Archive for Rational Mechanics and Analysis最新文献

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.

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.

In this work, starting from the predictions of the Post-Newtonian theory for a system of N infalling masses from the infinite past (i^-), we formulate and solve a scattering problem for the system of linearised gravity around Schwarzschild in a double null gauge, as introduced in Dafermos (Acta Math 222:1–214, 2019). The scattering data are posed on a null hypersurface (underline{mathcal {C}}) emanating from a section of past null infinity (mathcal {I}^{-}), and on the part of (mathcal {I}^{-}) that lies to the future for this section. Along (underline{mathcal {C}}), we implement the Post-Newtonian theory-inspired hypothesis that the gauge-invariant components of the Weyl tensor and (a.k.a. (Psi _0) and (Psi _4)) decay like (r^{-3}), (r^{-4}), respectively, and we exclude incoming radiation from (mathcal {I}^{-}) by demanding the News function to vanish along (mathcal {I}^{-}). We also show that compactly supported gravitational perturbations along (mathcal {I}^{-}) induce very similar data, with , decaying like (r^{-3}), (r^{-5}). After constructing the unique solution to this scattering problem, we then provide a complete analysis of the asymptotic behaviour of projections onto fixed spherical harmonic number (ell ) near (mathcal {I}^{-}), spacelike infinity (i^0) and future null infinity (mathcal {I}^{+}), crucially exploiting a set of approximate conservation laws enjoyed by and . Having obtained a clear understanding of the asymptotics of linearised gravity around Schwarzschild, we also give constructive corrections to popular historical notions of asymptotic flatness such as Bondi coordinates or asymptotic simplicity. In particular, confirming earlier heuristics authorized by Damour and Christodoulou, we find that the peeling property is violated both near (mathcal {I}^{-}) and near (mathcal {I}^{+}), with for example near (mathcal {I}^{+}) only decaying like (r^{-4}) instead of (r^{-5}). We also find that the resulting solution decays slower towards (i^0) than often assumed, with both decaying like (r^{-3}) towards (i^0). The issue of summing up the estimates obtained for fixed angular modes in (ell ) in order to obtain asymptotics for the full solution is dealt with in forthcoming work.

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 study the dynamics of the two dimensional Navier–Stokes equations linearized around a shear flow on a (non-square) torus which possesses exactly two non-degenerate critical points. We obtain linear inviscid damping and vorticity depletion estimates for the linearized flow that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The main task is to understand the associated Rayleigh and Orr–Sommerfeld equations, under the natural assumption that the linearized operator around the shear flow in the inviscid case has no discrete eigenvalues. The key difficulty is to understand the behavior of the solution to Orr–Sommerfeld equations in three distinct regimes depending on the spectral parameter: the non-degenerate case when the spectral parameter is away from the critical values, the intermediate case when the spectral parameter is close to but still separated from the critical values, and the most singular case when the spectral parameter is inside the viscous layer.

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.

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.

Contraction-driven self-propulsion of a large class of living cells can be modeled by a Keller-Segel system with free boundaries. The ensuing “active” system, exhibiting both dissipation and anti-dissipation, features stationary and traveling wave solutions. While the former represent static cells, the latter describe propagating pulses (solitary waves) mimicking the autonomous locomotion of the same cells. In this paper we provide the first proof of the asymptotic nonlinear stability of both of these solutions, static and dynamic. In the case of stationary solutions, the linear stability is established using the spectral theorem for compact, self-adjoint operators, and thus linear stability is determined classically, solely by eigenvalues. For traveling waves the picture is more complex because the linearized problem is non-self-adjoint, opening the possibility of a “dark” area in the phase space which is not “visible” in the purely eigenvalue/eigenvector approach. To establish linear stability in this case we employ spectral methods together with the Gearhart-Prüss-Greiner (GPG) theorem, which controls the entire spectrum via bounds on the resolvent operator. For both stationary and small-velocity traveling wave solutions, nonlinear stability is then proved for appropriate parameter values by showing that the nonlinear part of the problem is dominated by the linear part and then employing a Grönwall inequality argument. The developed novel methodology can prove useful also in other problems involving non-self-adjoint (non-Hermitian or non-reciprocal) operators which are ubiquitous in the modeling of “active” matter.