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Global hypoellipticity and solvability for a class of evolution operators in time-periodic weighted Sobolev spaces

Fernando de Ávila Silva, Matteo Bonino, Sandro Coriasco

TL;DR

This work develops a framework of Gevrey time-periodic weighted Sobolev spaces on $\\mathbb{T}^n\times\mathbb{R}^d$ to study global hypoellipticity and solvability of time-periodic evolution operators with polynomially growing coefficients. The authors leverage SG-calculus and eigenfunction expansions of elliptic SG-operators to characterize function spaces and to reduce partial differential equations to decoupled ODE systems for each eigenmode. They establish precise criteria: global hypoellipticity holds if either the imaginary part of the frequency is nonzero or the real part satisfies a Diophantine condition (Condition (A)); global solvability holds if either the imaginary part is nonzero or Condition (B) is satisfied. The results connect spectral properties, periodicity, and ultradifferentiable regularity to provide sharp, implementable conditions for regularity and solvability of a broad class of time-periodic evolutions in unbounded settings. This has implications for understanding the propagation of regularity in non-compact domains under periodic forcing and for broader applications of SG-calculus in evolution equations.

Abstract

We study the hypoellipticity and solvability properties of a class of time-periodic evolution operators, with coefficients globally defined on $\mathbb{R}^d$ and growing polynomially with respect to the space variable. To this aim, we introduce a class of time-periodic weighted Sobolev spaces, whose elements are characterised in terms of suitable Fourier expansions, associated with elliptic operators.

Global hypoellipticity and solvability for a class of evolution operators in time-periodic weighted Sobolev spaces

TL;DR

This work develops a framework of Gevrey time-periodic weighted Sobolev spaces on to study global hypoellipticity and solvability of time-periodic evolution operators with polynomially growing coefficients. The authors leverage SG-calculus and eigenfunction expansions of elliptic SG-operators to characterize function spaces and to reduce partial differential equations to decoupled ODE systems for each eigenmode. They establish precise criteria: global hypoellipticity holds if either the imaginary part of the frequency is nonzero or the real part satisfies a Diophantine condition (Condition (A)); global solvability holds if either the imaginary part is nonzero or Condition (B) is satisfied. The results connect spectral properties, periodicity, and ultradifferentiable regularity to provide sharp, implementable conditions for regularity and solvability of a broad class of time-periodic evolutions in unbounded settings. This has implications for understanding the propagation of regularity in non-compact domains under periodic forcing and for broader applications of SG-calculus in evolution equations.

Abstract

We study the hypoellipticity and solvability properties of a class of time-periodic evolution operators, with coefficients globally defined on and growing polynomially with respect to the space variable. To this aim, we introduce a class of time-periodic weighted Sobolev spaces, whose elements are characterised in terms of suitable Fourier expansions, associated with elliptic operators.
Paper Structure (14 sections, 28 theorems, 202 equations)

This paper contains 14 sections, 28 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Eigenfunction expansions in weighted Sobolev spaces on $\mathbb{R}^d$
  3. Gevrey-periodic weighted Sobolev spaces on $\mathbb{T}^n\times\mathbb{R}^d$
  4. Gevrey classes on the torus
  5. Inductive limits of periodic Gevrey-Sobolev-Kato-type Banach spaces and their duals
  6. The spaces $\mathscr{F}=\mathscr{F}(\mathbb T ^n\times \mathbb{R}^d)$ and $\mathscr{F}'=(\mathscr{F}(\mathbb T ^n\times \mathbb{R}^d))'$
  7. Eigenfunctions expansions in $\mathscr{F}$
  8. Eigenfunctions expansions in $\mathscr{F}'$
  9. Periodic evolution equations and hypoellipticity on $\mathbb{T}\times\mathbb{R}^d$
  10. Periodic evolution equations and solvability on $\mathbb{T}\times\mathbb{R}^d$
  11. The calculus of SG pseudo-differential operators
  12. Eigenvalues of elliptic SG-operators
  13. Eigenvalues asymptotics
  14. Subsequences of the eigenvalues sequence $\{\lambda_j\}_{j\in\mathbb{N}^\ast}$

Key Result

Proposition 2.1

Let $P \in \mathrm{Op}(S^{m,\mu}(\mathbb{R}^d))$ be an elliptic, normal SG-operator with order components $m,\mu>0$. Denote by $P_0$ the orthogonal projection on $\ker P$ and let $\widetilde{P}=P+P_0$. Then, and

Theorems & Definitions (64)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 54 more