**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 Cornelia Mihaila.

Logarithmic correction in multi-dimensional periodic Fisher-KPP equations

Fisher-KPP equations are a type of reaction-diffusion equations that model population dynamics. Their behavior is characterized by the invasion of an unstable state by a stable one, which leads to the phenomenon of spreading. In one dimension, it has been shown that localized data give rise to solutions that lag behind the slowest traveling front by a logarithmic term. In this talk, we will discuss the analogous periodic problem in R^n, where we determine the asymptotic rate of spreading of the solutions and the logarithmic correction along every unit direction.

Axial Symmetry for Fractional Capillarity Droplets

A classical result of Wente, motivated by the study of sessile capillarity droplets, shows the axial symmetry of every hypersurface which meets a hyperplane at a constant angle and has mean curvature depending only on the distance from that hyperplane. We will prove an analogous result for the fractional mean curvature operator.

Recent Advances in Statistical Inference for Stochastic PDEs

We consider a parameter estimation problem for finding the drift and volatility coefficient for a large class of parabolic Stochastic PDEs driven by space-time noise (white in time, and possible colored in space). In the first part of the talk, we focus on spectral approach and derive several classes of estimators based on the first N Fourier modes of a sample path observed continuously on a finite time interval. We will briefly review the maximum likelihood estimators, and then discuss a new class of estimators for the drift parameter, called the Trajectory Fitting Estimator (TFE). The TFE can be viewed as an analog to the least square estimator from the time series analysis. We will discuss consistency and asymptotic normality of such estimators as number of the Fourier modes tends to infinity. Some nonlinear SPDEs will be discussed too. In the second part of the talk we will discuss the parameter estimation problems for discretely sampled SPDEs. We will present some general results on derivation of consistent and asymptotically normal estimators based on computation of the p-variations of stochastic processes and their smooth perturbations, that consequently are conveniently applied to SPDEs. Both the drift and the volatility coefficients are estimated using two sampling schemes - observing the solution at a fixed time and on a discrete spatial grid, and at a fixed space point and at discrete time instances of a finite interval. The theoretical results will be illustrated via numerical examples.

Dynamics of living tissues and Hele-Shaw limit

Tissue growth, as in solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics. Several levels of mathematical descriptions are commonly used, including possibly elastic behaviours, visco-elastic laws, nutrients, active movement, surrounding tissue, vasculature remodeling and several other features. We will focuss on the links between two types of mathematical models. The `compressible' description describes the cell population density and a more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form. Including additional features also opens other questions as circumstances in which instabilities may develop.

Singular control, HJB and MFG equations

TBA

Description:An embedded corrector problem for homogenization of elliptic equations

In this work, we present new alternatives for the approximation of the homogenized matrix for diffusion problems with highly-oscillatory coefficients. These different approximations all rely on the use of an "embedded corrector problem", where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients are constant. The motivation for considering such embedded corrector problems is the fact that, for some particular and practically relevant heterogeneous materials (e.g. piecewise constant coefficients), they can be very efficiently solved. Several strategies are presented to appropriately determine the outside constant coefficients. We then prove that the different approximations we introduce all converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity. Some works in progress will also be mentioned. Joint works with E. Cancès, V. Ehrlacher, B. Stamm and S. Xiang. Organizer: CAMP/Nonlinear PDE

Some existence results for damped compressible radiation hydrodynamics

Radiation hydrodynamics is a widely used model in astrophysics simulations. In many situation, a diffusion approximation of the radiative part is sufficient to decribe correctly phenomena of interest. In this talk, I will review some existence results on such systems, with an additional damping term in the momentum equation and small data. It is based on the Kawashima-Shizuta method for hyperbolic-diffusion systems. Similar results are also proved for models including magnetic effects. Another widely used model is the compressible Euler model coupled to Poisson equation. Here, extending a method due to Grassin-Serre, we also prove global existence for this system, we prove existence of a global solution for small data. These are joint works with B. Ducomet (Univ. Créteil), R. Danchin (Univ. Créteil) and S. Necasova (Acad. Sc. Czech Republic, Praha)

