We consider the conformal Einstein equations for massless colli-sionless gas cosmologies which admit an isotropic singularity. After developing the general theory, we restrict to spatially-homogeneous cosmologies. We show that the Cauchy problem for these equations is well-posed with data consisting of the limiting particle distribution function at the singularity.

We discuss a Hilbert space method that allows to prove analytical well-posedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most common linear cases we can furthermore apply a recent result due to Crouzeix–Haase, thus extending several known results and obtaining optimal analyticity angle.

Initial-boundary value problems for 1-dimensional ‘completely integrable’ equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the pr...

This study deals with asymptotic models for the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. We present a new Green-Naghdi type model in the Camassa-Holm (or medium amplitude) regime. This model is fully justified, in the sense that it is consistent, well-pose...

In the present paper, we generalize the concept of well-posedness to a generalized hemivariational inequality, give some metric characterizations of the α-well-posed generalized hemivariational inequality, and derive some conditions under which the generalized hemivariational inequality is strongly α-well-posed in the generalized sense. Also, we show that the α-well-posedness of the generalized...

We discuss optimal well-posedness results for parabolic boundary value problems on non-compact Riemannian manifolds. We explain, in particular, the

For the backwards heat equation, stabilized by an a priori initial bound, an estimator is determined for intermediate values which is optimal with respect to the bound and the observation accuracy. It is shown how this may be implemented computationally with error estimates for the computed approximation which can be made arbitrarily close to the uncertainty level induced by the ill-posedness o...

We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey’s very-well-poised 6ψ6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via...

We introduce a notion of wave maps with a target in the subRiemannian Heisenberg group and study their relation with Riemannian wave maps with range in Lagrangian submanifolds. As an application we establish existence and eventually ill-posedness of the corresponding Cauchy problem.