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1. 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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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. 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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4. 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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5. 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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