Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus
type: Item
instance of: scholarly article
title: Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus
zbMATH Open document ID: 1395.35074
published in: Journal of Differential Equations
publication date: 2018-01-11T00:00:00Z
review text: Let \(L={\partial\over\partial t}+ \sum^N_{j=1} (a_j+ ib_j)(t){\partial\over\partial x_j}\) be a vector field defined on \(\mathbb T^{N+1}\simeq \mathbb R^{N+2}/2\pi \mathbb Z^{N+1}\), where \(a_j\), \(b_j\) are real-valued functions and belonging to the Gevrey class \(G^s(\mathbb T^1)\), \(s> 1\), for \(j= 1,\dots, N\). The author presents a complete characterization for the \(s\)-global solvability and \(s\)-global hypoellipticity of \(L\). His results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets. For \(s>1\) introduce the following exponential Diophantine conditions for a pair \((\alpha,\beta)\in \mathbb R^N\times\mathbb R^N\): \((EDC)^s_1\): For each \(\varepsilon> 0\) there exists a positive constant \(C_\varepsilon\) such that \[ |\tau+\langle\xi,\alpha+i\beta\rangle|\geq C_\varepsilon\exp\{-\varepsilon(|\xi|+ |\tau|)^{1/s}\}, \] for all \((\xi,\tau)= (\xi_1,\dots,\xi_N,\tau)\in \mathbb Z^{N+1}\setminus\{0\}\). \((EDC)^s_2\): For each \(\varepsilon>0\) there exists a positive constant \(C_\varepsilon\) such that \[ |\tau+\langle\xi,\alpha+i\beta\rangle|\geq C_\varepsilon\exp\{-\varepsilon|\xi|+ |\tau|)^{1/s}\}, \] for all \((\xi,\tau)= (\xi_1,\dots, \xi_N,\tau)\in \mathbb Z^{N+1}\) such that \(\tau+\langle\xi,\alpha+\beta\rangle\neq 0\). Define \(\alpha= (a_1,\dots, a_N),\quad \beta=(\beta_1,\dots, \beta_N)\), \(a_{j,0}= {1\over 2\pi} \int^{2\pi}_0 a_j(t)\,dt,\quad b_{j0}= {1\over 2\pi} \int^{2\pi}_0 a_j(t)\,dt\), \(\alpha_0= (a_{10},\dots, a_{N0})\), \(\beta_0= (b_{10},\dots, b_{N0})\). The author has obtained the following two theorems. Theorem 1. Let \(L\) be given by above. Then, \(L\) is \(s\)-global solvable \((s>1)\) if and only if one of the following situations occurs: {\parindent=1cm \begin{itemize}\item[(I)] For each \(j=1,\dots,N\), \(b_j\) vanishes identically, and \((\alpha_0, 0)\) satisfies \((EDC)^s_2\). \item[(II)] At least one \(b_j\) dones not vanishes identically, \(b_{k0}=0\) fo…
reviewed by: Kunihiko Kajitani
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6824720
zbMATH Keywords: Gevrey solvability
zbMATH Keywords: Gevrey hypoellipticity
zbMATH Keywords: Fourier series
zbMATH Keywords: Diophantine properties of the coefficients
author: Adalberto P. Bergamasco
author: Paulo L. Dattori da Silva
author: Rafael Borro Gonzalez
MaRDI profile type: Publication
full work available at URL: j.jde.2017.11.022
OpenAlex ID: W2772329677
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DOI: 10.1016/J.JDE.2017.11.022
description: scientific article; zbMATH DE number 6824720