Browsing by Subject "Wellposedness"

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    Dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line
    (Element d.o.o., 2021-09) Özsarı, Türker; Alkan, Kıvılcım; Kalimeris, Konstantinos
    In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schr¨odinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.
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    Well-posedness and stability of planar conewise linear systems
    (2021-09) Namdar, Daniyal
    Planar conewise linear systems constitute a subset of piecewise linear systems. The state space of a conewise linear system is a nite number of convex polyhedral cones lling up the space. Each cone is generated by a positive linear combination of a nite set of vectors, not all zero. In each cone the dynamics is that of a linear system and any pair of neighboring cones share the same dynamics at the common border, which is itself a cone of one lower dimension. Each cone with its linear dynamics is called a mode of the conewise system. This thesis focuses on the simplest case of planar systems that is composed of a nite number of cones of dimension two; with borders that are cones of dimension one, that is rays. Stability of such conewise linear systems is well understood and there are a number of necessary and su cient conditions. Somewhat surprisingly, their well-posedness is not so well understood or studied except for the special case where there are two modes only, i.e, the bimodal case. A graphical necessary and su cient condition is here derived for the wellposedness of a planar conewise linear system of arbitrary number of modes and the well-known condition for stability is re-stated on this same graph. This graphical result is expected to provide some guidance to well-posedness studies of conewise systems in a higher dimension.