Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation

We study the stochastic mass-conserving Allen-Cahn equation posed on a bounded two-dimensional domain with additive spatially smooth space-time noise. This equation associated with a small positive parameter 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 droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco 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 sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of droplets in $L^2$ and $H^1$, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales

Layer dynamics for the one dimensional ε-dependent Cahn–Hilliard/Allen–Cahn equation

The final publication is available at Springer via http://dx.doi.org/10.1007/s00526-021-02085-4We study the dynamics of the one-dimensional ε-dependent Cahn-Hilliard / Allen-Cahn 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 mass-conservation constraint in place of one of the zero-mass flux boundary conditions at x = 1. Motivated by the study of Carr and Pego on the layered metastable patterns of Allen-Cahn in [10], and by this of Bates and Xun in [5] for the Cahn-Hilliard equation, we implement an N-dimensional, and a mass-conservative N−1-dimensional 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 Cahn-Hilliard / Allen-Cahn operator, and specify wide families of ε-dependent 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 Allen-Cahn, or, when the mass is conserved, close to the Cahn-Hilliard solution

Crank-Nicolson finite element discretizations for a two-dimenional linear Schroedinger-type equation posed in noncylindrical domain

First published in Mathematics of Computation online 2014 (84 (2015), 1571-1598), published by the American Mathematical SocietyMotivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundary-value problem for a general Schr¨odinger-type equation posed on a two space-dimensional 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

The sharp interface limit for the stochastic Cahn-Hilliard Equation

We study the two and three dimensional stochastic Cahn-Hilliard 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 Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength

Malliavin Calculus for the stochastic Cahn- Hilliard/Allen-Cahn equation with unbounded noise diffusion

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 Cahn-Hilliard and Allen-Cahn type operators with a multiplicative, white, space-time 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 Allen-Cahn 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

Existence and regularity of solution for a Stochastic CahnHilliard / Allen-Cahn equation with unbounded noise diffusion

The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption-spin flip mechanisms in the context of surface processes. For such an equation we consider a multiplicative space-time 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 sub-linear 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 Cahn-Hilliard equation in the case of bounded noise diffusion. Our results are also valid for the stochastic Cahn-Hilliard 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 bi-Laplacian operator seems to be dominant in the combined model

Proline-rich tyrosine kinase 2 mediates gonadotropin-releasing hormone signaling to a specific extracellularly regulated kinase-sensitive transcriptional locus in the luteinizing hormone beta-subunit gene

G protein-coupled receptor regulation of gene transcription primarily occurs through the phosphorylation of transcription factors by MAPKs. This requires transduction of an activating signal via scaffold proteins that can ultimately determine the outcome by binding signaling kinases and adapter proteins with effects on the target transcription factor and locus of activation. By investigating these mechanisms, we have elucidated how pituitary gonadotrope cells decode an input GnRH signal into coherent transcriptional output from the LH β-subunit gene promoter. We show that GnRH activates c-Src and multiple members of the MAPK family, c-Jun NH(2)-terminal kinase 1/2, p38MAPK, and ERK1/2. Using dominant-negative point mutations and chemical inhibitors, we identified that calcium-dependent proline-rich tyrosine kinase 2 specifically acts as a scaffold for a focal adhesion/cytoskeleton-dependent complex comprised of c-Src, Grb2, and mSos that translocates an ERK-activating signal to the nucleus. The locus of action of ERK was specifically mapped to early growth response-1 (Egr-1) DNA binding sites within the LH β-subunit gene proximal promoter, which was also activated by p38MAPK, but not c-Jun NH(2)-terminal kinase 1/2. Egr-1 was confirmed as the transcription factor target of ERK and p38MAPK by blockade of protein expression, transcriptional activity, and DNA binding. We have identified a novel GnRH-activated proline-rich tyrosine kinase 2-dependent ERK-mediated signal transduction pathway that specifically regulates Egr-1 activation of the LH β-subunit proximal gene promoter, and thus provide insight into the molecular mechanisms required for differential regulation of gonadotropin gene expression

Κινητική ετερογενών αντιδράσεων του οργανικού κλάσματος φυσικών και ανθρωπογενών αιωρούμενων σωματιδίων ΑΣ2.5

