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Minimal solutions to nonlinear systems of PDEs

Projektbeschreibung

Minimale Lösungen für die Lösung nichtlinearer partieller Differentialgleichungen

Das über die Marie-Skłodowska-Curie-Maßnahmen finanzierte Projekt MinSol-PDEs wird eine systematische Untersuchung minimaler Lösungen für eine große Klasse nichtlinearer partieller Differentialgleichungen durchführen. Ein Teil der Forschungsarbeiten wird sich mit Phasenübergangsproblemen befassen, die durch die Allen-Cahn-Gleichung beschrieben werden. Die Hauptidee besteht darin, die Gleichung auf ein Hamilton-System zu reduzieren, um neue Klassen von Minimallösungen zu konstruieren und die Bedingungen zu verstehen, die eine Variablenreduktion beinhalten. Ein weiterer Teil der Forschung wird sich auf die Painlevé-Gleichung konzentrieren, die in so unterschiedlichen Bereichen wie Zufallsmatrizen, integrierbaren Systemen und Supraleitung eine bedeutsame Rolle spielt. Hauptziel ist die Klassifizierung und Untersuchung der minimalen Lösungen von Painlevé-Systemen in niedrigen Dimensionen.

Ziel

The aim of this proposal is to provide a systematic study of minimal solutions for a large class of nonlinear systems of PDE. Namely we will construct minimal solutions with predefined characteristics and investigate their qualitative properties, addressing the fundamental challenges that appear in the case of systems and which cannot be tackled with tools from the scalar case.


The first part focuses on phase transition problems described by the Allen-Cahn system. This is a hot and difficult topic linking PDE with the theory of minimal surfaces. The main idea is to reduce the Allen-Cahn system to a Hamiltonian system in order to construct new classes of minimal solutions, and understand the conditions implying the reduction of variables (vector analog of the celebrated De Giorgi conjecture).

In the second part, our focus is on the Painlevé equation which plays a crucial role in areas as diverse as random matrices, integrable systems, and superconductivity. The objective is to classify and investigate the minimal solutions of Painlevé-type systems in low dimensions. These have direct applications in the study of vortices in liquid crystals and Bose-Einstein condensates. The proposed approach connects the Painlevé equation with a singular problem, easier to study.

The fellow has a strong research record on the Allen-Cahn system (a book + 6 papers), and has also worked on the Ginzburg-Landau model of liquid crystals. On the one hand, he will develop his own innovative approaches to the proposed problems, and transfer his expertise to the host. On the other hand, at BCAM and through a secondment, he will link his previous research on liquid crystals to other alternative models (for which the supervisor is a world-leading expert), and to the theory of Bose-Einstein condensates. He will also acquire new skills in simulation and computation. The achievement of this project will reinforce Fellow's reputation and support him in obtaining a strong academic position.

Koordinator

BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS
Netto-EU-Beitrag
€ 160 932,48
Adresse
AL MAZARREDO 14
48009 Bilbao
Spanien

Auf der Karte ansehen

Region
Noreste País Vasco Bizkaia
Aktivitätstyp
Research Organisations
Links
Gesamtkosten
€ 160 932,48