The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Chris Henderson.
Anisotropic surface energies are a natural generalization of the perimeter that arise in models for equilibrium shapes of crystals. We discuss some recent results for anisotropic isoperimetric problems concerning the strong quantitative stability of minimizers, bubbling phenomena for critical points, and a weak Alexandrov theorem for non-smooth anisotropies. Part of this talk is based on joint work with Delgadino, Maggi, and Mihaila.
Tumor growth modeling is the investigation of the complex dynamics of cancer progression using a mathematical formulation. Internal dynamics of tumor cells, their interactions with each other and with their surrounding tissue, transfer of chemical substances and many phenomena are typically encoded in mathematical models. This mathematical formulation relies on biological and clinical observations. At the same time, it is currently clear that mathematics could make a huge contribution to many areas of experimental cancer investigation since there is now a wealth of experimental data which requires systematic analysis.
In this research project we investigate the evolution of cancerous cells relying on a non-linear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the role of nutrient and drug application in the evolution of cancerous cells. We construct a convergent numerical scheme to approximate solutions of the nonlinear system by employing compactness methods in the spirit of P.L. Lions. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The talk will present results obtained in collaboration with D. Donatelli and F. Weber.
For 2D Euler flows, it is known that the L^\infty norm of the gradient of vorticity can grow at most double exponentially in time. In recent years, this bound has been proven to be sharp by Kiselev and Sverak on the unit disc. We will examine the possibility of growth in the 3D axisymmetric setting for flows without swirl component.
I will present results of accelerated spreading in reaction-transport equations which combine a linear kinetic equation (free transport & scattering) with a nonlinear logistic growth term. The diffusive limit of this equation is the celebrated Fisher's reaction-diffusion equation which exhibits spreading at finite speed. A similar statement holds true when the stationary velocity distribution is compactly supported. However, acceleration occurs if the latter is not compactly supported. Weak bounds are derived in the case of a Gaussian velocity distribution. More precise asymptotics can be captured using an analog of the approximation of geometric optics that yields a nonlocal Hamilton-Jacobi equation.
This is a joint work with Emeric Bouin, Emmanuel Grenier and Grégoire Nadin.
In this talk I will present some recent results concerning the analysis of the level set formulation of the crystalline mean curvature flow. The crystalline mean curvature, understood as the first variation of an anisotropic surface energy with an anisotropy whose Wulff shape is a polytope, is a singular quantity, nonlocal on the flat parts of the crystal surface. Therefore the level set equation is not a usual PDE and does not fit into the classical viscosity solution framework. Its well-posedness in dimensions higher than two was an open problem until very recently. In a joint work with Yoshikazu Giga (U. of Tokyo), we introduce a new notion of viscosity solutions for this problem and establish its well-posedness for compact crystals in an arbitrary dimension, including a comparison principle and the stability with respect to approximation by a smooth anisotropic mean curvature flow.