In this article we propose a novel geometric model to study the motion of a physical flag. In our approach, a flag is viewed as an isometric immersion from the square with values in satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a spirit similar to Arnold's geometric picture for the motion of an incompressible fluid.

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. The spectral profiles of these operators describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers, for the first time, non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel's work on one dimensional intervals arXiv:1502.04404v1. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

We obtain a new formula for the Loewner energy of Jordan curves on the sphere, which is a Kähler potential for the essentially unique Kähler metric on the Weil-Petersson universal Teichmüller space, as the renormalised energy of moving frames on the two domains of the sphere delimited by the given curve.

In this article, we study the propagation of defect measures for Schrödinger operators on a Riemannian manifold (, ) of dimension with having conormal singularities along a hypersurface in the sense that derivatives along vector fields tangential to preserve the regularity of . We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surface whenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangential to at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold.

The Dean-Kawasaki equation-a strongly singular SPDE-is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of independent diffusing particles in the regime of large particle numbers . The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean-Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean-Kawasaki equation may approximate the density fluctuations of non-interacting diffusing particles to arbitrary order in (in suitable weak metrics). In other words, the Dean-Kawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.

We provide a suitable generalisation of Pansu's differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures , then must be absolutely continuous with respect to the Haar measure of the group.

We show convergence of the Navier-Stokes/Allen-Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility in the Allen-Cahn equation tends to zero in a subcritical way, i.e., for some and . The proof proceeds by showing via a relative entropy argument that the solution to the Navier-Stokes/Allen-Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger-Kantorovich problem (), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton-Jacobi equation arising in the dual dynamic formulation of , which are sufficiently strong to construct a characteristic transport-growth flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic -convexity with respect to the Hellinger-Kantorovich distance. Examples of geodesically convex functionals are provided.

We prove the existence of steady stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel . These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash-Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin's cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.

We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain is obtained as the integral of a cost function (, ) depending on the solution of a certain PDE problem on . The main feature of these functionals is that the minimality of a domain cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions , where is the solution of the PDE with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal from the inwards/outwards optimality of and then we use the stability of with respect to variations with smooth vector fields in order to study the blow-up limits of the state function . By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove regularity of the regular part of the free boundary when the data are smooth.

We establish the future nonlinear stability of a large class of FLRW models as solutions to the Einstein-Dust system. We consider the case of a vanishing cosmological constant, which, in particular implies that the expansion rate of the respective models is linear, i.e. has zero acceleration. The resulting spacetimes are future globally regular. These solutions constitute the first generic class of future regular Einstein-Dust spacetimes not undergoing accelerated expansion and are thereby the slowest expanding generic family of future complete Einstein-Dust spacetimes currently known.

We consider a gas of bosons interacting through a hard-sphere potential with radius in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term and shows that corrections are smaller than , for a sufficiently large constant . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order , in agreement with the Lee-Huang-Yang prediction.

We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner-Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis' results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.

The skew mean curvature flow is an evolution equation for dimensional manifolds embedded in (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions . In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension . This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.

We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side for . In the case of the heat equation, we also show the optimal regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are in the parabolic obstacle problem and in the parabolic Signorini problem.

We consider the parabolic-elliptic Keller-Segel system in dimensions , which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071-1098, 1999), and is conjectured to be nonlinearly radially stable. We prove this conjecture for . Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we use a technique we recently established in Glogić and Schörkhuber (Comm Part Differ Equ 45(8):887-912, 2020). To the best of our knowledge, this provides the first result on stable self-similar blowup for the Keller-Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.

In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.

The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundary is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl's law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as "slow" exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result, in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings.

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

We consider the pressureless Euler-Poisson equations with quadratic confinement. For spatial dimension , we give a necessary and sufficient condition for the existence of radial global smooth solutions, which is formulated explicitly in terms of the initial data. This condition appears to be much more restrictive than the critical-threshold conditions commonly seen in the study of Euler-type equations. To obtain our results, the key observation is that every characteristic satisfies a periodic ODE system, and the existence of a global smooth solution requires the period of every characteristic to be identical.

For any (analytic) axisymmetric toroidal domain we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) state in . Each vector field in this set is Morse-Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS state. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.