Stability for Cauchy problem of first order linear PDEs on $ \mathbb{T}^m $ with forced frequency possessing finite uniform Diophantine exponent

Abstract

In this paper, we studied the stability of the Cauchy problem for a class of first-order linear quasi-periodically forced PDEs on the $ m $-dimensional torus: $ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+(\xi+f(x, \omega t, \xi))\cdot \partial_x u = 0, \\ u(x, 0) = u_0(x), \ \end{array} \right. \end{eqnarray*} $ where $ \xi\in \mathbb{R}^m, x\in \mathbb{T}^m, \omega\in\mathbb{R}^d, $ in the case of multidimensional Liouvillean forced frequency. We proved that for each compact set $ \mathcal{O}\in\mathbb{R}^m $ there exists a Cantor subset $ \mathcal{O}_\gamma $ of $ \mathcal{O} $ with positive Lebesgue measure such that if $ \xi\in\mathcal{O}_\gamma, $ then for a perturbation $ f $ being small in some analytic Sobolev norm, there exists a bounded and invertible quasi-periodic family of linear operator $ \Psi(\omega t) $, such that the above PDEs are reduced by the transformation $ v: = \Psi(\omega t)^{-1}[u] $ into the following PDE: $ \begin{eqnarray*} \partial_t v+ (\xi+ m_\infty(\omega t))\cdot\partial_x v = 0, \end{eqnarray*} $ provided that the forced frequency $ \omega\in \mathbb{R}^d $ possesses finite uniform Diophantine exponent, which allows Liouvillean frequency. The reducibility can immediately cause the stability of the above Cauchy problem, that is, the analytic Sobolev norms of the Cauchy problem are controlled uniformly in time. The proof is based on a finite dimensional Kolmogorov-Arnold-Moser (KAM) theory for quasi-periodically forced linear vector fields with multidimensional Liouvillean forced frequency. As we know, the results on Liouvillean frequency existing in the literature deal with two-dimensional frequency and exploit the theory of continued fractions to control the small divisor problem. The results in this paper partially extend the analysis to higher-dimensional frequency and impose a weak nonresonance condition, i.e., the forced frequency $ \omega $ possesses finite uniform Diophantine exponent. Our result can be regarded as a generalization of analytic cases in the work [R. Feola, F. Giuliani, R. Montalto and M, Procesi, Reducibility of first order linear operators on tori via Moser's theorem, J. Funct. Anal., 2019] from Diophantine frequency to Liouvillean frequency.

Keywords

• rotations reducibility
• stability of cauchy-problem
• hyperbolic pdes
• liouvillean frequency
• uniform diophantine exponent