Archive for Rational Mechanics and Analysis最新文献

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.

We prove that in the chiral limit of the Bistritzer–MacDonald Hamiltonian, there exist magic angles at which the Hamiltonian exhibits flat bands of multiplicity four instead of two. We analyse the structure of Bloch functions associated with the bands of arbitrary multiplicity, compute the corresponding Chern number to be ( -1 ), and show that there exist infinitely many degenerate magic angles for a generic choice of tunnelling potential, including the Bistritzer–MacDonald potential. Moreover, we demonstrate for generic tunnelling potentials that flat bands have only twofold or fourfold multiplicities.

In this paper, we consider the asymptotic stability of the 2D Taylor-Couette flow in the exterior disk, with a small kinematic viscosity ( nu ll 1 ) and a large rotation coefficient ( |B| ). Due to the degeneracy of the Taylor-Couette flow at infinity, we cannot expect the solution to decay exponentially in a space-time decoupled manner. As stated in a previous work (Li et al. in Linear enhanced dissipation for the 2D Taylor-Couette flow in the exterior region: A supplementary example for Gearhart-Pr(ddot{u})ss type lemma. arXiv:2501.14187), even space-time coupled exponential decay cannot be expected, and at most, we can obtain space-time coupled polynomial decay. To handle the space-time coupled decay multiplier, the previous time-independent resolvent estimate methods no longer work. Therefore, this paper introduces time-dependent resolvent estimates to deal with the space-time coupled decay multiplier ( Lambda _k ). We remark that the choice of ( Lambda _k ) is not unique, here we just provide one way to construct it. Finally, as an application, we derive a transition threshold bound of (frac{1}{2}), which is the same as that for the Taylor-Couette flow in the bounded region.

In 2018, Bourgain pioneered a novel perturbative harmonic-analytic approach to the stochastic homogenization theory of discrete elliptic equations with weakly random i.i.d. coefficients. The approach was subsequently refined to show that homogenized approximations of ensemble averages can be derived to a precision four times better than almost sure homogenized approximations, which was unexpected by the state-of-the-art homogenization theory. In this paper, we grow this budding theory in various directions: first, we prove that the approach is robust by extending it to the continuum setting with exponentially mixing random coefficients. Second, we give a new proof via Malliavin calculus in the case of Gaussian coefficients, which avoids the main technicality of Bourgain’s original approach. This new proof also applies to strong Gaussian correlations with power-law decay. Third, we extend Bourgain’s approach to the study of fluctuations by constructing weak correctors up to order 2d, which also clarifies the link between Bourgain’s approach and the standard corrector approach to homogenization. Finally, we draw several consequences from those different results, both for quantitative homogenization of ensemble averages and for asymptotic expansions of the annealed Green’s function.

We consider nonlinear elliptic equations of the form (Delta u = f(u,nabla u)) for the suitable analytic nonlinearity f, in the vinicity of infinity in (mathbb {R}^d), which is on the complement of a compact set. We show that there is a one-to-one correspondence between the nonlinear solution u defined there and the linear solution (u_L) to the Laplace equation such that, in an adequate space, (u - u_Lrightarrow 0) as (|x|rightarrow +infty ). This is a kind of scattering operator. Our results apply in particular for the energy critical and supercritical pure power elliptic equation and for the 2d (energy critical) harmonic maps and the H-system. Similar results are derived for solutions defined on the neighborhood of a point in (mathbb {R}^d). The proofs are based on a conformal change of variables, and studied as an evolution equation (with the radial direction playing the role of time) in spaces with analytic regularity on spheres (the directions orthogonal to the radial direction).

We establish the Gauss-Green formula for extended divergence-measure fields (i.e., vector-valued measures whose distributional divergences are Radon measures) over open sets. We prove that, for almost every open set, the normal trace is a measure supported on the boundary of the set. Moreover, for any open set, we provide a representation of the normal trace of the field over the boundary of the open set as the limit of measure-valued normal traces over the boundaries of approximating sets. Furthermore, using this theory, we extend the balance law from classical continuum physics to a general framework in which the production on any open set is measured with a Radon measure and the associated Cauchy flux is bounded by a Radon measure concentrated on the boundary of the set. We prove that there exists an extended divergence-measure field such that the Cauchy flux can be recovered through the field, locally on almost every open set and globally on every open set. Our results generalize the classical Cauchy’s Theorem (that is only valid for continuous vector fields) and extend the previous formulations of the Cauchy flux (that generate vector fields within (L^{p})). Thereby, we establish the equivalence between entropy solutions of the multidimensional nonlinear partial differential equations of divergence form and of the mathematical formulation of physical balance laws via the Cauchy flux through the constitutive relations in the axiomatic foundation of Continuum Physics.

We study the asymptotic behavior of the volume preserving mean curvature and the Mullins–Sekerka flat flow in three dimensional space. Motivated by this, we establish a 3D sharp quantitative version of the Alexandrov inequality for (C^2)-regular sets with a perimeter bound.