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.

We consider the Kelvin-Helmholtz system describing the evolution of a vortex-sheet near the circular stationary solution. Answering previous numerical conjectures in the 1990s physics literature, we prove an almost global existence result for small-amplitude solutions. We first establish the existence of a linear stability threshold for the Weber number, which represents the ratio between the square of the background velocity jump and the surface tension. Then, we prove that, for almost all values of the Weber number below this threshold, any small solution lives for almost all times, remaining close to the equilibrium. Our analysis reveals a remarkable stabilization phenomenon: the presence of both non-zero background velocity jump and capillarity effects enables us to prevent nonlinear instability phenomena, despite the inherently unstable nature of the classical Kelvin-Helmholtz problem. This long-time existence would not be achievable in a setting where capillarity alone provides linear stabilization, without the richer modulation induced by the velocity jump. Our proof exploits the Hamiltonian nature of the equations. More specifically, we employ Hamiltonian Birkhoff normal form techniques for quasi-linear systems together with a general approach for paralinearization of non-linear singular integral operators. This approach allows us to control resonances and quasi-resonances at arbitrary order, ensuring the desired long-time stability result.

In this work 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 = frac{22-8sqrt{7}}{9} > 0). We construct solutions in mathbb {R}^3times [0,T] with a finite T>0 and with an external forcing which is in L^1_t([0, T]) C_x^{1,epsilon }cap L^{infty }_{t} L^{2}_{x}, such that for each time t in [0, T), the velocity u is in the space C^infty cap L^2 and such that as the time t approaches the blow-up moment T, the integral int _0^t |nabla u| text {d}s tends to infinity. Since this result lays in a well-posedness class, the blow-up is generated by the dynamics of the equation and not by the force itself. This is the first blow-up result for hypodissipative Navier–Stokes in a well-posedness class.

We establish the nonlinear stability on a timescale O(varepsilon ^{-2}) of a linearly, stably stratified rest state in the inviscid Boussinesq system on mathbb {R}^2. Here, varepsilon >0 denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation.At the core of this result is a dispersive effect due to anisotropic internal gravity waves. At the linearized level, this gives rise to amplitude decay at a rate of t^{-1/2}, as observed in Elgindi and Widmayer (SIAM J. Math. Anal. 47(6):4672–4684, 2015). We establish a refined version of this, and propagate nonlinear control via a detailed analysis of nonlinear interactions using the method of partial symmetries developed in Guo et al. (Invent. Math. 231(1):169–262, 2023).

We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a "droplet state" which is metastable, i.e.persists on a much longer time scale than the time scale of convergence, before eventually diffusing to 0. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is O(1) as the inverse temperature . Meanwhile the rate of leakage away from its centre of mass is . Furthermore, the quasi-stationary distribution is localised on a length scale of order . Our proofs rely on understanding the large -asymptotics of the first two eigenvalues of the generator, which we study using techniques from semiclassical analysis. We thus provide a partial answer to a question posed by Carrillo et al. (seeAggregation-diffusion equations: dynamics, asymptotics, and singular limits. Active particles. Advances in theory, models, and applications, modeling and simulation in science, engineering and technology, vol 2, pp 65-108, Birkhäuser/Springer, Cham, 2019, Section 3.2.2) in the microscopic setting.

Abstract We derive the ‘fast response’ formula for the linear response, the parameter derivatives of long-time-averaged statistics, of hyperbolic deterministic and chaotic systems. The expression is pointwisely defined, so we can compute the linear response in high-dimensions via Monte-Carlo-type algorithms. This has two parts, where the shadowing contribution is computed by the nonintrusive shadowing algorithm. The unstable contribution is expressed by renormalized second-order tangent equations; importantly, it does not contain any distributional derivatives. The algorithm’s cost is solving u , the unstable dimension, and many first-order and second-order tangent equations along a long orbit; the main error is the sampling error of the orbit. We numerically demonstrate the algorithm on a 21-dimensional example, which is difficult for previous methods.

A central limit theorem is shown for moderately interacting particles in the whole space. The interaction potential approximates singular attractive or repulsive potentials of sub-Coulomb type. It is proven that the fluctuations become asymptotically Gaussians in the limit of infinitely many particles. The methodology is inspired by the classical work of Oelschläger on fluctuations for the porous-medium equation. The novelty of this work is that we can allow for attractive potentials in the moderate regime and still obtain asymptotic Gaussian fluctuations. The key element of the proof is the mean-square convergence in expectation for smoothed empirical measures associated to moderately interacting N-particle systems with rate N^{-1/2-varepsilon } for some varepsilon >0. To allow for attractive potentials, the proof uses a quantitative mean-field convergence in probability with any algebraic rate and a law-of-large-numbers estimate as well as a systematic separation of the terms to be estimated in a mean-field part and a law-of-large-numbers part.

Abstract We consider the free boundary problem for a 3-dimensional, incompressible, irrotational liquid drop of nearly spherical shape with capillarity. We study the problem from the beginning, extending some classical results from the flat case (capillary water waves) to the spherical geometry: the reduction to a problem on the boundary, its Hamiltonian structure, the analyticity and tame estimates for the Dirichlet-Neumann operator in Sobolev class, and a linearization formula for it, both with the method of the good unknown of Alinhac and by a geometric approach. Then, also thanks to the analyticity of the operators involved, we prove the bifurcation of traveling waves, which are nontrivial (i.e., nonspherical) fixed profiles rotating with constant angular velocity. To the best of our knowledge, this is the first example of global-in-time nontrivial solutions of the free boundary problem for the capillary liquid drop.

We consider the problem of transfer of energy to high frequencies in a quasilinear Schrödinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such an operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre’s commutator theory.