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1. 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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2. 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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3. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition

   https://doi.org/10.3934/krm.2022003  

   Dong, Hongjie ; Guo, Yan ; Yastrzhembskiy, Timur ( January 2022 , Kinetic and Related Models)

   We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the \begin{document}$ S_p $\end{document} estimate of [7], we prove regularity in the kinetic Sobolev spaces \begin{document}$ S_p $\end{document} and anisotropic Hölder spaces for such weak solutions. Such \begin{document}$ S_p $\end{document} regularity leads to the uniqueness of weak solutions.

    

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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. A spectral target signature for thin surfaces with higher order jump conditions

   https://doi.org/10.3934/ipi.2022020  

   Cakoni, Fioralba ; Lee, Heejin ; Monk, Peter ; Zhang, Yangwen ( January 2022 , Inverse Problems and Imaging)

   In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in \begin{document}$ {\mathbb R}^m $\end{document}, \begin{document}$ m = 2, 3 $\end{document} from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open \begin{document}$ m-1 $\end{document} dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [20]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach

    

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