Oberseminar Analysis Winter 2022/23

Organizers: S. Conti, H. Koch, S. Müller, B. Niethammer, M. Rumpf, C. Thiele, J.J.L. Velázquez, I. Cristian

Previous weeks:

• Thursday, November 03 at 14:15 in Lipschitz Saal
  
  Prof. Dr. Phan Thành Nam (LMU Munich)
  Correlation energy of the electron gas in the mean-field regime
  

  Show Abstract

  For a large fermionic system interacting via Coulomb potential in the high density limit, the correlation energy is expected to be given by the Gell-Mann−Brueckner formula $c_1 \rho \log(\rho)+c_2 \rho$. We will discuss a rigorous upper bound in the mean-field regime. While the analogue of this result for smooth potentials has been well understood via a bosonization method, the analysis for singular potentials requires nontrivial refinements to capture the exchange contribution which is absent in the purely bosonic picture. The talk is based on joint work with Martin Ravn Christiansen and Christian Hainzl.

  
  Download iCal file (.ics) Further Information
  

• Thursday, November 10 at 14:15 in SemR 1.008
  
  Corentin Le Bihan (ENS de Lyon)
  Dynamical correlation of the Gibbs measure of a gas in low density scaling
  

  Show Abstract

  We consider a gas of $N$ particles in a box of dimension, interacting pairwise with a potential $\alpha V(\frac{r}{\varepsilon})$. We want to understand the behavior of the system in the limit $N\rightarrow\infty$, with a suitable scaling for $\alpha$ and $\varepsilon$. If we choose $\varepsilon=1$, $\alpha=\frac{1}{N}$, it is the mean field limit: particles interact weakly at long distance. We are interested in the low density limit $\varepsilon=N^{-\frac{1}{2}}$, $\alpha=1$. Then the distance crossed by a particle is constant. This limit is well understood since the work of Lanford: the empirical law of the system converges in law to solution of the Boltzmann equation. It is a kind of Law of Large Numbers. Sadly this convergence occurs only for short time. In order to go longer, we need look at the fluctuations around the equilibrium, which are supposed to follow a linearized version of Boltzmann equation. The talk will present the idea of the proof of Bodineau, Gallagher, Saint Raymond and Simonella for hard sphere potential and the idea of an improvement in case of real interaction potential.

  
  Download iCal file (.ics) Further Information
  

• Thursday, November 17 at 14:15 in Lipschitz Saal
  
  Prof. Dr. Vesa Julin (University of Jyväskylä)
  Consistency of the flat flow solution to the volume preserving mean curvature flow
  

  Show Abstract

  We consider the flat flow solution to the volume preserving mean curvature flow starting from $C^{1,1}$-regular set and show that it coincides with the classical solution as long as the latter exists. The proof is based on sharp regularity estimates for the minimizing movements scheme that are stable with respect to the time discretization.

  
  Download iCal file (.ics) Further Information
  

• Thursday, November 24 at 14:15 in Lipschitz Saal
  
  Lucas Ertzbischoff (École polytechnique)
  Local well-posedness for thick spray equations
  

  Show Abstract

  We consider a coupled system between kinetic and fluid equations, describing a cloud of particles immersed within a gas. We are interested in the so-called "thick spray" case, where the volume fraction for the particles is not negligible compared to that of the fluid.

  Because of the coupling between both phases, this system displays several losses of derivatives. In particular, and contrary to some other fluid-kinetic models, its rigorous study is almost completely absent.

  Based on ideas used for singular Vlasov equations, I will show how to build a local Sobolev theory for this system and its variants (in the compressible Navier-Stokes case), when the initial data satisfies a Penrose type stability condition.

  This is a joint work with D. Han-Kwan (CNRS and Université de Nantes).

  
  Download iCal file (.ics) Further Information
  

• Thursday, December 08 at 14:15 in Lipschitz Saal
  
  Dr. Jinyeop Lee (LMU Munich)
  Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential
  

  Show Abstract

  We consider a gas of weakly interacting bosons in three dimensions subject to an external potential in the mean field regime. Assuming that the initial state of our system is a product state, we show that in the trace topology of one-body density matrices, the dynamics of the system can be described by the solution to the corresponding Hartree type equation. Using a dispersive estimate for the Hartree type equation, we obtain an error term that is uniform in time. Moreover, the dependence of the error term on the particle number is optimal.

