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1. 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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2. 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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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. 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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