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

Abstract

<abstract><p>We prove the well-posedness of a Cauchy problem of the kind:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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. $\end{document} </tex-math></disp-formula></p> <p>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</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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 $\end{document} </tex-math></disp-formula></p> <p>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:</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \{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) $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>