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Abstract We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the$$n$$ $n$-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for$$n \geq 5$$ $n\ge 5$. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution. 

Abstract We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani,$$\begin{aligned} \begin{aligned} \partial _t \theta - \nabla ^\perp \log (10+(-\Delta )^{\frac{1}{2}})\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta -{\nabla }^{\perp }log(10+{(-\Delta )}^{\frac{1}{2}})\theta ·\nabla \theta =0,\end{array}\end{array}$and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities,arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae et al. (Comm Pure Appl Math 65(8):1037–1066, 2012). This well-posedness result can be applied to describe the long-time dynamics of the$$\delta $$ $\delta$-SQG equations, defined by$$\begin{aligned} \begin{aligned} \partial _t \theta + \nabla ^\perp (10+(-\Delta )^{\frac{1}{2}})^{-\delta }\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta +{\nabla }^{\perp }{(10+{(-\Delta )}^{\frac{1}{2}})}^{-\delta }\theta ·\nabla \theta =0,\end{array}\end{array}$for all sufficiently small$$\delta >0$$ $\delta >0$depending on the size of the initial data. For the same range of$$\delta $$ $\delta$, we establish global well-posedness of smooth solutions to the dissipative SQG equations. 

This is the first part of the four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs. Scattering Dichotomy for the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space-time. The primary subject of this paper, however, is another PDE, namely the energy critical Yang-Mills heat flow on the 4-dimensional Euclidean space. Our first goal is to establish sharp criteria for global existence and asymptotic convergence to a flat connection for this system in H1, including the Dichotomy Theorem (i.e., either the above properties hold or a harmonic Yang-Mills connection bubbles off) and the Threshold Theorem (i.e., if the initial energy is less than twice that of the ground state, then the above properties hold). Our second goal is to use the Yang-Mills heat flow in order to define the caloric gauge, which will play a major role in the analysis of the hyperbolic Yang-Mills equation in the subsequent papers.