Many dynamical processes in scientific disciplines such as physics, biology and chemistry are captured by nonlinear evolution equations, which are partial differential equations of the form du/dt = Au + N(u), where t represents time, A is a linear operator and N(u) denotes a nonlinearity. Examples are reaction-diffusion systems, the nonlinear Schrödinger equation or the incompressible Navier-Stokes equation. This course provides a concise introduction into the world of nonlinear evolution equations. We will address:
- local existence and uniqueness
- regularity
- an introduction to (analytic) semigroup theory
- the variation of constants formula (mild solutions)
- global existence and blow-up
- the maximum principle
- the method of characteristics and shocks
- Hamiltonian/integrable systems
- systems with a gradient structure
(- amplitude equations)
(- diffusive dynamics)
Due to the complexity of nonlinear evolution equations, many techniques have been developed for prototype examples first and could then be transferred to a larger class of equations. To avoid getting `lost' in technicalities, we take a similar perspective in this course and prove our results for prototype examples such as the Burgers, the Kolmogorov-Petrovsky-Piscounov, the nonlinear Schrödinger, the Korteweg-de Vries or the Nagumo equations first, before setting up a general framework.