On the parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force

Colloidal Transport in Locally Periodic Evolving Porous Media---An Upscaling Exercise

“…Here, we only sketch the proof idea for the application of the Banach-fixed-point theorem to this setting ensuring local-in-time well-posedness of the FBP. We transform the moving domain Ω ε 0 (t) into the fixed domain Ω ε 0 (0) by some diffeomorphism depending not only on the choice of r ε (t) but also on its regularity and uniform boundedness in suitable norms; see also [3], where the authors have fixed the free boundaries in a multiple-connected domain related to Ostwald ripening with kinetic undercooling. We refer the reader to [32] where a simple connected domain has been treated.…”

The multi-dimensional stochastic Stefan financial model for a portfolio of assets

“…When the solid phase at time t consists of 2 well separated spherical domains of radii R 1 (t) > R 2 (t) and thus of curvatures 1 R1(t) < 1 R2(t) , in later times as separation evolves, the growth of the larger sphere is expected (here this of radius R 1 ) at the expense of the smaller. In fact this is rigorously proved for n = 3 and zero volatility in [25,7] for the Stefan problem of type (2), and for a more general case where kinetic undercooling acts on the Gibbs Thomson condition.…”

“…Various deterministic parabolic Stefan problems have been extensively used for describing the phase separation of alloys and a relevant mathematical theory is already well established. See for example the results of Niethammer, X. Chen and Reitich, Antonopoulou Karali and Yip in [25,13,7], or for the quasi-static problem in [1,2,3,4,26,10,9,11]. Note that the quasi-static problem approximates the parabolic one when the diffusion tends to infinity as in the case of a very large trading activity.…”

“…Moreover, she proved well posedeness for the static (elliptic) problem with undercooling, [26]. Antonopoulou, Karali and Yip in [7] proved well posedeness for the full parabolic problem with undercooling and obtained the modified dynamics of radii; cf. also the work of X. Chen and Reitich for the two-phases deterministic Stefan problem, [13], and in [2,3,4] for a quasi-static version.…”

“…We estimate the areas of zero trading with diameter approximating the minimum of the n spreads for orders from the limit order books. In dimensions n = 3, for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by Niethammer in [25], and in [7]. We propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices with radii representing the half of the minimum spread.…”