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1. 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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2. 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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3. 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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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. Hodge and Teichmüller

   https://doi.org/10.3934/jmd.2022007  

   Kahn, Jeremy; Wright, Alex (January 2022, Journal of Modern Dynamics)

   We consider the derivative \begin{document}$$ D\pi $$\end{document} of the projection \begin{document}$$ \pi $$\end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form \begin{document}$$ \eta $$\end{document} determines a relative cohomology class \begin{document}$$ [\eta]_\Sigma $$\end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of \begin{document}$$ D\pi([\eta]_\Sigma) $$\end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory. 

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