Research

Research Interests

The research activity of the mathematical analysis group focuses on partial differential equations, calculus of variations and spectral theory of differential operators. Particular attention is payed to nonlinear hyperbolic equations in fluid dynamics, free discontinuity problems and models for phase transitions, damage and adhesive contacts.

Keywords

    • PE1_8 Analysis

    • PE1_11 Theoretical aspects of partial differential equations

    • PE1_20 Application of mathematics in sciences

Major Research Topics

  1. Nonlinear Hyperbolic Equations in Fluid Dynamics

    1. The research concerns the analysis of free boundary problems for nonlinear equations of hyperbolic type in Fluid Dynamics and Magneto-Hydrodynamics (MHD). We have studied the existence and stability conditions of compressible vortex sheets, of contact discontinuities and current-vortex sheets in MHD, the existence and stability of plasma-vacuum interface problems.

  2. Free discontinuity problems

    1. The research concerns the study of free discontinuity problems in spaces of functions of bounded variation. The main applications are in material science, in particular to fracture (with attention to the variational model of crack propagation by Francfort and Marigo) and to plasticity (classical and strain-gradient). A recent new application is suggested by shape optimization problems with Robin boundary conditions.

  1. Models for phase transitions, damage, and adhesive contact

    1. Various models describing phase change phenomena, as well as damage, and contact with adhesion between solids, are analyzed. For the associated evolutionary PDE systems, we address the issues of existence, uniqueness and regularity of solutions, as well as their long-time behaviour.

  1. Abstract Evolution Equations in Banach and Metric Spaces

    1. The focus is on evolution equations set in Banach and metric spaces. In this realm, we address gradient flows, doubly nonlinear equations, and rate-independent systems. We investigate the existence and approximation of solutions, as well as further fine properties.

  1. Spectral properties of differential operators

    1. This research deals with spectral and scattering properties of various differential operators arising from quantum physics (such as Schroedinger or Pauli operators). The main attention is payed to the analysis of the discrete spectrum, the long time behavior of the associated semi-groups and their connections with certain functional inequalities.

Major Research Project

  1. MIUR/PRIN 2015

    1. Project name: Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications

    2. Period (months): 36

  2. MIUR/PRIN 2012

    1. Project name: Nonlinear hyperbolic partial differential equations, dispersive and transport equations: theoretical and applicative aspects

    2. Period (months): 36

  1. MIUR/PRIN 2009

    1. Project name: Equations of Fluid Dynamics of Hyperbolic Type and Conservation Laws

    2. Period (months): 24

  1. MIUR/PRIN 2007

    1. Project name: Equations of Fluid Dynamics and Conservation Laws

    2. Period (months): 24

  1. MIUR/PRIN 2005

    1. Project name: Fluid Dynamics and Conservation Laws

    2. Period (months): 24

  1. FLR 2002-2014

    1. Project name: Teoria delle Equazioni alle Derivate Parziali

    2. Start period: 2002

  1. MIUR/COFIN 2000

    1. Project name: Equations of hyperbolic type in Fluid Dynamics and Continuum Mechanics

    2. Period (months): 24

  1. MIUR/COFIN 1998

    1. Project name: Hyperbolic equations in Fluid Dynamics and Continuum Mechanics

    2. Period (months): 24

Strategic collaborations

    • Babadjian Jean-François, Université de Paris 6 "Pierre et Marie Curie", Laboratoire "Jacques-Louis Lions"

    • Bauzet Caroline, Aix-Marseille Univ. CNRS, Centrale Marseille, LMA, France

    • Berselli Carlo Luigi, Università di Pisa, Dipartimento di Matematica Applicata "Ulisse Dini"

    • Bonetti Elena, Università di Milano

    • Bucur Dorin, Université de Savoie, Laboratoire de Mathématiques CNRS UMR 5127

    • Coulombel Jean-François, Université Lille 1, Laboratoire Paul Painlevé

    • Ekholm Tomas, KTH, Stockholm, Department of Mathematics

    • Exner Pavel, Doppler Institute for Mathematical Physics and Applied Mathematics, Prague

    • Francfort Gilles, Courant Institute

    • Frank Rupert, Caltech, Pasadena, Department of Mathematics

    • Frémond Michel, Università di Roma "Tor Vergata"

    • Garello Gianluca, Torino University - Department of Mathematics

    • Knees Dorothee, Kassel University

    • Laptev Ari, Imperial College, London, Department of Mathematics

    • Lebon Frédéric, , Aix-Marseille Univ. CNRS, Centrale Marseille, LMA, France

    • Licht Christian, LMGC Univ. Montpelier, CNRS, Montpelier, France & Mahidol Univ., Bangkok

    • Mielke Alexander, Weierstrass Institute for Applied Analysis and Stochastics

    • Musesti Alessandro, Università Cattolica "del Sacro Cuore"

    • Rocca Elisabetta, Università di Pavia

    • Savaré Giuseppe, Università di Bocconi Milano

    • Shibata Yoshihiro, Waseda University, Department of Mathematics

    • Thomas Marita, Weierstrass Institute for Applied Analysis and Stochastics

    • Trakhinin Yuri, Sobolev Institute of Mathematics, Novosibirsk

    • Weidl Timo, Stuttgart University, Department of Mathematics

    • Zanini Chiara, Politecnico di Torino