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Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus. (English) Zbl 1395.35074
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:
(I)
For each \(j=1,\dots,N\), \(b_j\) vanishes identically, and \((\alpha_0, 0)\) satisfies \((EDC)^s_2\).
(II)
At least one \(b_j\) dones not vanishes identically, \(b_{k0}=0\) for each \(k= 1,\dots, N\), \(\alpha_0\in \mathbb Z^N\), and the sublevel sets \[ \Biggl\{t\in \mathbb T^1;\,\int^t_0 (\xi,\beta(\tau))\,d\tau< r\Biggr\},\;r\in \mathbb R,\;\xi\in \mathbb Z^N, \] are connected.
(III)
\(b_{j0}\neq 0\) for at least one \(j\in\{1,\dots,N\}\), and the following conditions hold:
(III.1)
dim span\(\{b_1,\dots,b_N\}=1\);
(III.2)
the functions \(b_j\) do not change sign;
(III.3)
the pair \((\alpha_0,\beta_0)\) satisfies \((EDC)^s_2\).

Theorem 2. Let \(L\) be given by above. Then, \(L\) is \(s\)-global hypoelliptic \((s>1)\) if and only if the following conditions are satisfied:
(1)
each \(b_j\) does not change sign;
(2)
dim span\(\{b_1,\dots,b_N\}\leq 1\);
(3)
\((\alpha_0,\beta_0)\) satisfies \((EDC)^s_1\).

MSC:
35H10 Hypoelliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B10 Periodic solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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