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Addendum to the article: On the Dirichlet to Neumann Problem for the 1dimensional Cubic NLS Equation on the HalfLineAntonopoulou, Dimitra; Kamvissis, Spyridon (IOPSCIENCE Published jointly with the London Mathematical Society, 20160831)We present a short note on the extension of the results of [1] to the case of nonzero initial data. More specifically, the defocusing cubic NLS equation is considered on the halfline with decaying (in time) Dirichlet data and sufficiently smooth and decaying (in space) initial data. We prove that for this case also, and for a large class of decaying Dirichlet data, the Neumann data are sufficiently decaying so that the Fokas unified method for the solution of defocusing NLS is applicable.

CrankNicolson finite element discretizations for a twodimenional linear Schroedingertype equation posed in noncylindrical domainAntonopoulou, Dimitra; Karali, Georgia D.; Plexousakis, Michael; Zouraris, Georgios; University of Chester (AMS, 20141105)Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundaryvalue problem for a general Schr¨odingertype equation posed on a two spacedimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain, and we approximate its solution by a Crank–Nicolson finite element method. For the proposed numerical method, we derive an optimal order error estimate in the L2 norm, and to support the error analysis we prove a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments are presented which verify the optimal order of convergence of the method.

Existence and regularity of solution for a Stochastic CahnHilliard / AllenCahn equation with unbounded noise diffusionAntonopoulou, Dimitra; Karali, Georgia D.; Millet, Annie; University of Chester (Elsevier, 20151024)The CahnHilliard/AllenCahn equation with noise is a simpliﬁed mean ﬁeld model of stochastic microscopic dynamics associated with adsorption and desorptionspin ﬂip mechanisms in the context of surface processes. For such an equation we consider a multiplicative spacetime white noise with diﬀusion coeﬃcient of linear growth. Applying technics from semigroup theory, we prove local existence and uniqueness in dimensions d = 1,2,3. Moreover, when the diﬀusion coeﬃcient satisﬁes a sublinear growth condition of order α bounded by 1 3, which is the inverse of the polynomial order of the nonlinearity used, we prove for d = 1 global existence of solution. Path regularity of stochastic solution, depending on that of the initial condition, is obtained a.s. up to the explosion time. The path regularity is identical to that proved for the stochastic CahnHilliard equation in the case of bounded noise diﬀusion. Our results are also valid for the stochastic CahnHilliard equation with unbounded noise diﬀusion, for which previous results were established only in the framework of a bounded diﬀusion coeﬃcient. As expected from the theory of parabolic operators in the sense of Petrovsk˘ıı, the biLaplacian operator seems to be dominant in the combined model.

Galerkin methods for a Schroedingertype equation with a dynamical boundary condition in two dimensionsAntonopoulou, Dimitra; University of Chester (EDP Sciences / SMAI, 20150630)In this paper, we consider a twodimensional Schodingertype equation with a dynamical boundary condition. This model describes the longrange sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a CrankNicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution.

Layer Dynamics for the one dimensional $\eps$dependent CahnHilliard / AllenCahn EquationAntonopoulou, Dimitra; Karali, Georgia; Tzirakis, Konstantinos; University of Chester; University of Crete; IACM/FORTH (Springer, 20210827)We study the dynamics of the onedimensional εdependent CahnHilliard / AllenCahn equation within a neighborhood of an equilibrium of N transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a massconservation constraint in place of one of the zeromass flux boundary conditions at x = 1. Motivated by the study of Carr and Pego on the layered metastable patterns of AllenCahn in [10], and by this of Bates and Xun in [5] for the CahnHilliard equation, we implement an Ndimensional, and a massconservative N−1dimensional manifold respectively; therein, a metastable state with N transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized CahnHilliard / AllenCahn operator, and specify wide families of εdependent weights δ(ε), µ(ε), acting at each part of the operator, for which the dynamics are stable and rest exponentially small in ε. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to AllenCahn, or, when the mass is conserved, close to the CahnHilliard solution.

Malliavin Calculus for the stochastic Cahn Hilliard/AllenCahn equation with unbounded noise diffusionAntonopoulou, Dimitra; Farazakis, Dimitris; Karali, Georgia D.; University of Chester; Foundation for Research and Technology; University of Crete (Elsevier, 20180508)The stochastic partial di erential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of CahnHilliard and AllenCahn type operators with a multiplicative, white, spacetime noise of unbounded di usion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution u. In dimension one, according to the regularity result in [5], u admits continuous paths a.s. Using this property, and inspired by a method proposed in [8], we construct a modi ed approximating sequence for u, which properly treats the new second order AllenCahn operator. Under a localization argument, we prove that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous, establishing thus that a density exists.

Motion of a droplet for the Stochastic mass conserving AllenCahn equationAntonopoulou, Dimitra; Bates, Peter W.; Bloemker, Dirk; Karali, Georgia D.; University of Chester (SIAM, 20160216)We study the stochastic massconserving AllenCahn equation posed on a smoothly bounded domain of R2 with additive, spatially smooth, spacetime noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply It^o calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco [2] for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a su ciently small noise strength, we establish stochastic stability of a neighborhood of the manifold of boundary droplet states in the L2 and H1norms, which means that with overwhelming probability the solution stays close to the manifold for very long timescales.

