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1. 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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2. 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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3. Realizing arbitrary $$d$$-dimensional dynamics by renormalization of $C^d$-perturbations of identity

   https://doi.org/10.3934/dcds.2021129  

   Fayad, Bassam; Saprykina, Maria (January 2022, Discrete & Continuous Dynamical Systems)

   Any \begin{document}$ C^d $$\end{document} conservative map \begin{document}$$ f $$\end{document} of the \begin{document}$$ d $$\end{document}-dimensional unit ball \begin{document}$$ {\mathbb B}^d $$\end{document}, \begin{document}$$ d\geq 2 $$\end{document}, can be realized by renormalized iteration of a \begin{document}$$ C^d $$\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}$$ {\mathbb B}^d $$\end{document}, arbitrarily close to identity in the \begin{document}$$ C^d $$\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$$ C^d $$\end{document} change of coordinates is exactly \begin{document}$$ f $$\end{document}$. 

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4. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

   https://doi.org/10.3934/krm.2022012  

   Nazarov, Fedor; Zumbrun, Kevin (January 2022, Kinetic and Related Models)

   We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$$ (0, +\infty) $$\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $$\end{document} stable manifolds of such equations, showing that \begin{document}$$ L^2_{loc} $$\end{document} solutions that remain sufficiently small in \begin{document}$$ L^\infty $$\end{document} (i) decay exponentially, and (ii) are \begin{document}$$ C^\infty $$\end{document} for \begin{document}$$ t>0 $$\end{document}, hence lie eventually in the \begin{document}$$ H^1 $$\end{document}$ stable manifold constructed by Pogan and Zumbrun. 

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5. On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

   https://doi.org/10.3934/dcdsb.2021305  

   Massatt, David (January 2022, Discrete and Continuous Dynamical Systems - B)

   We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data \begin{document}$$ u_{01} \in L^2 $$\end{document} and \begin{document}$$ u_{02} \in H^{-1 + \eta} $$\end{document} for \begin{document}$$ \eta > 0 $$\end{document}. 

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