Archive for Rational Mechanics and Analysis最新文献

We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie hyperbolic system an incompressible liquid crystal model, where the direction function of liquid crystal molecules satisfies a wave map equation with an acoustical metric. This improves the almost global existence result by Huang-Jiang-Zhao (J Funct Anal, 288:110858, 2025). As a byproduct, we obtain the sharp (same as the linear solution) (L^infty _x)-decay estimates for both the heat part and the wave part. Moreover the nonlinear wave part scatters to a linear solution as time goes to infinity. This paper’s main contribution is the discovery of a novel null structure within the velocity equation’s wave-type quadratic self-interaction. This structure compensates the insufficient decay rate in 2D, which previously hindered the establishment of global regularity for small data.

In this paper, we investigate the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble.

We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier–Stokes system in the whole three-dimensional space. Our primary goal is to establish that small initial velocities with critical Sobolev regularity (H^{1/2}) and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier–Stokes equations (NS) that has been proved by Fujita and Kato in [11]. Assuming also that the initial velocity is in (L^1,) we establish that the total energy (E_0) of the system decays to 0 with the same rate (t^{-3/2}) as for the weak solutions of (NS), see [22, 24]. Our results partly rely on the use of a higher order energy functional (E_1) that controls the regularity (H^1) of the velocity. This idea seems to originate from the recent paper [18] by Li, Shou and Zhang, devoted to the inhomogeneous Vlasov-Navier–Stokes system. Here we show that (E_1) decays with the rate (t^{-5/2}) which, in particular, allows us to prove that the density of the particles has a strong limit when the time goes to infinity.

In Ammari et al. (SIAM J Math Anal 52:5441–5466, 2020), the first author with collaborators proved the existence of Dirac dispersion cones at subwavelength scales in bubbly honeycomb phononic crystals. In this paper, we study the time-evolution of wave packet, which are spectrally concentrated near such conical points. We prove that the wave packet dynamic is governed by a time-dependent effective Dirac system, which still depends, but in a simple way, on the subwavelength scale.

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.