Tag - Applied mathematics

Jincheng Yang: Layer separation and energy dissipation for 3D NSE at high Reynolds number

7th May, 2024

In this talk, we consider the 3D incompressible Navier-Stokes equation in a bounded domain, with a canonical example of Poiseuille flow in mind. We provide an unconditional upper bound for the boundary layer separation and energy dissipation of Leray-Hopf weak solutions, uniformly in high Reynolds numbers. We estimate layer separation by measuring the energy norm of the discrepancy between a (turbulent) low-viscosity Leray-Hopf solution and a fixed (laminar) regular Euler solution with similar initial conditions and body force. This is accomplished by a new nonlinear boundary vorticity estimate.

Maurizio Grasselli: Non-local Navier-Stokes-Cahn-Hilliard systems with unmatched densities

16th April, 2024

H. Abels, H. Garcke, and G. Grün (2012) proposed a diffuse interface model to describe liquid-liquid phase separation in incompressible binary fluids of different densities. This model consists of the Navier-Stokes system which is non-linearly coupled with an advective Cahn-Hilliard equation. In this talk, however, instead of taking the usual free energy functional, we consider a non-local version. Therefore, the resulting Cahn-Hilliard equation is a second-order (spatially) non-local equation. This system was already analysed by S. Frigeri (2016) who established the existence of a global weak solution. I intend to present some further results in dimension two which I have obtained jointly with C.G. Gal, A. Giorgini, and A. Poiatti (2023). These results are mainly concerned with strong solutions, uniqueness, and convergence to a single equilibrium. Some related open issues will also be discussed.

Matteo Fornoni: Optimal control for non-local Cahn-Hilliard tumour growth models

2nd April, 2024

We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in the presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. The optimal control problem aims to identify a suitable therapy, capable of guiding the evolution of the tumour towards a predefined target. We first establish novel regularity results for the PDE system by applying maximal regularity theory in weighted Lp spaces. Such a technique enables us to prove the local existence and uniqueness of a regular solution in a quite general framework, which also includes chemotaxis effects to some extent, but restricts us to regular double-well potentials. Then, by leveraging the time-regularisation properties of the weighted spaces and some global boundedness estimates, we further extend the solution to a global one. In a second version of the model, we add a viscous regularisation term which allows us to prove the existence and uniqueness of a global regular solution under more general hypotheses. Indeed, we can also include singular double-well potentials and cross-diffusive chemotactic effects, at the expense of some additional hypotheses on the controls. These results provide the foundation for addressing the optimal control problem in both cases. Specifically, we prove the existence of an optimal therapy and then, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. We finally discuss some open questions and future research directions.

Hendrik Weber: Noise, differential equations and quantum fields

27th March, 2024

Stochastic Analysis is concerned with solving differential equations in the presence of highly irregular random noise terms. The field has evolved from the foundational works by Itô in the 1940s and its method are used today in numerous modelling contexts. In the first half of this talk I will present my personal take on some of this history and some of the key ideas used. In the second half, I will discuss exciting developments of the last 10 years that show how methods developed for stochastic differential equations allow to give a new perspective on the classical problem to rigorously construct quantum fields.

Katharina Schratz: Resonances as a computational tool

27th March, 2024

A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying non-linear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.

Joaquim Serra: Stable phase transitions: open questions and new results

26th March, 2024

Surface tension and similar forces lead to area-minimizing interfaces in some physical phenomena, observable at macroscopic scales. However, this principle of surface area minimization does not uniformly apply across all scales, as the underlying physical energies often vary with scale. For example, describing a soap film as an area-minimizing surface becomes implausible at scales comparable to 5 nanometers, the size of a soap molecule. Similarly, the Allen-Cahn energy (i.e., scalar Ginzburg-Landau) exhibits scale-dependent behavior that mirrors area minimization only at larger scales. The regularity theory for absolute energy-minimizing minimal surfaces has been successfully extended to several scale-dependent models, including Allen-Cahn. Yet, extending these results to all stable configurations, which represent the states observable in nature, poses significant challenges. In the talk, I will discuss the pressing open questions and the latest findings regarding stable phase transitions in 3-dimensional environments.

Jason Matthiopoulos: Space use and time lags in rewilding applications

25th March, 2024

Two key questions in biodiversity conservation relate to how we manage land/sea-scapes for wildlife and at what point in time we can expect to observe the consequences of our actions. Mathematically, these questions relate to (multiple) steady states and transient dynamics. I will consider two theoretical and inferential examples under each of these questions. I will first examine how the use of suburban and agroforestry ecosystems can be optimised for biodiversity gains. I will then explore frameworks for the analysis of delayed impacts in large-scale bird community data and very fine scale grazer rewilding experiments. The convergence of ideas from dynamical systems and modern statistical inference is a very exciting development in quantitative ecology. Thorny rewilding applications are one of the top areas where such theoretical advances can gain real traction and produce tangible impact.