- Titel
- On the influence of quenched randomness on the large-scale behavior in drift-diffusion equations and variational interface models
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-965513
- Datum der Einreichung
- 02.10.2024
- Datum der Verteidigung
- 25.03.2025
- Abstract (EN)
- In this thesis we study the influence of quenched randomness on the large-scale behavior in three mathematical models. First, we study a drift-diffusion process with a random drift that is assumed to be divergence-free. Moreover, we consider two variational interface models. Motivated by recent progress on the two-dimensional Random Field Ising Model, we study a perimeter functional perturbed by an external random field. Moreover, we analyze the limiting behavior of heterogeneous Allen-Cahn functionals with random coefficients. Chapter 2 is concerned with the large-time behavior of the aforementioned drift-diffusion process, which we study in the critical situation of a white noise-type drift in two space dimensions. Upon taking thermal averages, we show that the resulting process displays intermittent behavior. More specifically, we study the thermally averaged Jacobian matrix with respect to the starting point, which we like to understand as a measure of sensitivity for the dependence on the initial data. We show that normalized second moments of this process are not equi-integrable. The result is obtained through the analysis of an approximation that implements the idea of a scale-by-scale homogenization technique. This approximation shows intermittent behavior, which can be monitored through its second and fourth moment. In a weak form, this intermittency is transferred to the original process. Next, we consider the first interface model in Chapter 3, where we study the influence of a small-amplitude random field on an isotropic perimeter functional. Again, we consider a critical setting with a white noise-type field in two space dimensions. In the presence of an ultra-violet regularization, we establish a large-scale regularity theory for local minimizers. More specifically, we show that the averaged normal of any minimizing surface admits a well-controlled modulus of continuity. This shows that typically minimizing surfaces are flat on exponentially large length scales when the amplitude of the noise vanishes. Lastly, in Chapter 4, we consider an Allen-Cahn model for diffuse interfaces of width epsilon with microscopic heterogeneities on some length scale delta. We distinguish two regimes. In what we call the homogenization regime, we establish a criterion on the relation between the scales epsilon and delta that ensures that in the joint limit epsilon, delta to zero the homogenization limit delta to zero happens before the sharp-interface limit epsilon to zero. In the so-called rare events regime, we show the sharpness of the previously mentioned criterion in case of an one-dimensional random checkerboard by determining the extremal behavior and the transition layer towards the homogenization regime.
- Freie Schlagwörter (EN)
- quenched randomness, critical drift-diffusion equation, Random Field Ising Model, Allen-Cahn functional
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- angenommene Version / Postprint / Autorenversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-965513
- Veröffentlichungsdatum Qucosa
- 08.04.2025
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Deutsch
- Lizenz / Rechtehinweis
CC BY 4.0