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On a compact manifold , we consider the affine space of non-self-adjoint perturbations of some invertible elliptic operator acting on sections of some Hermitian bundle by some differential operator of lower order.
We construct and classify all complex-analytic functions on the Fréchet space vanishing exactly over noninvertible elements, having minimal growth at infinity along complex rays in and which are obtained by local renormalization, a concept coming from quantum field theory, called renormalized determinants. The additive group of local polynomial functionals of finite degrees acts freely and transitively on the space of renormalized determinants. We provide different representations of the renormalized determinants in terms of spectral zeta-determinants, Gaussian free fields, infinite products and renormalized Feynman amplitudes in perturbation theory in position space à la Epstein–Glaser.
Specializing to the case of Dirac operators coupled to vector potentials and reformulating our results in terms of determinant line bundles, we prove our renormalized determinants define some complex-analytic trivializations of some holomorphic line bundle over . This relates our results to a conjectural picture from some unpublished notes by Quillen from April 1989.
For incomplete sub-Riemannian manifolds and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric.
This paper is devoted to a systematic study of certain geometric integral inequalities which arise in continuum combinatorial approaches to -improving inequalities for Radon-like transforms over polynomial submanifolds of intermediate dimension. The desired inequalities relate to and extend a number of important results in geometric measure theory.
We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and we obtain results on multipliers and left compositions in this setting.
We reveal new classes of solutions to hydrodynamic Euler alignment systems governing collective behavior of flocks. The solutions describe unidirectional parallel motion of agents and are globally well-posed in multidimensional settings subject to a threshold condition similar to the one-dimensional case. We develop the flocking and stability theory of these solutions and show long-time convergence to a traveling wave with rapidly aligned velocity field.
In the context of multiscale models introduced by Shvydkoy and Tadmor (Multiscale Model. Simul.19:2 (2021), 1115–1141) our solutions can be superimposed into Mikado formations — clusters of unidirectional flocks pointing in various directions. Such formations exhibit multiscale alignment phenomena and resemble realistic behavior of interacting large flocks.
We show that two classically known properties of positive supersolutions of uniformly elliptic PDEs, the boundary point principle (Hopf lemma) and global integrability, can be quantified with respect to each other. We obtain an extension up to the boundary of the De Giorgi–Moser weak Harnack inequality, optimal with respect to the norms involved, for equations in divergence form.
We introduce a truncated version of the cubic Szegő equation, an integrable model for deterministic turbulence. In this truncation, a majority of the Fourier mode couplings are eliminated, while the signature features of the model are preserved, namely, a Lax pair structure and a hierarchy of finite-dimensional dynamically invariant manifolds. Despite the impoverished structure of the interactions, the turbulent behaviors of our new equation are stronger in an appropriate sense than for the original cubic Szegő equation. We construct explicit analytic solutions displaying exponential growth of Sobolev norms. We furthermore introduce a family of models that interpolate between our truncated system and the original cubic Szegő equation, along with other related deformations. These models possess Lax pairs and invariant manifolds, and display a variety of turbulent cascades. We additionally mention numerical evidence, in some related systems, for an even stronger type of turbulence in the form of a finite-time blow-up.
We study the asymptotic decay properties for defocusing semilinear wave equations in with pure power nonlinearity. By applying new vector fields to the null hyperplane, we derive improved time decay of the potential energy, with a consequence that the solution scatters both in the critical Sobolev space and energy space for all . Moreover, combined with Brezis–Gallouet–Wainger-type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate when and with rate for all . This in particular implies that the solution scatters in energy space when .
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