Projects

Motivation

Many gases we encounter in nature and industry are not single substances but mixtures of multiple interacting components. Examples include:

• Atmospheric air, composed of nitrogen, oxygen, carbon dioxide, and trace gases.

• Combustion processes, where fuel, oxygen, and exhaust gases interact dynamically.

• Plasma and semiconductor gases, where precise control of diffusion and heat transfer is essential.

Understanding such systems is key for applications ranging from climate and environmental modeling to industrial processes and energy technologies.

Challenges

Modeling multicomponent gases is not straightforward. Different species interact through diffusion, friction, and heat transfer, and their collective behavior must respect the fundamental principles of mechanics and thermodynamics (such as conservation of mass, momentum, energy, and entropy production). Capturing these interactions requires systems of nonlinear partial differential equations (PDEs). However, modern challenges arise in ensuring that such PDE systems are:

• Thermodynamically consistent, meaning they align with physical laws.

• Mathematically well-posed, meaning they admit solutions that are unique and physically meaningful.

• Reliable for long-term prediction, which requires understanding their asymptotic behavior.

Goals

Before applying these models to computations and simulations, it is essential to ensure that the theoretical foundations are sound. My PhD research focused on:

• Using relaxation and asymptotic techniques to derive simplified models and analyzing these models to make sure they are thermodynamically consistent.

• Proving global-in-time existence  of weak solutions and weak-strong uniqueness to the non-isothermal Maxwell-Stefan system and uniqueness of weak solutions to the isothermal Maxwell-Stefan system.

• Showing that the system converges to an equilibriums state as time passes.

• Showing that weak solutions to the Maxwell-Stefan system are regular enough to induce a non-chaotic flow.

Outcomes

This work provided new insights into the mathematical structure of multicomponent gas mixtures. It connected thermodynamic principles with rigorous PDE analysis, laying a foundation for future computational and applied investigations.

The project resulted in the following publications:

1. Georgiadis, S., Tzavaras, A.E. Asymptotic derivation of multicomponent compressible flows with heat conduction and mass diffusion, ESAIM: Math. Model. Numer. Anal. 57 (2023), 69–106.  (see  https://doi.org/10.1051/m2an/2022065 or arXiv[2112.13625])

2. Georgiadis, S., Tzavaras, A.E. Alignment via friction for nonisothermal multicomponent fluid systems, Acta Appl. Math. 191, 3 (2024). (see https://doi.org/10.1007/s10440-024-00655-0 or arXiv[2311.10546])

3. Georgiadis, S., Jüngel, A. Global existence of weak solutions and weak-strong uniqueness for nonisothermal Maxwell-Stefan systems, Nonlinearity 37, 075016 (2024). (see DOI: 10.1088/1361-6544/ad4c49 or arXiv[2303.17693])

4. Georgiadis, S. , Jüngel, A., Tzavaras, A.E. Non-isothermal multicomponent flows with mass diffusion and heat conduction, In: Parés, C., Castro, M.J., Morales de Luna, T., Muñoz-Ruiz, M.L. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Volume I. HYP 2022. SEMA SIMAI Springer Series, vol 34. Springer, Cham. (see https://doi.org/10.1007/978-3-031-55260-1_19 or arXiv[2301.08928])

5. Berselli, L.C., Georgiadis, S., Tzavaras, A.E. Absence of anomalous dissipation for weak solutions of the Maxwell-Stefan system. Nonlinearity 38, 025018 (2025). (see https://iopscience.iop.org/article/10.1088/1361-6544/ada7b8 or arXiv[2407.10134])

6. Georgiadis, S., Kim, H., Tzavaras, A.E. Renormalized solutions for the Maxwell-Stefan system with an application to uniqueness of weak solutions. (see arXiv[2311.10465])

Motivation

A central principle of fluid mechanics and thermodynamics is that physical systems conserve energy or satisfy an entropy balance. However, when working with weak solutions to nonlinear PDEs such as the Euler or Navier–Stokes equations, it is not always guaranteed that these conservation laws hold. Determining when weak solutions respect such identities is essential for distinguishing between physically relevant flows and those that may exhibit anomalous dissipation, known as turbulence (chaotic) flow.

Challenges

The main difficulty lies in the fact that weak solutions generally lack the regularity needed to justify classical energy or entropy balances. Small-scale irregularities may produce artificial energy loss, and existing results often required restrictive assumptions (for example, on the gradients of solutions) that limit their applicability. Extending conservation criteria to broader function spaces, and to more complex systems beyond the Euler/Navier–Stokes framework, remains a major challenge.

Goals

• To identify sharp regularity thresholds under which weak solutions conserve energy or entropy.

• To extend classical results (such as Onsager-type conditions for Euler flows) to wider ranges of space–time regularity.

• To apply these methods to more complex systems, such as Navier–Stokes–Cahn–Hilliard models and multicomponent diffusion Maxwell–Stefan systems.

• To clarify the connection between mathematical weak solutions and their physical admissibility.

Outcomes

This work provided a unified perspective on when weak solutions can be trusted to preserve fundamental thermodynamic structures, bridging mathematical analysis with the physical interpretation of turbulence and diffusion phenomena. To this degree:

• We proved new energy conservation criteria for incompressible Euler flows, revealing a compensation mechanism between space and time regularity. This clarified when limits of Navier–Stokes solutions (when viscosity vanishes) coverge to energy-conserving Euler solutions.

