Communications on Pure and Applied Mathematics最新文献

We establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well-posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD.

We study the (characteristic) Cauchy problem for the Maxwell-Bloch equations of light-matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically-motivated initial-boundary conditions are satisfied. In particular, we present a proper Riemann-Hilbert problem that generates the unique causal solution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading-order asymptotics to self-similar solutions that satisfy a system of ordinary differential equations related to the Painlevé-III (PIII) equation. We identify these solutions and show that they are related to a family of PIII solutions recently discovered in connection with several limiting processes involving the focusing nonlinear Schrödinger equation. We fully explain a resulting boundary layer phenomenon in which, even for smooth initial data (an incident pulse), the solution makes a sudden transition over an infinitesimally small propagation distance. At a formal level, this phenomenon has been described by other authors in terms of the PIII self-similar solutions. We make this observation precise and for the first time we relate the PIII self-similar solutions to the Cauchy problem. Our analysis of the asymptotic behavior satisfied by the optical field and medium density matrix reveals slow decay of the optical field in one direction that is actually inconsistent with the simplest version of scattering theory. Our results identify a precise generic condition on an optical pulse incident on an initially-unstable medium sufficient for the pulse to stimulate the decay of the medium to its stable state.

The Theorem of Bonnet–Myers implies that manifolds with topology $��M^{�n�-�1�}�\times �S^1�$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $�T^n$ does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so-called m-intermediate curvature), and use stable weighted slicings to show that for $��n�\le �7�$ and $��1�\le �m�\le �n�-�1�$ the manifolds $��N^n�=�M^{�n�-�m�}�\times �T^m�$ do not admit a metric of positive m-intermediate curvature.

Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topological K-theory of finite dimensional simplicial complexes.

Let X be a compact, Kähler, Calabi-Yau threefold and suppose $��X�\mapsto �\underline{X}�⇝�X_t�$ , for $��t�\in �\Delta �$, is a conifold transition obtained by contracting finitely many disjoint $��(�-�1�,�-�1�)�$ curves in X and then smoothing the resulting ordinary double point singularities. We show that, for $��\left|�t�\right|�\ll �1�$ sufficiently small, the tangent bundle $��T^{�1�,�0�}�X_t�$ admits a Hermitian-Yang-Mills metric $�H_t$ with respect to the conformally balanced metrics constructed by Fu-Li-Yau. Furthermore, we describe the behavior of $�H_t$ near the vanishing cycles of $�X_t$

Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure-preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure-preserving), which can achieve high-order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high-dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046-dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >10⁵ that has a 295,122-dimensional state space.

We study the question of the integrability of Einstein deformations and relate it to the question of the desingularization of Einstein metrics. Our main application is a negative answer to the long-standing question of whether or not every Einstein 4-orbifold (which is an Einstein metric space in a synthetic sense) is limit of smooth Einstein 4-manifolds. We more precisely show that spherical and hyperbolic 4-orbifolds with the simplest singularities cannot be Gromov-Hausdorff limits of smooth Einstein 4-metrics without relying on previous integrability assumptions. For this, we analyze the integrability of deformations of Ricci-flat ALE metrics through variations of Schoen's Pohozaev identity. Inspired by Taub's conserved quantity in General Relativity, we also introduce conserved integral quantities based on the symmetries of Einstein metrics. These quantities are obstructions to the integrability of infinitesimal Einstein deformations “closing up” inside a hypersurface – even with change of topology. We show that many previously identified obstructions to the desingularization of Einstein 4-metrics are equivalent to these quantities on Ricci-flat cones. In particular, all of the obstructions to desingularizations bubbling off Eguchi-Hanson metrics are recovered. This lets us further interpret the obstructions to the desingularization of Einstein metrics as a defect of integrability.

Our goal in this work is to better understand the relationship between replica symmetry breaking, shattering, and metastability. To this end, we study the static and dynamic behaviour of spherical pure p-spin glasses above the replica symmetry breaking temperature $�T_s$. In this regime, we find that there are at least two distinct temperatures related to non-trivial behaviour. First we prove that there is a regime of temperatures in which the spherical p-spin model exhibits a shattering phase. Our results holds in a regime above but near $�T_s$. We then find that metastable states exist up to an even higher temperature $�T_{�B�B�M�}$ as predicted by Barrat–Burioni–Mézard which is expected to be higher than the phase boundary for the shattering phase . We develop this work by first developing a Thouless–Anderson–Palmer decomposition which builds on the work of Subag. We then present a series of questions and conjectures regarding the sharp phase boundaries for shattering and slow mixing.

We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three-dimensional incompressible Euler equations. The flow is axi-symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting introduced by A. Friedman and B. Turkington (Trans. Amer. Math. Soc., 1981), which produced a maximizer of the kinetic energy under constraints on vortex strength, impulse, and circulation. We match the set of maximizers with the Hill's vortex via the uniqueness result of C. Amick and L. Fraenkel (Arch. Rational Mech. Anal., 1986). The matching process is done by an approximation near exceptional points (so-called metrical boundary points) of the vortex core. As a consequence, the stability up to a translation is obtained by using a concentrated compactness method.