Prescribing analytic singularities for solutions of a class of vector fields on the torus

Abstract

We consider the operator $L=\partial _t+(a(t)+ib(t))\partial _x$ acting on distributions on the two-torus $\mathbb T^2,$ where $a$ and $b$ are real-valued, real analytic functions defined on the unit circle $\mathbb T^1.$ We prove, among other things, that when $b$ changes sign, given any subset $\Sigma$ of the set of the local extrema of the local primitives of $b,$ there exists a singular solution of $L$ such that the $t-$projection of its analytic singular support is $\Sigma ;$ furthermore, for any $\tau \in \Sigma$ and any closed subset $F$ of $\mathbb T^1_x$ there exists $u\in \mathcal D’(\mathbb T^2)$ such that $Lu\in C^\omega (\mathbb T^2)$ and $\operatorname {sing supp_A}(u)=\{\tau \}\times F.$ We also provide a microlocal result concerning the trace of $u$ at $t=\tau .$