Archive for Rational Mechanics and Analysis最新文献

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.

The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be strongly well-posed. Of central importance is a fully nonlinear boundary condition for the director field, which, in this generality, is necessary to guarantee that the system fulfills physical principles. The system is shown to be locally, strongly well-posed in the (L_p)-setting. More precisely, the existence and uniqueness of a local, strong (L_p)-solution to the general system is proved and it is shown that the director d satisfies (|d|_2equiv 1) provided this holds for its initial data (d_0). In addition, the solution is shown to depend continuously on the data. The results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.

We are concerned with inverse problems for supersonic potential flows past infinite axisymmetric Lipschitz cones. The supersonic flows under consideration are governed by the steady isentropic Euler equations for axisymmetric potential flows, which give rise to a singular geometric source term. We first study the inverse problem for the stability of an oblique conical shock as an initial-boundary value problem with both the generating curve of the cone surface and the leading conical shock front as free boundaries. We then establish the existence and asymptotic behavior of global entropy solutions with bounded BV norm of the inverse problem, under the condition that the Mach number of the incoming flow is sufficiently large and the total variation of the pressure distribution on the cone is sufficiently small. To this end, we first develop a modified Glimm-type scheme to construct approximate solutions by self-similar solutions as building blocks to balance the influence of the geometric source term. Then we define a Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions. Meanwhile, the approximate generating curves of the cone surface are also constructed. Next, when the Mach number of the incoming flow is sufficiently large, by asymptotic analysis of the reflection coefficients in those interaction estimates, we prove that appropriate weights can be chosen so that the corresponding Glimm-type functional decreases in the flow direction. Finally, we determine the generating curves of the cone surface and establish the existence of global entropy solutions containing a strong leading conical shock, besides weak waves. Moreover, the entropy solution is proved to approach asymptotically the self-similar solution determined by the incoming flow and the asymptotic pressure on the cone surface at infinity.

We study the quantitative pointwise behavior of solutions to the Boltzmann equation for hard potentials and Maxwellian molecules, which generalize the hard sphere case introduced by Liu and Yu (Commun Pure Appl Math 57:1543–1608, 2004). The large time behavior of the solution is dominated by fluid structures, similar to the hard sphere case (Liu and Yu in Commun Pure Appl Math 57:1543–1608, 2004; Liu and Yu in Bull Inst Math Acad Sin (N.S.) 6:151–243, 2011). However, unlike the hard sphere case, the spatial decay here depends on the potential power (gamma ) and the initial velocity weight. A key challenge in this problem is the loss of velocity weight in linear estimates, which makes standard nonlinear iteration infeasible. To address this, we develop an Enhanced Mixture Lemma, demonstrating that mixing the transport and gain part of the linearized collision operator can generate arbitrary order regularity and decay in both space and velocity variables. This allows us to decompose the linearized solution into fluid (arbitrary regularity and velocity decay) and particle (rapid space-time decay, but with loss of velocity decay) parts, making it possible to solve the nonlinear problem through this particle-fluid duality.

This paper analyzes weak solutions of the quantum hydrodynamics (QHD) system with a collisional term posed on the one-dimensional torus. The main goal of our analysis is to rigorously prove the time-relaxation limit towards solutions to the quantum drift-diffusion (QDD) equation. The existence of global in-time, finite energy weak solutions can be proved by straightforwardly exploiting the polar factorization and wave function lifting tools previously developed by the authors. However, the sole energy bounds are not sufficient to show compactness and then pass to the limit. For this reason, we consider a class of more regular weak solutions (termed GCP solutions), determined by the finiteness of a functional involving the chemical potential associated with the system. For solutions in this class and bounded away from vacuum, we prove the time-relaxation limit and provide an explicit convergence rate. Our analysis exploits compactness tools and does not require the existence (and smoothness) of solutions to the limiting equations or the well-preparedness of the initial data. As a by-product of our analysis, we also establish the existence of global in time (H^2) solutions to a nonlinear Schrödinger–Langevin equation and construct solutions to the QDD equation as strong limits of GCP solutions to the QHD system.