More Like this

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. 

   more » « less
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. 

   more » « less
3. A proof by foliation that lawson's cones are $$ A_{\Phi} $$-minimizing

   https://doi.org/10.3934/dcds.2021077  

   Mooney, Connor; Yang, Yang (January 2021, Discrete & Continuous Dynamical Systems)

   We give a proof by foliation that the cones over \begin{document}$$ \mathbb{S}^k \times \mathbb{S}^l $$\end{document} minimize parametric elliptic functionals for each \begin{document}$$ k, \, l \geq 1 $$\end{document}. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals. 

   more » « less
4. On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

   https://doi.org/10.3934/dcdsb.2021305  

   Massatt, David (January 2022, Discrete and Continuous Dynamical Systems - B)

   We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data \begin{document}$$ u_{01} \in L^2 $$\end{document} and \begin{document}$$ u_{02} \in H^{-1 + \eta} $$\end{document} for \begin{document}$$ \eta > 0 $$\end{document}. 

   more » « less
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}$. 

   more » « less