On the continuous dependence on the coefficients of evolutionary equations
Marcus Waurick · arXiv (Cornell University) · 2016
In an abstract Hilbert space setting, we discuss many linear phenomena of mathematical physics. The functional analytic framework presented is used to address continuous dependence of the solution operators $\mathcal{S}(\mathcal{M})$ of certain (linear partial differential) equations on the coefficients $\mathcal{M}$. For this, we introduce a particular class of coefficients $\mathcal{M}$ and study the (nonlinear) mapping $\mathcal{M}\mapsto \mathcal{S}(\mathcal{M})$. We provide criteria that guarantee the continuity of $\mathcal{S}(\cdot)$ under the norm, the strong, and the weak operator topology. We exemplify our findings in non-autonomous electro-magnetic theory, thermodynamics and acoustics.