The multidimensional Stochastic Stefan Financial Model for a portfolio of assetsAntonopoulou, Dimitra; Bitsaki, Marina; Karali, Georgia; University of Chester; University of Crete (American Institute of Mathematical Sciences (AIMS), 20210401)The financial model proposed in this work involves the liquidation process of a portfolio of n assets through sell or (and) buy orders placed, in a logarithmic scale, at a (vectorial) price with volatility. We present the rigorous mathematical formulation of this model in a financial setting resulting to an ndimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading, the socalled solid phase. We will focus on a case of financial interest when one or more markets are considered. In particular, our aim is to estimate for a short time period the areas of zero trading, and their diameter which approximates the minimum of the n spreads of the portfolio assets for orders from the n limit order books of each asset respectively. In dimensions n = 3, and 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] in a more general setting. There in, when the initial moving boundary consists of well separated spheres, a first order approximation system of odes had been rigorously derived for the dynamics of the interfaces and the asymptotic pro le of the solution. In our financial case, we propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices, while each sphere may correspond to a different market; the relevant radii represent the half of the minimum spread. We apply It^o calculus and provide second order formal asymptotics for the stochastic version dynamics, written as a system of stochastic differential equations for the radii evolution in time. A second order approximation seems to disconnect the financial model from the large diffusion assumption for the trading density. Moreover, we solve the approximating systems numerically.

Numerical approximation of the Stochastic CahnHilliard Equation near the Sharp Interface LimitAntonopoulou, Dimitra; Banas, Lubomir; Nurnberg, Robert; Prohl, Andreas; University of Chester; University of Bielefeld; Imperial College London; University of TuebingenAbstract. We consider the stochastic CahnHilliard equation with additive noise term that scales with the interfacial width parameter ε. We verify strong error estimates for a gradient flow structureinheriting timeimplicit discretization, where ε only enters polynomially; the proof is based on highermoment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For γ sufficiently large, convergence in probability of iterates towards the deterministic HeleShaw/MullinsSekerka problem in the sharpinterface limit ε → 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ γ) on the geometric evolution in the sharpinterface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) MullinsSekerka problem. The computational results indicate that the limit for γ ≥ 1 is the deterministic problem, and for γ = 0 we obtain agreement with a (new) stochastic version of the MullinsSekerka problem.

On the Dirichlet to Neumann Problem for the 1dimensional Cubic NLS Equation on the halflineAntonopoulou, Dimitra; Kamvissis, Spyridon; Department of Mathematics, University of Chester, UK(D.A) and Department of Mathematics and Applied Mathematics, University of Crete, Greece (S.K) (IOPSCIENCE Published jointly with the London Mathematical Society, 20150724)Initialboundary value problems for 1dimensional `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 wellposed problem. In the case of cubic NLS, knowledge of the Dirichet data su ces to make the problem wellposed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also su ciently decaying and that, hence, the Fokas method can be applied.

A Posteriori Analysis for SpaceTime, discontinuous in time Galerkin approximations for parabolic equations in a variable domainAntonopoulou, Dimitra; Plexousakis, Michael; University of Chester; University of Crete (ECP sciences, 20190424)This paper presents an a posteriori error analysis for the discontinuous in time spacetime scheme proposed by Jamet for the heat equation in multidimensional, noncylindrical domains [25]. Using a Cl ementtype interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of twodimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with spacetime dependent coe cients but posed on a cylindrical domain. We formulate a discontinuous in time space{time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of [19] for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso [36], proposed for adaptive, RungeKutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

The sharp interface limit for the stochastic CahnHilliard EquationAntonopoulou, Dimitra; Bloemker, Dirk; Karali, Georgia D.; Universiy of Chester (IMS Journals, 20180219)We study the two and three dimensional stochastic CahnHilliard equation in the sharp interface limit, where the positive parameter \eps tends to zero, which measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the solutions of stochastic CahnHilliard converge to a solution of a HeleShaw problem with stochastic forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic HeleShaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.

SpaceTime Discontinuous Galerkin Methods for the '\eps'dependent Stochastic AllenCahn Equation with mild noiseAntonopoulou, Dimitra; Department of Mathematics, University of Chester, UK (Oxford University Press, 20190408)We consider the $\eps$dependent stochastic AllenCahn equation with mild space time noise posed on a bounded domain of R^2. The positive parameter $\eps$ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki in [15]. The noise although smooth becomes white on the sharp interface limit as $\eps$ tends to zero. We construct a nonlinear dG scheme with spacetime finite elements of general type which are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer's Theorem. We first derive abstract error estimates and then for the case of piecewise polynomial finite elements we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\eps$. Finally, we present a linear approximation of the nonlinear scheme for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in 2+1dimensional subdomains for a nonlinear parabolic problem. In addition, this problem involves noise. These type of schemes avoid any RungeKutta type discretization for the evolutionary variable and seem to be very effective when applied to equations of such a difficulty.