engleski

Non-stationary Friedrichs systems

Symmetric positive systems (Friedrichs systems) of first-order linear partial differential equations were introduced by Kurt Otto Friedrichs (1958) in order to treat the equations that change their type, like the equations modelling transonic fluid flow. More recently, Ern, Guermond and Caplain (CPDE, 2007) expressed the theory in terms of operators acting in abstract Hilbert spaces and proved wellposedness result in this abstract setting. Although their setting can be used to represent various boundary value problems for (partial) differential equations, some evolution (non-stationary) problems, such as the initial-boundary value problem for the non-stationary Maxwell system, or the Cauchy problem for the symmetric hyperbolic system, can not be treated within their framework. We develop an abstract theory for non-stationary Friedrichs systems that can address these problems as well. More precisely, we consider an abstract Cauchy problem in a Hilbert space, that involves a time independent abstract Friedrichs operator. We use the semigroup theory approach, and prove that the operator involved satisfies conditions of the Hille-Yosida generation theorem. We also address the semilinear problem and apply new results to symmetric hyperbolic systems, the unsteady Maxwell system, the unsteady div-grad problem, and the wave equation. The theory can be extended to the complex Banach space setting, with application to the Dirac system. This is a joint work with Marko Erceg (University of Zagreb).

Friedrichs systems

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