engleski

Friedrichs systems (with complex coefficients)

Symmetric positive systems of first-order linear partial differential equations were introduced by Kurt Otto Friedrichs (thus they are often called Friedrichs' systems today) in an attempt to treat equations that change their type, like the equations modelling transonic fluid flow. A Friedrichs system consists of a certain first order system of partial differential equation and an admissible boundary condition. Friedrichs showed that this class of problems encompasses a wide variety of classical and neoclassical initial and boundary value problems for various linear partial differential equations. More recently, Ern, Guermond and Caplain (2007) suggested another approach to Friedrichs' theory, which was inspired by their interest in the numerical treatment of Friedrichs systems. They expressed it in terms of operators acting on abstract Hilbert spaces and proved well-posedness result in this abstract setting. We (Antonić & Burazin, 2010) rewrote their cone formalism in terms of an indefinite inner product space, which in a quotient by its isotropic part gives a Krein space. This new viewpoint allowed us to show that the three sets of intrinsic boundary conditions are actually equivalent, which facilitates further investigation of their precise relation to the original Friedrichs boundary conditions. Although some evolution (non-stationary) problems can be treated within this framework, their theory is not suitable for problems like the initial-boundary value problem for the non-stationary Maxwell system, or the Cauchy problem for the symmetric hyperbolic system. This motivates the interest in non-stationary Friedrichs systems. Some numerical treatment of such problems was already done by Burman, Ern and Fernandez (2010), while the existence and uniqueness result was recently provided by Burazin and Erceg. Most classical papers deal with Friedrichs systems in real space setting. In this talk we shall address the extensions of both stationary and non-stationary theory to complex spaces, as well as the two-field theory, commenting on the difficulties encountered in the semilinear case, as well as in the Banach space setting. Finally, we shall investigate the applicability of these extensions to some examples, like the Dirac or Maxwell system.

Friedrichs system

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