The course of Evolution equations is an introduction to solving evolution partial differential equations from the point of view of semigroup theory. These equations appear among other in the modeling of non-stationary systems from physics (e.g.: fluid machanics, quantum mechanics, electromagnetism, relativity), or biology (reation-diffusion). In such contexts, these equations describe quantity temporal evolution (speed, pressure, wave function, concentration) which also depend of a spatial variable.
The first mathematics question is about the equation's well-posedness: Does every configuration give an solution of the equation? What is the more adapted functional framework? Which sense to such a solution? Is the solution unique? Is it always defined or on the contrary, does it stop to exist in finite time?
We will present some fundamental results of the volution equation theory. About the independent linear equations, we will study the theory of semigroups on Banach's spaces, which provide an general abstract framework responding to the questions above. We will then explain how linear equations of heat, waves and of Schrödinger in this framework. To do this, we'll use abstract functional analysis results and adapted function spaces (). In the second part of the course, we will describe a strategy for the general solutions of semi-linear Banach fixed-point evolution problems. For the heat and non-linear equations, we will shox results of local and global existence. Finally, we will identify some finite-time explosion phenomena.
Lecture notes (in English) are available and the course may be taught in English depending on the audience
Program
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H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011).
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T. Cazenave and A. Haraux. An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1998).
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K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York (2000).
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L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition (2010).
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A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York (1983).
- Teaching coordinator: Carrapatoso Kleber