On abstract grad-div systems

Picard, Rainer and Seilder, Stefan and Trostorff, Sascha and Waurick, Marcus (2016) On abstract grad-div systems. Journal of Differential Equations, 260 (6). pp. 4888-4917. ISSN 0022-0396 (https://doi.org/10.1016/j.jde.2015.11.033)

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For a large class of dynamical problems from mathematical physics the skew-selfadjointness of a spatial operator of the form ⁎A=(0−C⁎C0), where C:D(C)⊆H0→H1 is a closed densely defined linear operator between Hilbert spaces H0,H1, is a typical property. Guided by the standard example, where C=grad=(∂1⋮∂n) (and ⁎−C⁎=div, subject to suitable boundary constraints), an abstract class of operators C=(C1⋮Cn) is introduced (hence the title). As a particular application we consider a non-standard coupling mechanism and the incorporation of diffusive boundary conditions both modeled by setting associated with a skew-selfadjoint spatial operator A.