On Kolmogorov Fokker Planck operators with linear drift and time dependent measurable coefficients

Abstract

We prove the well-posedness of a Cauchy problem of the kind: $ \left\{\begin{array}{@{}l@{}c} \mathcal{L}u = f, & \text{ in }D'(\mathbb{R}^N\times(0,+\infty)),\\ u(x,0) = g(x),&\forall x\in\mathbb{R}^N, \end{array}\right. $ where $ f $ is Dini continuous in space and measurable in time and $ g $ satisfies suitable regularity properties. The operator $ \mathcal{L} $ is the degenerate Kolmogorov-Fokker-Planck operator $ \mathcal{L} = \sum\limits_{i,j = 1}^q a_{ij}(t)\partial_{x_ix_j}^2+ \sum\limits_{k,j = 1}^N b_{kj}x_k\partial_{x_j}-\partial_t $ where $ \{a_{ij}\}_{ij = 1}^q $ is measurable in time, uniformly positive definite and bounded while $ \{b_{ij}\}_{ij = 1}^N $ have the block structure: $ \{b_{ij}\}_{ij = 1}^N = \left( \begin{matrix}{} \mathbb{O} & \dots & \mathbb{O} & \mathbb{O} \\ \mathbb{B}_1 & \dots & \mathbb{O} & \mathbb{O} \\ \vdots & \ddots& \vdots & \vdots \\ \mathbb{O} & \dots & \mathbb{B}_\kappa & \mathbb{O} \end{matrix} \right) $ which makes the operator with constant coefficients hypoelliptic, 2-homogeneous with respect to a family of dilations and traslation invariant with respect to a Lie group.

Keywords

• kolmogorov-fokker-planck operators
• nonhomogeneous cauchy problem
• duhamel method
• dini continuity
• measurable time dependent coefficients