2008/2009

Nonlinear Partial Differential Equations

Mathematics

Principal investigators

Helge Holden

Professor
Norwegian University of Science and Technology (NTNU)
Year at CAS

Kenneth H. Karlsen

Professor
University of Oslo (UiO)
Year at CAS

Abstract

The research program will focus on the mathematical discipline nonlinear partial differential equation. Differential equations have their origin in the quest to describe nature by mathematics, and these equations are perfect tools to describe physical phenomena that vary in space and time. In fact, all the fundamental laws of nature are given by differential equations (Newton's law describes gravitation, Maxwell's equations describe electromagnetism, Navier-Stokes' equations describe fluid flow, etc). Thus they model a wide variety of phenomena, and have been extensively studied by mathematicians, physicists, engineers, and others. Differential equations constitute a core mathematical discipline, cf. the Abel Prize 2005. However, several key questions have not been answered, or only been partly answered. Central issues are:
- Do the equations have a solution? It is still not known if the equations that describe the flow of air around an airplane have a solution.
- Given that the equation has a solution, is it unique? For flow of oil in petroleum reservoirs the equations give several possible distinct values for the saturation at the same point in space.
- Is the solution stable? Will a small change in one place in space have dramatic consequences in a completely different location ("the butterfly effect").
- If the equation has a unique and stable solution, how can one determine the solution? Given the measured weather data for today, how can one predict the weather for tomorrow?

Variations of these questions for carefully selected nonlinear partial differential equations constitute the focus for the research program, in particular the interplay between theoretical results (mathematical properties of the solution) and numerical computations (how to compute the solution). Differential equations have been an intensive research area in mathematics in Norway the last two decades, and there are now strong research groups in Oslo, Trondheim, and Bergen. The research program will both benefit from the strong national activities and further strengthen it. Furthermore, applications to other areas of science will have impact there. The generic nature of mathematics means that the methods and techniques of this program can be applied to several seemingly distinct areas. We will focus on four applications, namely flow in porous media (e.g., flow of oil in a petroleum reservoir), mixtures of solids and fluids (e.g., sedimentation), gas dynamics (e.g., flow of gas around obstacles), water waves (e.g., breaking of waves).

Fellows

Debora Amadori

Dr.
University of L'Aquila
Year at CAS

Boris Andreianov Andreianov

Dr.
University of Franche-Comté
Year at CAS

Yann Brenier

University of Nice
Year at CAS

Alberto Bressan

Professor
Pennsylvania State University
Year at CAS

Jose Antonio Carrillo de la Plata

Professor
Catalan Institution for Research and Advanced Studies (ICREA)
Year at CAS
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Gui-Qiang G. Chen

Professor
University of Oxford
Year at CAS

Giuseppe Maria Coclite

Assistant Professor
University of Bari
Year at CAS

Eduard Feireisl

Professor
Institute of Mathematics of the Czech Academy of Sciences
Year at CAS

Hermano Frid

Professor
Instituto de Matemática Pura e Aplicada (IMPA)
Year at CAS
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Fritz Gesztesy

Professor
University of Missouri
Year at CAS

Harald Hanche-Olsen

Associate Professor
Norwegian University of Science and Technology (NTNU)
Year at CAS
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Espen Robstad Jakobsen

Associate Professor
Norwegian University of Science and Technology (NTNU)
Year at CAS
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Kristian Jenssen

Associate Professor
Pennsylvania State University
Year at CAS

Henrik Kalisch

Associate Professor
University of Bergen (UiB)
Year at CAS

Trygve Klovning Karper

Research Fellow
Norwegian University of Science and Technology (NTNU)
Year at CAS
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Sergiu Klainerman

Professor
Princeton University
Year at CAS
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Young-Sam Kwon

Dr.
University of Maryland
Year at CAS

Philippe Gerard LeFloch

Professor
Pierre and Marie Curie University
Year at CAS
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Lars Peter Lindqvist

Professor
Norwegian University of Science and Technology (NTNU)
Year at CAS
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Hailiang Liu

Professor
Iowa State University
Year at CAS

Evgeniy Yuz'evich Panov

Professor
Novgorod State University
Year at CAS

Xavier Marcel Raynaud

University of Oslo (UiO)
Year at CAS

Nils Henrik Risebro

Professor
University of Oslo (UiO)
Year at CAS

Christian Rohde

Professor
University of Stuttgart
Year at CAS

Hilde Sande

Norwegian University of Science and Technology (NTNU)
Year at CAS
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Sigmund Selberg

Associate Professor
Norwegian University of Science and Technology (NTNU)
Year at CAS
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Denis Serre

Professor
ENS de Lyon
Year at CAS

Wen Shen

Assistant Professor
Pennsylvania State University
Year at CAS

Eitan Tadmor

Professor
University of Maryland
Year at CAS
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Gerald Teschl

Professor
University of Vienna
Year at CAS

John David Towers

Professor
Mira Costa College
Year at CAS
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Konstantina Trivisa

Professor
University of Maryland
Year at CAS
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Yin Zhaoyang

Professor
Zhongshan University
Year at CAS

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