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Abstract:
In this talk we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier-Stokes equations where the dissipative term is given by \( |\nabla|^\alpha \) for any \( \alpha \in [0, \alpha_0) \) \( (\alpha_0 = 0.09 \dots) \).
Author together with collaborator Moritz Reintjes recently introduced the Regularity Transformation Equations (RT-equations), an elliptic, non-invariant system of equations which determine the Jacobians of coordinate transformations which (locally) lift the regularity of a connection to one derivative above the regularity of its Riemann curvature tensor. Our existence theory for the RT-equations generalizes celebrated results of Kazden-DeTurck, valid for Riemannian metrics, to arbitrary non-Riemannian connections, including the metrics and connections of General Relativity. Authors have found applications of the RT-equations, including extending Uhlenbeck compactness from Riemannian to non-Riemannian connections on vector bundles, extending existence and uniqueness of ODEs one derivative below the threshold for Picard's method, and an application to the Strong Cosmic Censorship Conjecture (Reintjes). In this talk, I discuss our forthcoming paper in which we use the theory of the RT-equations to give a necessary and sufficient condition for determining when a singularity appearing in a connection or metric in geometry can be regularized by coordinate transformation; we establish the consistency of the \( C^\infty \), highest possible regularity to which a singular connection can be regularized by coordinate transformation; and we describe an explicit procedure (based in the RT-equations) for constructing the coordinate transformations which lift a singular connection to its essential regularity. Results apply both locally and globally, and we show that there always exists a maximal \( C^\infty \) atlas on a manifold which globally preserves the essential regularity of any connection. Our necessary and sufficient condition relies on our existence theory for the RT-equations which we currently require connection regularity in \( L^p \), \( p > n \), sufficient to address shock wave singularities in GR, but not yet black hole singularities. Extending our existence theory to the case \( p < n \) is an important topic of authors' current research.
This talk is concerned with the properties of the co-normal derivative of (adjoint) solutions to elliptic and parabolic PDEs in divergence form, that is, \( L u = -\text{div}(A \nabla u) = 0 \) or \( L u = -\partial_t u - \text{div}(A \nabla u) = 0 \) in some domain \( \Omega \). Specifically, the properties of the co-normal derivative on a subset of the boundary where the solution \( u \) vanishes. A prototypical situation is when \( \Omega \) is the upper half-space (\( \{(x, \lambda) : x \in \mathbb{R}^n, \lambda > 0\} \) or \( \{(t, x, \lambda) : t \in \mathbb{R}, x \in \mathbb{R}^n, \lambda > 0\} \)) and \( u \) is the Green function of \( L \) with a pole at infinity. In that case, the co-normal derivative is the elliptic/parabolic measure.
In this talk, I will introduce the co-normal derivative and discuss some sufficient conditions on the coefficients \( A \) for the co-normal derivative to be quantitatively absolutely continuous with respect to the surface measure or even have a density that is \( \dot{C}^{\alpha}_{\text{loc}} \) (locally Hölder continuous) on the boundary. The method will unify these regimes by refining the work of David, Li, and Mayboroda, and combining it with some of my recent work with Toro and Zhao, and Egert and Saari. For simplicity, in the talk \( \Omega \) will be assumed to be the upper half-space, but more exotic domains can be considered.
The Landau equation is one of the main equations in kinetic theory. It models the evolution of the density of particles when they are assumed to repel each other by Coulomb potentials. It is a limit case of the Boltzmann equation with very soft potentials. In the space-homogeneous case, we show that the Fisher information is monotone decreasing in time. As a consequence, we deduce that for any initial data the solutions stay smooth and never blow up, closing a well-known open problem in the area.
Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present a recent result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. There are several essential difficulties in establishing such blowup result. We use the dynamic rescaling formulation and turn the problem of proving finite time singularity into a problem of proving stability of an approximate self-similar profile. A crucial step is to establish linear stability of the approximate self-similar profile. We decompose the solution operator into a leading order operator with the desired stability property plus a finite rank perturbation operator that can be estimated with computer assistance. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations.
The talk deals with functionals in the calculus of variations which are the sum of a perimeter or total variation term and a \( \mu \)-volume term. Here, \( \mu \) is a possibly lower-dimensional signed measure which has the role of a given right-hand side in corresponding Euler equations. Lower semicontinuity and existence results will be shown to depend crucially on certain (small-volume) isoperimetric conditions for \( \mu \). These conditions admit a wide class of measures up to the critical case of area measures on hypersurfaces and are partially optimal and interesting in themselves. The general theory will be illustrated with examples. Some of the results have been obtained in a joint work with E. Ficola (Hamburg).
Our understanding of cosmological processes, like many other predictions of physical theories, are based on studying regimes where the equations of motion reduce to a finite dimensional dynamical system. An example of a conclusion derived from such reductions is the idea of a big bang cosmology in general relativity. Such reductions are physically justified by the working assumption that when viewed from the largest scales, the inhomogeneities average out and the matter content can be approximated by a homogeneous compressible fluid. Jointly with Shih-Fang Yeh, we probe whether this working assumption is justified mathematically. Our results show that on the cosmological timescale, some big bang solutions are susceptible to instabilities generated through nonlinear self-interactions of the constituent matter when inhomogeneities are present. The goal of this talk is to present the mathematical context of this result and briefly describe the mechanism driving the instability, focusing on the relevance of the conformal (or causal) geometry of the big bang solutions. (No prior familiarity with mathematical relativity is assumed.)
Extension operators are at the core of studying function spaces, allowing us to reduce numerous problems on domains to those on full space. While this theme has witnessed a huge number of contributions over the past century, very little is known on extension operators that preserve certain differential constraints. In this talk, we will give a rather complete picture for divergence-type constraints, where we put a special focus on the borderline case \( p = 1 \) and thereby answer a borderline case left open by Kato, Mitrea, Ponce, and Taylor. This is joint work with Stefan Schiffer (MPI MIS Leipzig).
Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of expressing general data, \( f \), as the divergence of uniformly bounded vector fields, \( U \). We present a multi-scale approach to construct uniformly bounded solutions of: \[ \text{div}(U) = f \] for general \( f \)'s in the critical regularity space \( L^2 \). The study of this equation and related problems was motivated by results of Bourgain and Brezis. The intriguing critical aspect here is that although the problems are linear, the construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations, formulated as inverse problems in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations: \[ U = \sum_j u_j \] which we introduced earlier in the context of image processing, and yield a multi-scale decomposition of “objects” \( U \).