Analysis & PDE最新文献

We consider a system of quasilinear wave equations on the product space ${ℝ}^{1+3}�\times {\mathbb{𝕊}}^1$, which we want to see as a toy model for the Einstein equations with additional compact dimensions. We show global existence of solutions for small and regular initial data with polynomial decay at infinity. The method combines energy estimates on hyperboloids inside the light cone and weighted energy estimates outside the light cone.

This article introduces $L^p$ versions of the support function of a convex body $K$ and associates to these canonical $L^p$-polar bodies $K^{\circ ,p}$ and Mahler volumes ${\mathcal{ℳ}}_p(K)$. Classical polarity is then seen as $L^{\infty }$-polarity. This one-parameter generalization of polarity leads to a generalization of the Mahler conjectures, with a subtle advantage over the original conjecture: conjectural uniqueness of extremizers for each $p�\in �(0,\infty )$. We settle the upper bound by demonstrating the existence and uniqueness of an $L^p$-Santaló point and an $L^p$-Santaló inequality for symmetric convex bodies. The proof uses Ball’s Brunn–Minkowski inequality for harmonic means, the classical Brunn–Minkowski inequality, symmetrization, and a systematic study of the ${\mathcal{ℳ}}_p$ functionals. Using our results on the $L^p$-Santaló point and a new observation motivated by complex geometry, we show how Bourgain’s slicing conjecture can be reduced to lower bounds on the $L^p$-Mahler volume coupled with a certain conjectural convexity property of the logarithm of the Monge–Ampère measure of the $L^p$-support function. We derive a suboptimal version of this convexity using Kobayashi’s theorem on the Ricci curvature of Bergman metrics to

We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in $L\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->L{(\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->\log <!--FUNCTION\; APPLICATION-->L)}^{1+𝜖}$ for any $𝜖�>0$.

We prove that a Radon measure $\mu$ on ${ℝ}^n$ can be written as $\mu �={\sum <!--FUNCTION\; APPLICATION-->�}_{i=0}^n{\mu }_i$, where each of the ${\mu }_i$ is an $i$-dimensional rectifiable measure if and only if, for every Lipschitz function $f�:{ℝ}^n�\to �ℝ$ and every $𝜀�>0$, there exists a function $g$ of class $C^1$ such that $\mu (\{x�\in {ℝ}^n�:�g(x)\ne f(x)\})�<�𝜀$.

We prove the uniqueness of several excited states to the ODE $ÿ(t)�+�(2∕t)ẏ(t)�+�f(y(t))�=0$, $y(0)�=�b$, and $ẏ(0)�=0$, for the model nonlinearity $f(y)�=y^3�-�y$. The $n$-th excited state is a solution with exactly $n$ zeros and which tends to $0$ as $t�\to \infty$. These represent all smooth radial nonzero solutions to the PDE $\mathrm{\Delta }u�+�f(u)�=0$ in $H^1$. We interpret the ODE as a damped oscillator governed by a double-well potential, and the result is proved via rigorous numerical analysis of the energy and variation of the solutions. More specifically, the problem of uniqueness can be formulated entirely in terms of inequalities on the solutions and their variation, and these inequalities can be verified numerically.

We prove existence results and lower bounds for the resonances of Schrödinger operators associated to smooth, compactly support potentials on hyperbolic space. The results are derived from a combination of heat and wave trace expansions and asymptotics of the scattering phase.

We revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L_t^1{W}^{1,p}$ for all $p�<�\infty$ in space dimensions $d�\ge 2$ whose transport equations admit nonunique weak solutions belonging to $L_t^p{C}^k$ for all $p�<�\infty$ and $k�\in �\mathbb{N}$. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna–Lions theory is essential. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions $d�\ge 2$.

Onsager’s conjecture states that the conservation of energy may fail for three-dimensional incompressible Euler flows with Hölder regularity below $\frac{1}{3}$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy nonconserving solutions to the three-dimensional incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents $[0,\frac{1}{3})$.

Our construction improves the author’s previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a convex integration scheme. A crucial point is to avoid loss of powers in frequency in the estimates of the iteration. This goal is achieved using localization techniques of Isett and Oh (Arch. Ration. Mech. Anal. 221:2 (2016), 725–804) to modify the convex integration scheme.

We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov–Obukhov scaling for any $p�>3$ if their singular supports have space-time Lebesgue measure zero.

Given a sequence of genus $g�\ge 2$ curves converging to a punctured Riemann surface with complete metric of constant Gaussian curvature $�-1$, we prove that the Kodaira embedding using an orthonormal basis of the Bergman space of sections of a pluricanonical bundle also converges to the embedding of the limit space together with extra complex projective lines.

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is nondegenerate. Under a general condition, the Laplacian acting on high tensor powers of the bundle exhibits gaps and clusters of eigenvalues. We prove that for each cluster the number of eigenvalues that it contains is given by a Riemann–Roch number. We also give a pointwise description of the Schwartz kernel of the spectral projectors onto the eigenstates of each cluster, similar to the Bergman kernel asymptotics of positive line bundles. Another result is that gaps and clusters also appear in local Weyl laws.