The Dirichlet–Cauchy problem for nonlinear hyperbolic equations in a domain with edges

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

“…In polygonal spatial domains or in domains composed of different homogeneous materials, high regularity of solutions in standard Sobolev scales is known to fail, due to diffractive solution components with point singularities. We refer to results obtained in recent years on the regularity of solutions, also in nonsmooth domains, which are phrased in terms of corner-weighted Sobolev spaces of Kondrat'ev type, see [28] and the references there, and the more recent Luong-Tung [30]. The solutions of linear, second-order wave equations in polygons has been shown in these references to consist of a smooth part plus a finite, time-dependent linear combination of singular "corner" solutions, with coefficients which depend regularly on the time-variable.…”

“…[26][27][28] and the references there) that these singularities are, in a sense, the hyperbolic counterpart of elliptic conical singularities, and that the solutions of linear, second order hyperbolic equations admit regularity results in scales of corner-weighted spaces. Based on the regularity results in [26][27][28]30], it was shown by some of the authors of the present manuscript in [35] that high-order, time-marching discretizations of linear, acoustic wave equations with FEM discretizations in the spatial domain can recover the maximal convergence rate afforded by the elemental polynomial degree, even in the presence of conical singularities.…”

Besov regularity of parabolic and hyperbolic PDEs

“…In this subsection we recall regularity results of linear hyperbolic equations of second order from [47].…”

“…Now Lemma 3.11 together with Theorem 6.2 give the following result concerning the Besov regularity of the solution to (3.19). Remark 6.4 There are more results in [47] compared to what we used in this section. In particular, in [47,Thm.…”

“…Remark 6.4 There are more results in [47] compared to what we used in this section. In particular, in [47,Thm. 3.5] weighted Sobolev regularity of nonlinear hyperbolic problems was investigated.…”