• For the Navier–Stokes–Cahn–Hilliard system with non-constant mobility, we established improved conditions for energy equality, replacing gradient assumptions with fractional regularity of the solutions themselves.

• We showed that weak solutions of the Maxwell–Stefan system satisfy an entropy identity without additional regularity, proving the absence of anomalous dissipation in this context.

The project resulted in the following publications:

1. Berselli, L.C., Georgiadis, S. Three results on the energy conservation for the 3D Euler equations, Nonlinear Differ. Equ. Appl. 31, 33 (2024). (see https://doi.org/10.1007/s00030-024-00924-9 or arXiv[2307.04410])

2. Georgiadis, S. Energy identity for the incompressible Cahn-Hilliard/Navier-Stokes system with non–degenerate mobility. Z. Angew. Math. Phys. 75, 174 (2024). (see https://doi.org/10.1007/s00033-024-02312-w or arXiv[2408.01749])

3. Berselli, L.C., Georgiadis, S., Tzavaras, A.E. Absence of anomalous dissipation for weak solutions of the Maxwell-Stefan system. Nonlinearity 38, 025018 (2025). (see https://iopscience.iop.org/article/10.1088/1361-6544/ada7b8 or arXiv[2407.10134])

Motivation

Many fluid systems exhibit phenomena that cannot be captured by classical diffusion models. In particular, capillarity effects — arising from surface tension and interfacial energies — play a fundamental role in multiphase flows, liquid-vapor transitions, and thin-film dynamics. These effects introduce nonlinear stresses depending on spatial variations of density, leading to higher-order diffusion terms. Such models also connect naturally with concepts from statistical mechanics, particle systems, and optimal transport, linking fluid dynamics with deep structures from analysis and geometry.

Challenges

Mathematically, these systems involve fourth-order nonlinear PDEs with degeneracies, which makes their analysis highly nontrivial. The main difficulties include:

• Ensuring existence of weak solutions for all time horizons, despite strong nonlinearities.

• Understanding regularity and vacuum formation, since solutions may vanish or concentrate, leading to singular behaviors.

• Identifying the right energy and entropy structure that guarantee dissipation and allow rigorous compactness arguments

• Connecting different regimes, such as porous-media equations, thin-film models, and quantum drift-diffusion, within a unified  analytical framework.

Goals

The central objective of this project was to establish a theoretical foundation for degenerate capillarity-driven PDE systems by:

• Studying generalized models where the capillarity coefficient depends on the density, covering a wide class of diffusion and thin-film type equations.

• Proving the global-in-time existence of weak solutions under broad assumptions on the model parameters.

• Developing new entropy methods to obtain uniform a priori estimates and control higher-order terms.

• Exploring the variational structure of these systems, connecting them with gradient flows in Wasserstein space and optimal transport.

Outcomes

This research extended the range of parameters for which global weak solutions exist, resolving open cases and confirming conjectures from earlier work. The results highlight the rich interplay between nonlinear PDE analysis, entropy methods, and geometric variational principles, and they provide a rigorous mathematical framework for studying capillarity-driven fluids and degenerate diffusion. These insights pave the way for future studies on stability, uniqueness, and computational methods, with potential applications in fluid dynamics, materials science, and biological aggregation models.

The project resulted in the following publication:

1. Coming soon...

Motivation

I enjoy experimenting with machine learning on publicly available datasets. These mini-projects focus on analyzing and visualizing data rather than complex model development, serving as a practical playground for learning and exploring applied computational techniques. They're probably nothing special for the experts, but I need to start from somewhere, don't I? :)

CO2 Emissions

I trained a Multiple Linear Regression model to predict the CO2 emissions of cars, based on features such as engine size, number of cylinder and fuel consumption. The dataset was obtained from the Government of Canada.

This project serves as an introductory exploration of machine learning for environmental data, highlighting how statistical models can extract interpretable relations between vehicle characteristics and their emissions.

See the notebook here.

China's GDP

I trained two Nonlinear Regression models to capture and predict the trajectory of China's GDP growth over time. The dataset was obtained from the Federal Reserve Bank of St. Louis.

The goal of this project was to explore how simple yet nonlinear machine learning models can describe macroeconomic growth patterns, providing a hands-on example of predictive modeling with real-world economic data.

See the notebook here.

Breast cancer classification

I trained a Logistic Regression model to classify whether a breast tumor is benign or malignant (cancerous), based on features extracted from digitized medical images. The dataset was obtained from the UC Irvine Machine Learning Repository.

This project illustrates how machine learning methods can support medical diagnostics by detecting patterns in biomedical data. While the model itself is simple, it demonstrates the potential of statistical learning in healthcare applications.

See the notebook here.

Drug prescription

I trained a Decision Tree model to classify which treatment option (Drug A, B, C, X or Y) a patient with a certain medical condition should be assigned, based on features such as blood pressure, cholesterol level, and other health indicators. The dataset was obtained from the IBM website.

This project demonstrates how machine learning can be applied in clinical decision-making, offering interpretable models that capture relationships between patient characteristics and treatment recommendations.

See the notebook here.