On the cubic Dirac equation with potential and the Lochak--Majorana condition

We study a cubic Dirac equation on $\mathbb{R}\times\mathbb{R}^{3}$ \begin{equation*} i \partial _t u + \mathcal{D} u + V(x) u = \langle \beta u,u \rangle \beta u \end{equation*} perturbed by a large potential with almost critical regularity. We prove global existence and scattering for small initial data in $H^{1}$ with additional angular regularity. The main tool is an endpoint Strichartz estimate for the perturbed Dirac flow. In particular, the result covers the case of spherically symmetric data with small $H^{1}$ norm. When the potential $V$ has a suitable structure, we prove global existence and scattering for \emph{large} initial data having a small chiral component, related to the Lochak--Majorana condition.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1706.0484

On the ¥Phi33¥Phi^3_3-measure (Nonlinear and Random Waves)

This is a résumé of the construction part of the $¥Phi^3_3$-measure of the paper “Stochastic quantization of the $¥Phi^3_3$-model” by Oh, Tolomeo, and the author. In this note, we give an outline of the proof of the normalizability of the $¥Phi^3_3$-measure in the weakly nonlinear regime

On the stochastic nonlinear Schrödinger equations at critical regularities.

We consider the Cauchy problem for the defocusing stochastic nonlinear Schr\"odinger equations (SNLS) with an additive noise in the mass-critical and energy-critical settings. By adapting the probabilistic perturbation argument employed in the context of the random data Cauchy theory by the first author with B\'enyi and Pocovnicu (2015) to the current stochastic PDE setting, we present a concise argument to establish global well-posedness of the mass-critical and energy-critical SNLS.Comment: 23 pages. Appendix A added. Published in Stoch. Partial Differ. Equ. Anal. Compu

The Anti-Allergic Effects of the His-Ala-Gln Tripeptide and Constituent Amino Acids

The type‐1 allergy, as typified by allergic rhinitis, pollen, and food allergies, is defined as a hypersensitivity reaction, and its frequency is increasing worldwide. There is a need to develop therapeutic agents that either prevent sensitization to allergens or suppress the allergic response after initiation. It has been reported that various peptides show anti‐allergic effects, but there have been few reports concerning peptides derived from food. Previously, we studied the anti‐allergic effect of His‐Ala‐Gln (HAQ), which is present in CE90GMM, a peptide mixture derived from milk casein. In this chapter, special emphasis is placed on the anti‐allergic effects of the HAQ peptide in vitro and in vivo, and the effect of peptide binding, peptide sequence, number of amino acids, and the electron density of the amino acids is investigated

A remark on norm inflation for nonlinear wave equations

In this note, we study the ill-posedness of nonlinear wave equations (NLW). Namely, we show that NLW experiences norm inflation at every initial data in negative Sobolev spaces. This result covers a gap left open in a paper of Christ, Colliander, and Tao (2003) and extends the result by Oh, Tzvetkov, and the second author (2019) to non-cubic integer nonlinearities. In particular, for some low dimensional cases, we obtain norm inflation above the scaling critical regularity. We also prove ill-posedness for NLW, via norm inflation at general initial data, in negative regularity Fourier-Lebesgue and Fourier-amalgam spaces.Comment: 20 pages. Published in Dyn. Partial Differ. Eq