Oberseminar: Probability and variational methods in PDEs

Wintersemester 2021/22

Time: Thursday, 16:15-17:15
Room: Zoom (Meeting ID: 978 5204 7257, Passcode: 939484)

Speakers

• 09.12.2021: Daniel Heydecker (Max Planck Institute for Mathematics in the Sciences, Leipzig)

  Title: Large Deviations of Kac’s Random Walk
  Abstract: Kac introduced a many-particle Markov process, corresponding to the spatially homogeneous Boltzmann equation, and we consider the dynamical large deviations in the limit $N\to\infty$. With the expected rate function, the large deviations lower bound is only true on a restricted class of paths, and we find counterexamples to a global lower bound related to Lu and Wennberg’s energy non-conserving solutions to the Boltzmann equation. On the other hand, the class of paths where we have a matching upper and lower bounds is sufficiently rich to rederive the celebrated Boltzmann H-Theorem.

• 11.01.2022: Angkana Rüland (Heidelberg University)
  

  Title: On Scaling Laws for Some Matrix-Valued m-Well Problems
  Abstract: Highly non-(quasi)-convex, matrix-valued differential inclusions arise in numerous physical applications. One such example is the modelling of shape-memory alloys. In these settings, often the exact differential inclusions display a striking dichotomy between rigidity and flexibility in that

  • solutions of sufficiently high regularity obey the "characteristic equations" determined by the differential inclusion, the solutions are rigid in this sense,
  • while low regularity solutions are highly non-unique and hence extremely flexible.
  In order to investigate this dichotomy further, in this talk, I explore the effects of regularizations in the form of singular perturbation problems with vanishing regularization strength for these differential inclusions. Motivated by applications in shape-memory alloys, I discuss the role of scaling properties in the singular perturbation strength. In particular, I study two prototypical types of differential inclusions and illustrate how the structure of the underlying convex hulls results in the presented scaling laws. This is based on joint work with Jamie Taylor, Antonio Tribuzio, Christian Zillinger and Barbara Zwicknagl.

• 24.02.2022: Georgiana Chatzigeorgiou (Max Planck Institute for Mathematics in the Sciences, Leipzig)

  Title: On the theory of fluctuations in stochastic homogenization
  Abstract: This talk focuses on stochastic homogenization for linear elliptic equations in divergence form. In particular, we study weakly correlated Gaussian environments and emphasize on the recently developed theory of fluctuations. More precisely, it has been observed that the fluctuations of averages of the solution are captured by the so-called standard homogenization commutator Ξ, an object given in terms of the homogenization correctors. This suggests that a more delicate analysis of Ξ is needed. Our aim is to explain how Ξ decorrelates on large scales when it is averaged on balls which are far enough. We give a quantitative characterization of this decorrelation in terms of both the macroscopic scale and the distance between the balls showing that Ξ inherits the locality properties of the environment.

• 07.03.2022 (14:15-15:15): Florian Kunick (Max Planck Institute for Mathematics in the Sciences, Leipzig)
  

  Title: Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise
  Abstract: It is known that the thin-film equation has a gradient flow structure with respect to a (generalized) Wasserstein metric and the usual Dirichlet energy. Based on that, the fluctuation-dissipation theorem gives rise to a stochastic thin-film equation. This equation is a singular stochastic partial differential equation which is out of scope of the framework of regularity structures for now. In order to circumvent this issue, we discretize the gradient flow structure and rediscover a well-known discretization of the (deterministic) thin-film equation that preserves the so-called entropy estimate. In the stochastic setting this entropy estimate then yields positivity for the solution in the case that the mobility arises from the no-slip boundary condition. Moreover, we show that previous discretizations of the stochastic thin-film equation considered in the literature do not preserve positivity. This is joint work with Benjamin Gess, Rishabh Gvalani and Felix Otto.

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