Molecular simulation and mathematics

By modeling matter at the atomistic level, molecular simulations intend to provide a numerical microscope, able to study the microscopic origines of macroscopic properties of matter. Applications are numerous: protein structure prediction, drug design, defects propagation in crystals, etc. Molecular simulation is now a cornerstone in many scientific domains (biology, chemistry, physics). Despite the growth of computer processing power, it remains difficult to simulate sufficiently large systems over sufficiently long timescales to access all the quantities of interest. Mathematics play a fundamental role both to derive rigorously cheaper coarse-grained models, and to analyze and improve algorithms which aim at bridging the gap between the length and and time scales of molecular simulation and those relevant at the macroscopic level. The objective of this talk will be to present models used in classical molecular dynamics, and some mathematical questions raised by their simulations. Reference: T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica, 25, 681-880, (2016).

Metastability: a journey from stochastic processes to semiclassical analysis

We consider the exit event from a metastable state for the overdamped Langevin dynamics $dX_t = - \nabla f(X_t) dt + \sqrt{h} dB_t$. Using tools from semiclassical analysis, we prove that, starting from the quasi stationary distribution within the state, the exit event can be modeled using a jump Markov process parametrized with the Eyring-Kramers formula, in the small temperature regime $h \to 0$. We provide in particular sharp asymptotic estimates on the exit distribution which demonstrate the importance of the prefactors in the Eyring-Kramers formula. Numerical experiments indicate that the geometric assumptions we need to perform our analysis are likely to be necessary. These results also hold starting from deterministic initial conditions within the well which are sufficiently low in energy. From a modelling viewpoint, this gives a rigorous justification of the transition state theory and the Eyring-Kramers formula, which are used to relate the overdamped Langevin dynamics (a continuous state space Markov dynamics) to kinetic Monte Carlo or Markov state models (discrete state space Markov dynamics). From a theoretical viewpoint, our analysis paves a new route to study the exit event from a metastable state for a stochastic process. Reference: G. Di Gesù, D. Le Peutrec, T. Lelièvre and B. Nectoux, Sharp asymptotics of the first exit point density, https://arxiv.org/abs/1706.08728, to appear in Annals of PDE.

Computing the properties of multilayer 2D materials: a method based on non-commutative geometry

2D materials such as graphene have fascinating electronic and optical properties. Multilayer 2D materials are obtained by stacking several layers of possibly different 2D materials. Their study is one of the current hot topics in physics and materials science. The numerical simulation of such systems is made difficult by incommensurabilities originating from lattice mismatches and twisting angles. In this talk, I will first present the most common framework, based on Kubo formula, for deriving the frequency-dependent electrical conductivity tensor of a given material from its molecular structure. For periodic systems (perfect crystals), Bloch theory allows one to numerically compute the conductivity from Kubo formula in an efficient way. The situation is much more involved for aperiodic systems such as incommensurate multilayer 2D materials. However, it can be handled by relying on tools from non-commutative geometry introduced in the 80's and 90's by Jean Bellissard and co-workers following ideas of Alain Connes.

Mathematical theory and computational approaches for modern materials science

The talk, intended for a general audience, will survey some challenging mathematical and numerical problems in contemporary materials science. Questions such as the passage from the microscale to the macroscale, the insertion of uncertainties, defects and heterogeneities in the models, will be examined. We will discuss the interesting issues raised for mathematical analysis (theory of partial differential equations, homogenization theory) and for numerical analysis (finite element methods, Monte Carlo methods, etc).

March 19

Regularizing effects for HJ equations

TBA

Homogenization of MFG Problems

TBA

For questions, contact Cornelia Mihaila at: cmihaila [at] math [dot] uchicago [dot] edu.