# Fundamentals

Copyright: Fraunhofer ILTThe primary goals of our work are to investigate and improve technical systems with mathematical, physical and experimental methods and to share the established scientific and technical understanding through media-supported instructional methods.

**From Principles to Applications**

Our work is focused on expanding the principles of continuum physics and its industrial application. The time-dependent interactions of participating physical sub-processes (light propagation, heat conduction, hydrodynamics of melts, gas dynamics) is modelled with a Free Boundary Problem for a system of nonlinear partial differential equations that is too complicated for a detailed analysis. Even numerical analysis demands detailed knowledge of the solution structure.

**Inertial Manifolds**

The physical sub-processes involved are of dissipative nature. A typical property of dissipative infinite-dimensional dynamical systems is the existence of a finite-dimensional attractor, i.e. a subset of the phase space that attracts all trajectories. If the system possesses a single stationary solution, the attractor consists of only a single point. A non-trivial example would be the Lorenz attractor. From the theory of infinite-dimensional dynamical systems, it is known that in systems with multiple time scales and after decay of the fast degrees of freedom, a reduction in the dimension of the phase space occurs. Similar to this, the analysis of dissipative partial differential equations shows in many examples that the long-term dynamics of even such systems is concentrated on a manifold of finite dimension: the inertial manifold. An important task in the mathematical analysis of dynamic processes in laser material processing consists of identifying the degrees of freedom of the long-term dynamics and specifying their equations of motion in order to arrive at reliable statements regarding the solution structure.