We establish the existence of Lipschitz-continuous solutions to the Cauchy-Dirichlet problem for a class of evolutionary partial differential equations of the form in a space-time cylinder , subject to time-dependent boundary data prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data g along the lateral boundary . More precisely, we require that, for each fixed , the graph of over admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers.

Abstract Two-dimensional (2-D) incompressible, inviscid fluids produce fascinating patterns of swirling motion. How and why the patterns emerge are long-standing questions, first addressed in the 19th century by Helmholtz, Kirchhoff, and Kelvin. Countless researchers have since contributed to innovative techniques and results. However, the overarching problem of swirling 2-D motion and its long-time behavior remains largely open. Here we shed light on this problem via a link to isospectral matrix flows. The link is established through V. Zeitlin’s beautiful model for the numerical discretization of Euler’s equations in 2-D. When considered on the sphere, Zeitlin’s model offers deep connections between 2-D hydrodynamics and unitary representations of the rotation group; consequently, it provides a dictionary that maps hydrodynamical concepts to matrix Lie theory, which in turn gives connections to matrix factorizations, random matrices, and integrability theory, for example. Results about finite-dimensional matrices can then be transferred to infinite-dimensional fluids via quantization theory, which is here used as an analysis tool (albeit traditionally describing the limit between quantum and classical physics). We demonstrate how the dictionary is constructed and how it unveils techniques for 2-D hydrodynamics. We also give accompanying convergence results for Zeitlin’s model on the sphere.

Local solutions to the 3D stochastic quantisation equations of Yang–Mills–Higgs were constructed in Chandra (Invent Math 237:541–696, 2024), and it was shown that, in the limit of smooth mollifications, there exists a mass renormalisation of the Yang–Mills field such that the solution is gauge covariant. In this paper we prove the uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of Chandra (Invent Math 237:541–696, 2024), and is potentially important for the identification of the limit of other approximations, such as lattice dynamics. Our proof relies on systematic short-time expansions of singular stochastic PDEs and of regularised Wilson loops. We also strengthen the recently introduced state spaces of Cao (Comm Part Diff Equ 48:209–251, 2023); Cao (Comm Math Phys 405:3, 2024); Chandra (Invent Math 237:541–696, 2024) to allow for finer control on line integrals appearing in expansions of Wilson loops.

Given pin [1,infty ), we provide sufficient and necessary conditions on the non-negative measurable kernels (rho _t)_{tin (0,1)} ensuring convergence of the associated Bourgain–Brezis–Mironescu (BBM) energies (mathscr {F}_{t,p})_{tin (0,1)} to a variant of the p-Dirichlet energy on mathbb {R}^N as trightarrow 0^+ both in the pointwise and in the Gamma -sense. We also devise sufficient conditions on (rho _t)_{tin (0,1)} yielding local compactness in L^p(mathbb {R}^N) of sequences with bounded BBM energy. Moreover, we give sufficient conditions on (rho _t)_{tin (0,1)} implying pointwise and Gamma -convergence and equicoercivity of ({mathscr {F}}_{t,p})_{tin (0,1)} when the limit p-energy is of non-local type. Finally, we apply our results to provide asymptotic formulas in the pointwise and Gamma -sense for heat content-type energies both in the local and non-local settings.