My research focuses on the mathematical and numerical analysis of non-linear partial differential equations.
In particular, I study existence, uniqueness, and stability of solutions in the context of parabolic and elliptic equations with irregular data, as well as non-linear evolutionary systems derived from fluid thermomechanics.
In addition, I am interested in the numerical analysis of these types of equations, particularly the convergence of approximate solutions obtained by the finite volume method to the solution of these equations.
M. Aoun, O. Guibé. Finite volume scheme and renormalized solutions for nonlinear elliptic Neumann problem with $L^1$ data. (Published in Calcolo) [Hal]
M. Aoun. Existence and Uniqueness of renormalized solutions for parabolic Neumann problem with $L^1$ data. (Submitted 2024) [Hal]
M. Aoun. Existence of weak-renormalized for a nonlinear phase-field model system. (Submitted 2025) [Hal]
Title: Analysis and Numerical Analysis of PDEs derived from Fluid Thermomechanics.
Advisor: Olivier Guibé.
Defended on December 20, 2023.
Jury members:
Carole Rosier - University of Littoral Côte d'Opale - Jury President,
Alain Miranville - University of Poitiers - Reviewer,
Roger Lewandowski - University of Rennes 1 - Reviewer,
Boris P. Andreianov - University of Tours - Examiner,
Hasnaa Zidani - INSA of Rouen Normandy - Examiner,
Olivier Guibé - University of Rouen Normandy - Advisor.
Keywords: Nonlinear systems; Finite volume; Renormalized solutions; Convection-diffusion problem, L^1 data, existence of solutions; Phase-field; Navier-Stokes equations; Parabolic problems; Elliptic problems.
02/2025: Seminar of analysis, Denis Poisson Institute, University of Tour, France.
Problems with $L^1$ data and their application to coupled PDE systems.
07/2023: 15ème journée de la Fédération Normandie-Mathématiques, Le Havre, France.
Existence of Weak-Renormalized Solutions for a Non-Isothermal Solidification Problem.
06/2022: 45ème Congrès National d'Analyse Numérique, Evian-les-Bains, France.
Finite Volume Method and Renormalized Solutions for an Elliptic Problem with $L^1$ Data and Neumann Boundary Conditions, [abstract].
06/2022: PhD seminar, LMRS, University of Rouen Normandy, Saint-Étienne-du-Rouvray, France.
Finite Volume method and Renormalized solutions.
06/2021: 13ème Journée Normandie-Mathématiques, INSA of Rouen Normandy, France.
Elliptic Problems with $L^1$ Data and Neumann Boundary Conditions.
11/2024: Normandie Meetings on the Theoretical and Numerical Aspects of PDEs, Rouen, France.
11/2024: Workshop: Mathematics and Geosciences, Chambéry, France.
02/2022: Conference of the GDR of Analysis of Partial Differential Equations, Vannes, France.
11/2021: Autumn School 2021, Institute of Mathematics for the Planet Earth (IMPT), Lyon, France. (Morphological Impacts of Climate Change: Mathematical Problems Related to Glacier Melt, Landslides, Coastal Erosion, Floods, and Inundations).