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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. Thermodynamic formalism for dispersing billiards

   https://doi.org/10.3934/jmd.2022013  

   Baladi, Viviane ; Demers, Mark F. ( January 2022 , Journal of Modern Dynamics)

   For any finite horizon Sinai billiard map \begin{document}$ T $\end{document} on the two-torus, we find \begin{document}$ t_*>1 $\end{document} such that for each \begin{document}$ t\in (0,t_*) $\end{document} there exists a unique equilibrium state \begin{document}$ \mu_t $\end{document} for \begin{document}$ - t\log J^uT $\end{document}, and \begin{document}$ \mu_t $\end{document} is \begin{document}$ T $\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$ - \log J^uT $\end{document}.) We show that \begin{document}$ \mu_t $\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $\end{document} is analytic on \begin{document}$ (0,t_*) $\end{document}. In addition, \begin{document}$ P(t) $\end{document} is strictly convex if and only if \begin{document}$ \log J^uT $\end{document} is not \begin{document}$ \mu_t $\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$ t_a\ne t_b $\end{document} with \begin{document}$ \mu_{t_a} = \mu_{t_b} $\end{document}, then \begin{document}$ P(t) $\end{document} is affine on \begin{document}$ (0,t_*) $\end{document}. An additional sparse recurrence condition gives \begin{document}$ \lim_{t\downarrow 0} P(t) = P(0) $\end{document}.

    

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3. A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

   https://doi.org/10.1090/tran/8976  

   Livshyts, Galyna ( June 2023 , Transactions of the American Mathematical Society)

   We show that for any even log-concave probability measure$\mu$on$\mathbb {R}^n$, any pair of symmetric convex sets$K$and$L$, and any$\lambda \in [0,1]$,$\begin{equation*} \mu ((1-\lambda ) K+\lambda L)^{c_n}\geq (1-\lambda ) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*}$where$c_n\geq n^{-4-o(1)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures.

    

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4. 𝐿₁-distortion of Wasserstein metrics: A tale of two dimensions

   https://doi.org/10.1090/btran/143  

   Baudier, F. ; Gartland, C. ; Schlumprecht, Th. ( August 2023 , Transactions of the American Mathematical Society, Series B)

   By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid$\{0,1,\dots , n\}^2$has$L_1$-distortion bounded below by a constant multiple of$\sqrt {\log n}$. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if$\{G_n\}_{n=1}^\infty$is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number$\delta \in [2,\infty )$, then the 1-Wasserstein metric over$G_n$has$L_1$-distortion bounded below by a constant multiple of$(\log |G_n|)^{\frac {1}{\delta }}$. We proceed to compute these dimensions for$\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs$\{\mathsf {D}_n\}_{n=1}^\infty$has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over$\mathsf {D}_n$has$L_1$-distortion bounded below by a constant multiple of$\sqrt {\log | \mathsf {D}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of$L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not$L_1$-embeddable.

    

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5. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

   https://doi.org/10.3934/cpaa.2021169  

   Jolly, Michael S. ; Kumar, Anuj ; Martinez, Vincent R. ( January 2021 , Communications on Pure & Applied Analysis)

   This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \begin{document}$ \theta $\end{document} on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity \begin{document}$ u $\end{document} is of lower singularity, i.e., \begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}, where \begin{document}$ p $\end{document} is a logarithmic smoothing operator and \begin{document}$ \beta \in [0, 1] $\end{document}. We complete this study by considering the more singular regime \begin{document}$ \beta\in(1, 2) $\end{document}. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.

    

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