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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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##### References:
 [1] Albanese, A. A.; Popivanov, P., On the global solvability in Gevrey classes on the n-dimensional torus, J. Math. Anal. Appl., 297, 659-672, (2004) · Zbl 1058.35055 [2] Albanese, A. A.; Popivanov, P., Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions, Ann. Mat. Pura Appl., 185, 3, 395-409, (2007) · Zbl 1232.35035 [3] Albanese, A. A.; Zanghirati, L., Global hypoellipticity and global solvability in Gevrey classes on the n-dimensional torus, J. Differential Equations, 199, 256-268, (2004) · Zbl 1063.35059 [4] Bergamasco, A., Perturbations of globally hypoelliptic operators, J. Differential Equations, 114, 513-526, (1994) · Zbl 0815.35009 [5] Bergamasco, A., Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc., 351, 4113-4126, (1999) · Zbl 0932.35046 [6] Bergamasco, A.; Cordaro, P.; Petronilho, G., Global solvability for a class of complex vector fields on the two-torus, Comm. Partial Differential Equations, 29, 785-819, (2004) · Zbl 1065.35088 [7] Bergamasco, A.; Dattori da Silva, P. L., Solvability in the large for a class of vector fields on the torus, J. Math. Pures Appl., 86, 427-447, (2006) · Zbl 1157.35304 [8] Bergamasco, A. P.; Dattori da Silva, P. L.; Ebert, M. R., Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type, J. Differential Equations, 246, 4, 1673-1702, (2009) · Zbl 1173.35300 [9] Bergamasco, A.; Dattori da Silva, P. L.; Gonzalez, R., Existence and regularity of solutions to certain first-order partial differential operators on the torus, J. Fourier Anal. Appl., 23, 65-90, (2017) · Zbl 1357.35020 [10] Bergamasco, A.; Dattori da Silva, P. L.; Gonzalez, R.; Kirilov, A., Global solvability and global hypoellipticity for a class of complex vector fields on the 3-torus, J. Pseudo-Differ. Oper. Appl., 6, 341-360, (2015) · Zbl 1336.35124 [11] Dattori da Silva, P. L.; Fronza da Silva, M., Gevrey global solvability of non-singular real first-order differential operators, Ann. Mat. Pura Appl., 192, 245-253, (2013) · Zbl 1263.35065 [12] Gramchev, T.; Popivanov, P.; Yoshino, M., Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients, Proc. Japan Acad., 68, 53-57, (1992) · Zbl 0805.35023 [13] Greenfield, S.; Wallach, N., Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc., 31, 112-114, (1972) · Zbl 0229.35023 [14] Hörmander, L., Linear partial differential operators, (1963), Springer Berlin · Zbl 0171.06802 [15] Hounie, J., Globally hypoelliptic and globally solvable first order evolutions equations, Trans. Amer. Math. Soc., 252, 233-248, (1979) · Zbl 0424.35030 [16] Petronilho, G., Global s-solvability, global s-hypoellipticity and Diophantine phenomena, Indag. Math. (N.S.), 16, 1, 67-90, (2005) · Zbl 1065.35100 [17] Petronilho, G., On Gevrey solvability and regularity, Math. Nachr., 282, 3, 470-481, (2009) · Zbl 1172.35365 [18] Rodino, L., Linear partial differential operators in Gevrey spaces, (1993), World Scientific Publishing Co. Pte. Ltd. · Zbl 0869.35005
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