  We also consider a class of intermediate regimes between the mean field regime and the Gross–Pitaevskii regime, where the error term is uniform in time, but not optimal in the number of particles.

  
  Download iCal file (.ics) Further Information
  

• Thursday, December 15 at 14:15 in Lipschitz Saal
  
  Prof. Dr. Ulrich Menne (National Taiwan Normal University)
  Fine connectedness properties of varifolds
  

  Show Abstract

  The notion of indecomposability for varifolds is too restrictive a connectedness property to be always inherited by solutions to geometric variational problems phrased in terms of sets, $G$ chains, and immersions. In joint work with Christian Scharrer, a new weaker notion - indecomposability with respect to locally Lipschitzian real valued functions - has been introduced which is yet strong enough for the subsequent deduction of substantial geometric consequences therefrom.

  
  Download iCal file (.ics) Further Information
  

• Friday, January 13 at 14:15 in SemR 2.040
  
  Prof. Dr. Niels Benedikter (University of Milan)
  Dynamics of Interacting Fermions beyond Hartree-Fock Theory
  

  Show Abstract

  Almost 100 years after the publication of the Schrödinger equation, the quantum many-body problem is still a source of mathematical challenges. In certain scaling limits, the challenges in the computation of physically observable predictions may be overcome by deriving effective evolution equations. I will discuss the coupled mean-field and semiclassical scaling limit for high-density fermionic systems, and sketch the derivation of the time-dependent Hartree-Fock equation as the simplest effective evolution equation. I will then present more recent results based on bosonization that can be seen as a next-order correction to Hartree-Fock theory.

  
  Download iCal file (.ics) Further Information
  

• Thursday, January 19 at 14:15 in Lipschitz Saal
  
  Dr. Nicolas Clozeau (IST Austria)
  Artificial Boundary Conditions for Random Elliptic Systems with Correlated Coefficient Field
  

  Show Abstract

  In this talk, we are interested in numerical algorithms for computing the electrical field generated by a charge distribution localised on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We study an artificial boundary condition recently introduced by Otto, Lu and Wang suggesting optimal Dirichlet boundary conditions motivated by a multipole expansions established by Bella, Giunti and Otto. Previously showed to be optimal in random media having a finite range of dependence, we study here the effect of correlations on the algorithm. The correlations are encoded via a multi-scale spectral gap inequality which allow, for instance, to consider models of Gaussian fields with fat tails or various point processes such as Poisson tessellations and random parking which do not satisfy a finite range of dependence. With overwhelming probability, we obtain a (near) optimal convergence rate with respect to scaling in terms of $l$, $L$ and the size of the correlations.

  Joint work with Lihan Wang and Quinn Winters.

  
  Download iCal file (.ics) Further Information
  

• Thursday, March 16 at 14:15 in Lipschitz Saal
  
  Prof. Dr. Marta Lewicka (University of Pittsburgh)
  The Monge-Ampere system: flexibility in arbitrary dimension and codimension
  

  Show Abstract

  In this talk, we will be concerned with weak solutions obtained by means of convex integration, to the Monge-Ampere system. This system is a natural extension of the Monge-Ampere equation $\det\nabla^2 v=f$ in dimension $d=2$, to the linearization of the full Riemann curvature tensor in arbitrary dimension $d$, and arises in relation to:

  1. the problem of existence of isometric immersions and
  2. the context of nonlinear elasticity.

  Our main technical ingredient consists of the "stage" construction, in which we achieve the Hölder regularity $\mathcal{C}^{1,\alpha}$ of solutions to the aforementioned system, approximating its any subsolution, with exponents $\alpha<\frac{1}{1+ d(d+1)/k}$ where d is an arbitrary dimension and $k\geq 1$ is an arbitrary codimension in the problem.

  When $d=2$ and $k=1$, we recover the previous result by Lewicka-Pakzad for the Monge-Ampere equation. Our construction can be translated to the isometric immersion problem, where for $k=1$ we recover the result by Conti-Delellis-Szekelyhidi, and for large $k$ we quantify the result by Kallen.

  
  Download iCal file (.ics) Further Information