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1. The mean-field limit of the Lieb-Liniger model

   https://doi.org/10.3934/dcds.2022006  

   Rosenzweig, Matthew ( January 2022 , Discrete and Continuous Dynamical Systems)

   We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $\end{document} bosons interacting pairwise on the line via the \begin{document}$ \delta $\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$ \delta $\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$ N $\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$ L^2 $\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of \begin{document}$ N $\end{document}-body initial states.

    

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2. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

   https://doi.org/10.3934/eect.2022007  

   Triggiani, Roberto ; Wan, Xiang ( January 2022 , Evolution Equations and Control Theory)

   We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}, which, moreover, in the canonical case \begin{document}$ \gamma = 0 $\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$ \gamma = 0 $\end{document} or \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}, since \begin{document}$ \gamma \neq 0 $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$ g $\end{document} "smoother" than \begin{document}$ L^2(\Sigma) $\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}, and [44] for control less regular in space than \begin{document}$ L^2(\Gamma) $\end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].

    

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3. Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

   https://doi.org/10.3934/dcds.2022085  

   Mayer, Martin ; Ndiaye, Cheikh Birahim ( January 2022 , Discrete and Continuous Dynamical Systems)

   We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[21] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [29] to show solvability of the fractional Yamabe problem for conformal infinities of dimension \begin{document}$ 3 $\end{document} and fractional parameter in \begin{document}$ (\frac{1}{2}, 1) $\end{document} corresponding to a global case left by previous works.

    

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4. Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation

   https://doi.org/10.3934/dcdss.2022035  

   Promislow, Keith ; Wu, Qiliang ( January 2022 , Discrete and Continuous Dynamical Systems - S)

   Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter \begin{document}$ \varepsilon\ll1 $\end{document}, we show that this induces undulated bilayer solutions whose width perturbations decay on an \begin{document}$ O\!\left( \varepsilon^{-1/2}\right) $\end{document} inner length scale that is long in comparison to the \begin{document}$ O(1) $\end{document} scale that characterizes the bilayer width.

    

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5. Generative imaging and image processing via generative encoder

   https://doi.org/10.3934/ipi.2021060  

   Ong, Yong Zheng ; Yang, Haizhao ( January 2022 , Inverse Problems & Imaging)

   This paper introduces a novel generative encoder (GE) framework for generative imaging and image processing tasks like image reconstruction, compression, denoising, inpainting, deblurring, and super-resolution. GE unifies the generative capacity of GANs and the stability of AEs in an optimization framework instead of stacking GANs and AEs into a single network or combining their loss functions as in existing literature. GE provides a novel approach to visualizing relationships between latent spaces and the data space. The GE framework is made up of a pre-training phase and a solving phase. In the former, a GAN with generator \begin{document}$ G $\end{document} capturing the data distribution of a given image set, and an AE network with encoder \begin{document}$ E $\end{document} that compresses images following the estimated distribution by \begin{document}$ G $\end{document} are trained separately, resulting in two latent representations of the data, denoted as the generative and encoding latent space respectively. In the solving phase, given noisy image \begin{document}$ x = \mathcal{P}(x^*) $\end{document}, where \begin{document}$ x^* $\end{document} is the target unknown image, \begin{document}$ \mathcal{P} $\end{document} is an operator adding an addictive, or multiplicative, or convolutional noise, or equivalently given such an image \begin{document}$ x $\end{document} in the compressed domain, i.e., given \begin{document}$ m = E(x) $\end{document}, the two latent spaces are unified via solving the optimization problem

   \begin{document}$ z^* = \underset{z}{\mathrm{argmin}} \|E(G(z))-m\|_2^2+\lambda\|z\|_2^2 $\end{document}

   and the image \begin{document}$ x^* $\end{document} is recovered in a generative way via \begin{document}$ \hat{x}: = G(z^*)\approx x^* $\end{document}, where \begin{document}$ \lambda>0 $\end{document} is a hyperparameter. The unification of the two spaces allows improved performance against corresponding GAN and AE networks while visualizing interesting properties in each latent space.

    

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