Problems in Nonlinear Partial Differential Equations


November 14-15, 2019, VU University, Amsterdam
in honor of the 60th birthday of Joost Hulshof

[Joost Hulshof]

Titles (and abstracts) (tentative):

  • Sigurd Angenent In 1986 Huisken proved that under Mean Curvature Flow compact convex surfaces shrink to points, becoming asymptotically spherical just before they disappear. In this talk I will discuss what happens to convex Mean Curvature Flows as one follows their history backward in time, and show the underlying dynamical system, its fixed points and connecting orbits.

  • Michiel Bertsch Abstract: The talk will be focussed on the uniqueness issue for discontinuous viscosity solutions of first order Hamilton Jacobi equations in 1 spatial dimension, with Hamiltonians which grow at most linearly with respect to the gradient. In the 80's Ishii introduced semicontinuous viscosity sub- and supersolutions which satisfy a Comparison Theorem. Unfortunately the Comparison Theorem does not imply uniqueness of viscosity solutions if the initial function is not continuous. In this talk I present a first class of equations for which uniqueness of suitably defined discontinuous viscosity solutions can be proved. The proof is based on the "barrier effect" of spatial jump discontinuities. I shall also comment on counterexamples of uniqueness, observed in the 90's by Barles e.a. The talk is based on joint work with F. Smarrazzo, A. Terracina and A. Tesei.

  • Claude-Michel Brauner In combustion theory, the propagation of premixed flames is usually described by conventional thermal-diffusional models with standard Arrhenius kinetics. At the free interface, namely the flame front, the temperature and mass fraction gradients are discontinuous (thin flame). Models describing dynamics of thick flames with stepwise ignition-temperature kinetics have recently received considerable attention. There are differences with the Arrhenius kinetics, for example in the case of zero-order stepwise kinetics there are two free interfaces. The temperature and mass fraction gradients are this time continuous. In a pioneering paper with Hulshof and Lunardi (2000) and related works, we were able to associate the velocity with a combination of spatial derivatives up to the second order. Then, we reformulated the systems as fully nonlinear problems which are very suitable for local existence, stability analysis and numerical simulation as well. It is remarkable than the general method which was developed at first for solving thin flame problems, works perfectly on thick flame models with ignition-temperature kinetics.

  • Frank Bruggeman Earth is teeming with life. Mostly bacterial life. The molecular complexity of these simplest life forms is truly amazing. Different bacterial species exploit highly similar basic molecular mechanisms. They are subject to similar selective pressures. They vary in their capacities to cope with different nutrients and stresses, not in how they work. They all exploit enzyme networks for metabolism, self-repair, stress responses and self-fabrication. Thus, an universal theory about the behaviour of bacteria might exist. It is this theory that Joost has made important contributions too.

  • Arjen Doelman

  • Hans van Duijn We discuss the equations of nonlinear poroelasticity derived from mixture theory. They describe the quasi-static mechanical behaviour of a fluid saturated porous medium. The nonlinearity arises from the compressibility of the fluid and from the dependence of porosity and permeability on the volume strain. We point out some limitations of the model. In our approach we discretize the quasi-static formulation in time and first consider the incremental problem. For this we prove existence using Brezis? theory of pseudo-monotone operators. Generalizing Biot?s free energy to the nonlinear setting we construct a Lyapunov functional. This allows us to obtain bounds that are uniform with respect to the time step, yielding global stability. In the case when dissipative interface effects between fluid and solid are present, we consider the continuous time case in the limit when the time step tends to zero. This yields existence of a weak free energy solution. This is joint work with Andro Mikelic from the Institut Camille Jordan, Université Lyon 1, France.

  • John King Formal-asymptotic analyses of four examples associated with the porous-medium equation will be described.

  • Jan de Munck Trained as a physicist, with great interest in mathematics, I have always been looking for scientific problems where a mathematical modelling approach is a versatile ingredient. Employed at the VU Medical Center, I have found such problems in brain imaging, where a whole range of physical devices is available to measure different aspects of brain activity and structure. To be of clinical use, or to provide physiological insights, mathematical models are required to extract information from raw data. In my talk I will show a few examples focusing on cases where a decisive contribution was given by Joost Hulshof. A year ago, I made a career switch and continued my search for problems that deserve a mathematical modelling approach at PostNL. Indeed, the complex logistics of the daily transportation of millions of parcels from A to B, can be considered as an engine with different independent parts that secrete data providing information about its quality of execution. I will illustrate this idea with an example where we optimized process parameters in the evening distribution and with an open problem, focusing on the optimal collection and sorting of parcels. The talk will end with a ?weak law? and a ?strong law? on mathematical modelling.

  • Bert Peletier Traditionally, Michaelis-Menten type reactions are studied in an in vitro environment, i.e., they are viewed as closed systems. This is how on- and off-rates are usually calculated. However, in modelling pharmacological processes, which involve different enzymatic processes, such binding properties are used in an in vivo environment. In this lecture we discuss how the classical Michaelis-Menten dynamics is modified in the transition from an in vitro to in vivo environment.

  • Mark Peletier In 2014 Georg Prokert, Frieder Lippoth, and I started talking about moving-boundary problems for vesicles in viscous fluids. Indirectly this arose from the earlier work by Martijn Zaal and Joost on the modelling and analysis of osmosis. For the problem studied by Joost and Martijn the modelling setup was relatively clear, including its variational structure as a gradient flow, but we wanted to understand how to generalize the modelling in order to deal with various additional effects, such as viscosity of the solvent, solute diffusion, osmotic pressure differences across the membrane, and fluid transmission through the membrane. In the end we managed to set up a fairly simple modelling framework, based on gradient flows, in which these effects all combined in a natural way, and this was published in 2016. In this talk I want to explain this modelling setup, show how the different elements combine, and show how to derive the resulting differential equations.

  • Georg Prokert We consider a two-phase elliptic-parabolic moving boundary problem modelling an evaporation front in a porous medium. Our main result is a proof of short-time existence and uniqueness of strong solutions to the corresponding nonlinear evolution problem in an L^p -setting. It relies critically on nonstandard optimal regularity results for a linear elliptic- parabolic system with dynamic boundary condition. We identify a nontrivial well-posedness condition that can be interpreted as a ?linear combination? of the corresponding conditions for the Stefan and Hele-Shaw type problems to which the problem formally reduces in the single phases. (Joint work with F. Lippoth, Hannover)