Several kinds of spectral quantities associated with semigroup generators are involved in the problem of the return to the equilibrium for parabolic or hypoelliptic type linear evolution equations: the numerical range, the spectrum and the pseudo-spectrum (or ǫspectrum). The distinction between the three spectral objects becomes crucial when the generator is a parameter-dependent differential o...

We establish new regularity estimates, in terms of Sobolev spaces, of the solution f to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and f is assumed to have a certain amount of regularity in velocity. The result is that f is also regular in time and space, and this is related to a commutator identity introduced ...

The Heisenberg group Hn is apparently one of the meeting point of various mathematical and scientifical domains. This can be partially explained by the fact that the first Heisenberg group appears as the less elaborated Lie group after the Euclidean spaces. It includes only a few non-commutativity and its Lie algebra presents a unique non-trivial relation (that can be interpreted in quantic phy...

We prove a sharp Gevrey hypoellipticity for the operator $$\begin{aligned} D_{x}^{2}+\left( x^{2n+1}D_{y}\right) ^{2}+\left( x^{n}y^{m}D_{y}\right) ^{2}, \end{aligned}$$ in $$\Omega $$ open neighborhood of origin $${\mathbb {R}}^{2}$$ , where n and m are positive integers. The is non trivial generalization Métivier studied (C R Acad Sci Paris 292:401–404, 1981). However it has symplectic charac...

Ž Ž . Nq 1. q z s x, t g R , where the principal part is uniformly elliptic on R , q F N, and the constant matrix B is upper triangular and such that the operator obtained by freezing the coefficients a at any point z g R Nq 1 is hypoelliptic. We prove i j 0 p Ž . local L -estimates for the derivatives ­ u i, j s 1, . . . , q of a solution to the x x i j equation Lu s f , under the assumption t...

In the paper, for Cauchy problem on non-cutoff Boltzmann equation in torus, we establish global-in-time Gevrey smoothness velocity and space variables a class of low-regularity mild solutions near Maxwellians with index depending only angular singularity. This together [24] provides self-contained well-posedness theory both existence regularity global initial data low framework perturbations. F...

Let $G_1$ and $G_2$ be compact Lie groups, $X_1 \in \mathfrak{g}_1$, $X_2 \mathfrak{g}_2$ consider the operator \begin{equation*} L_{aq} = X_1 + a(x_1)X_2 q(x_1,x_2), \end{equation*} where $a$ $q$ are ultradifferentiable functions in sense of Komatsu, is real-valued. We characterize completely global hypoellipticity solvability $L_{aq}$ Komatsu. For this, we present a conjugation between consta...