The complexity of the composition of atmospheric aerosol makes the study of theirnatural and chemical properties a necessity. The present research dissertation is focused on the heterogeneous reaction kinetics of the non-volatile organic fraction of the natural and anthropogenic aerosol (PM2.5) under natural atmospheric conditions, in the city of Xanthi. The lifetime of the non-volatile chemical compounds in the PM2.5 was investigated in order to gain information regarding their potential dispersion and wet or dry deposition, based on the meteorological conditions at the time of their emission. In addition, their photochemical degradation can either create compounds more harmful to the human health or create deliquescent organic aerosol that may form cloud condensation nuclei and hence contribute to the change of climate. Specifically, the decomposition kinetics of non-volatile n-alkanes and PAHs were studied. The quartz filters that they were collected on, were exposed in two reactors in dark and natural light conditions for four days. Ambient air that was passed through the reactors continuously, with laminar flow, ensured a supply of exogenic oxidants to the surface of the filters. The operating conditions in the reactors (incoming radiation, air flow and temperature) were continuously monitored. Simultaneously, the atmospheric concentrations of ozone (Ο3), oxygen (Ο2), oxides of nitrogen (NO, NO2) and sulphur dioxide (SO2) were also monitored. The lifetimes of the non-volatile n-alkanes and PAHs were experimentally determined to be in the order of 3–8 days. The results from the light reactor exhibited approximately a 10% decrease in their lifetime in comparison to the dark reactor. The results obtained for the prevailing atmospheric conditions and for the duration of the experiments, were in agreement with values from the literature for n-alkanes, while for the PAHs the results agreed with literature values only in the cases were the substrate of these laboratory experiments was silica or graphite particles. Considering the fact that the experiments were conducted under various ambient conditions, with a temperature range of approximately 10K, an attempt was made to determine the activation energy and the pre-exponential factor through the Arrhenius equation, only for the n-alkanes. Finally, after the identification of the oxidation products that were detected, a first estimation of the decomposition mechanisms of the organic fragment of the aerosols in the study area will beconducted in the near future.Η πολυπλοκότητα της σύνθεσης των ατμοσφαιρικών αιωρουμένων σωματιδίων επιβάλλει την μελέτη των φυσικών και χημικών ιδιοτήτων τους. Η παρούσα διδακτορική έρευνα πραγματεύεται τη φωτοχημική κινητική των ετερογενών χημικών αντιδράσεων, στο οργανικό κλάσμα των ατμοσφαιρικών αιωρουμένων σωματιδίων (ΑΣ2.5), φυσικής και ανθρωπογενούς προέλευσης κάτω από πραγματικές συνθήκες στην πόλη της Ξάνθης. Η διάρκεια ζωής των χημικών ενώσεων στα ΑΣ2.5 μελετάται γιατί παρέχει πληροφορίες για την δυνητική τους διασπορά και ξηρά ή υγρή εναπόθεση, δεδομένων των μετεωρολογικών συνθηκών κατά την εκπομπή τους. Επίσης η φωτοχημική τους διάσπαση μπορεί είτε να δημιουργήσει νέες ενώσεις που είναι πιο επιβλαβείς για την ανθρώπινη υγεία, είτε να δημιουργήσει νέες υδατοδιαλυτές ενώσεις που μπορούν να λειτουργήσουν ως πυρήνες συμπύκνωσης συννέφων, δηλαδή μπορούν να συμβάλουν στην αλλαγή του κλίματος. Συγκεκριμένα εξετάστηκε η κινητική αποδόμησης των μη πτητικών κ-αλκανίων και πολυκυκλικών αρωματικών υδρογονανθρακών (ΠΑΥ) σε αστικά αερολύματα που συλλέχθηκαν από τον κέντρο της Ξάνθης. Τα φορτωμένα με ΑΣ2.5 φίλτρα χαλαζία τοποθετήθηκαν σε δύο αντιδραστήρες ο ένας λειτουργούσε σε συνθήκες φυσικού φωτισμού (Photo-reactor) και ο άλλος σε συνθήκες σκότους (Dark-reactor) για τέσσερις μέρες. Ο ατμοσφαιρικός αέρας που διέρχονταν διαμέσου των αντιδραστήρων εξασφάλιζε συνεχώς την παροχή εξωγενών οξειδωτικών ενώσεων ή ριζών στην επιφάνεια των φορτωμένων φίλτρων. Η ροή είχε ρυθμιστεί έτσι ώστε να είναι στρωτή. Οι συνθήκες λειτουργίας των αντιδραστήρων (εισερχόμενη ακτινοβολία, ροή ατμοσφαιρικού αέρα και θερμοκρασία) ελέγχονταν συνεχώς. Παράλληλα γινόταν μετρήσεις των ατμοσφαιρικών συγκεντρώσεων του όζοντος (Ο3), του οξυγόνου (Ο2), των οξειδίων του αζώτου (NO, NO2) και του διοξειδίου του θείου (SO2). Οι χρόνοι ζωής των μη πτητικών αλκανίων και των μη πτητικών ΠΑΥ προσδιορίστηκαν πειραματικά και ήταν της τάξεως των 3-8 ημερών. Τα αποτελέσματα από τον αντιδραστήρα που λειτουργούσε σε συνθήκες φυσικού φωτισμού (Photo-reactor) δείχνουν μείωση περίπου 10% στη διάρκεια ζωής των οργανικών ενώσεων σε σχέση με αυτά του αντιδραστήρα που λειτουργούσε σε συνθήκες σκότους (Darkreactor). Τα αποτελέσματα που ελήφθησαν για τις επικρατούσες ατμοσφαιρικές συνθήκες κατά τη διάρκεια των πειραμάτων ήταν σύμφωνα με τις τιμές της βιβλιογραφίας για τα αλκάνια, ενώ για τους ΠΑΥ ήταν σύμφωνα μόνο στην περίπτωση που το υπόστρωμα των πειραμάτων στην βιβλιογραφία ήταν σωματίδια διοξειδίου του πυριτίου ή γραφίτη. Δεδομένου ότι τα πειράματα διεξήχθησαν σε διάφορες περιβαλλοντικές συνθήκες, με εύρος θερμοκρασιών περίπου 10Κ, έγινε μια προσπάθεια υπολογισμού της ενέργειας ενεργοποίησης της αντίδρασης, καθώς και της προεκθετικής σταθεράς μέσω της εξίσωσης του Arrhenius, μόνο για την περίπτωση των Αλκανίων. Τέλος, με την ανίχνευση των οξειδωμένων ενώσεων στο μέλλον θα γίνει μία πρώτη εκτίμηση των μηχανισμών αποδόμησης του οργανικού κλάσματος των αιωρουμένων σωματιδίων της περιοχής μελέτης