk reaction diffusion equations arising in a model for cell migration and chemotaxis was presented by Mackenzie. Nürnberg simulated the dynamics of fluidic biomembranes in a bulk fluid through the coupling of bulk and surface Navier–Stokes equations which has in the latter the first variation of the Willmore energy as a forcing term. The accurate discretization of parabolic problems on moving domains with moving interfaces motivated by fluid-structure-interaction problems was discussed by Richter. Geometric Partial Differential Equations: Surface and Bulk Processes 3 While the abovementioned talks were primarily concerned with computational aspects, there were several presentations that focussed an modeling and analytical issues: Garcke suggested a mathematical model for tumour growth including chemotaxis and active transport and which couples the Cahn–Hilliard equation to Darcy flow. Sprekels analized optimal boundary control problems for CahnHilliard systems with dynamic boundary conditions. Stinner presented a model for multiphase flow taking into account the impact of surfactants on the separating interfaces involving certain coupling conditions at triple junctions. Venkataraman analysed various parameter limits for a receptor-ligand dynamics model giving rise to surface free boundary problems. In addition two talks presented interesting new ideas relevant for the efficient simulation of geometric PDEs: Fritz presented a systematic approach to the generation of good surface meshes using reparametrizations based on the harmonic map heat flow. Chopp suggested a locally adaptive time stepping strategy that can be applied in order to speed up the calculation of solutions to the Cahn–Hilliard problem. The organizers had deliberately invited experts covering all aspects of geometric pdes including analysis, numerical analysis, scientific computing, modeling and applications. This led to lively discussions and exchange of ideas during and after the talks which was continued throughout the afternoon breaks. On Tuesday and Wednesday evening two well attended ’Young Researcher sessions’ took place with three presentations in each session. In a more informal ’movie night’ on Thursday evening a host of participants took the opportunity to show impressive videos of numerical simulations related to the subject of the conference. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Moreover, the MFO and the workshop organizers would like to thank the Simons Foundation for supporting Robert Strehl in the “Simons Visiting Professors” program at the MFO. Geometric Partial Differential Equations: Surface and Bulk Processes 5 Workshop: Geometric Partial Differential Equations: Surface and Bulk Processes