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1. 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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2. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan ; Zhang, Yanzhi ( January 2022 , Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$ (- \Delta)^\frac{{ \alpha}}{{2}} $\end{document} for \begin{document}$ \alpha \in (0, 2) $\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$ {\mathcal O}(h^2) $\end{document}, while \begin{document}$ {\mathcal O}(h^4) $\end{document} for quadratic basis functions with \begin{document}$ h $\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$ \alpha \in (0, 2) $\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$ u \in C^{m, l}(\bar{ \Omega}) $\end{document} for \begin{document}$ m \in {\mathbb N} $\end{document} and \begin{document}$ 0 < l < 1 $\end{document}, our method has an accuracy of \begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $\end{document} for constant and linear basis functions, while \begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $\end{document} for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

    

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3. 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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4. Phase mixing for solutions to 1D transport equation in a confining potential

   https://doi.org/10.3934/krm.2022002  

   Chaturvedi, Sanchit ; Luk, Jonathan ( January 2022 , Kinetic and Related Models)

   Consider the linear transport equation in 1D under an external confining potential \begin{document}$ \Phi $\end{document}:

   \begin{document}$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $\end{document}

   For \begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} (with \begin{document}$ \varepsilon >0 $\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}, with an inverse polynomial decay rate \begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $\end{document}D under the external potential \begin{document}$ \Phi $\end{document}.

    

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5. Existence of global weak solutions of $ p $-Navier-Stokes equations

   https://doi.org/10.3934/dcdsb.2021051  

   Liu, Jian-Guo ; Zhang, Zhaoyun ( January 2022 , Discrete & Continuous Dynamical Systems - B)

   This paper investigates the global existence of weak solutions for the incompressible \begin{document}$ p $\end{document}-Navier-Stokes equations in \begin{document}$ \mathbb{R}^d $\end{document} \begin{document}$ (2\leq d\leq p) $\end{document}. The \begin{document}$ p $\end{document}-Navier-Stokes equations are obtained by adding viscosity term to the \begin{document}$ p $\end{document}-Euler equations. The diffusion added is represented by the \begin{document}$ p $\end{document}-Laplacian of velocity and the \begin{document}$ p $\end{document}-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\begin{document}$ p $\end{document} distances with constraint density to be characteristic functions.

    

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