相关论文: A remark on norm inflation for nonlinear wave equa…
In this paper, we prove that the Cauchy problem for a generalized Camassa-Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space $B^1_{\infty,1}(\R)$. It is shown in (J. Differ. Equ., 327:127-144,2022) that…
Space-based missions studying the cosmic microwave background (CMB) have progressively refined the parameter space in conventional models of inflation shortly ($\sim 10^{-37}$ seconds) after the big bang. While most inflationary scenarios…
The cosmological observations of cosmic microwave background and large-scale structure indicate that our universe has a nearly scaling invariant power spectrum of the primordial perturbation. However, the exact origin for this primordial…
We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy…
Primordial quantum fluctuations produced by inflation are conventionally assumed to be statistically homogeneous, a consequence of translational invariance. In this paper we quantify the potentially observable effects of a small violation…
The existence of the inflationary era in the early Universe seems to be strongly supported by recent CMB observations. However, only a few realistic inflation scenarios which have close relation to particle physics seem to have been known…
There has been much recent discussion, and some confusion, regarding the use of existing observational data to estimate the likelihood that next-generation cosmic microwave background (CMB) polarization experiments might detect a nonzero…
We study the focusing inhomogeneous nonlinear Schr\"odinger equation $$ i\partial_t u + \Delta u = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ with $b>0$ and $p>1$. Due to the spatial growth of the nonlinearity,…
We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven…
This proceedings contribution provides an overview of Loop Blow-up Inflation and updates its observational predictions and their comparison with the latest CMB and BAO data from combined analyses of SPT, Planck, ACT, and BICEP/Keck, as well…
We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated…
This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any…
Nonlinear electrodynamics with two dimensional parameters is studied. The range of electromagnetic fields when principles of causality, unitarity and the classical stability hold, are obtained. A singularity of the electric field at the…
We prove that the 2D Euler equations are not locally well-posed in $C^1$. Our approach relies on the technique of Lagrangian deformations and norm inflation of Bourgain and Li. We show that the assumption that the data-to-solution map is…
We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy…
The null energy condition (NEC) is a cornerstone of general relativity, and its violation could leave observable imprints in the cosmic gravitational wave spectrum. Theoretical models suggest that NEC violations during inflation can amplify…
The apparent alignment of the cosmic microwave background multipoles on large scales challenges the standard cosmological model. Scalar field inflation is isotropic and cannot account for the observed alignment. We explore the imprints, a…
Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree…
The flat directions LLe and udd within the minimal supersymmetric Standard Model provide all the necessary ingredients for a successful inflation with the right amplitude of the scalar density perturbations, negligible gravity waves and the…
The Wheeler-DeWitt equation provides the probability distribution for the curvature perturbation, the gauge invariant quantum fluctuation of the inflaton. From this, we can find a tower of power spectra which is not found in a perturbative…