Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and $VMO$ in space
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914169498894336 |
|---|---|
| author | Biagi, Stefano Bramanti, Marco |
| author_facet | Biagi, Stefano Bramanti, Marco |
| contents | We consider Kolmogorov-Fokker-Planck operators of the form $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-\partial_{t}u, $$ with $\left( x,t\right) \in\mathbb{R}^{N+1},N\geq q\geq1$. We assume that $a_{ij}\in L^{\infty}\left( \mathbb{R}^{N+1}\right) $, the matrix $\left\{ a_{ij}\right\} $ is symmetric and uniformly positive on $\mathbb{R}^{q}$, and the drift \[ Y=\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}-\partial_{t} \] has a structure which makes the model operator with constant $a_{ij}$ hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and $2$-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients $a_{ij}$ are $VMO$ w.r.t. the space variable, and only bounded measurable in $t$. We prove, for every $p\in\left( 1,\infty\right) $, global Sobolev estimates of the kind: \begin{align*} \Vert u\Vert _{W_{X}^{2,p}(S_{T})} \equiv & \sum_{i,j=1}^{q}\Vert u_{x_{i}x_{j}}\Vert_{L^{p}(S_{T})}
+\Vert Yu\Vert _{L^{p}(S_{T})} +\sum_{i=1}^{q}\Vert u_{x_{i}}\Vert _{L^{p}(S_{T})} +\Vert u\Vert _{L^{p}(S_{T})} \\ & \leq c\big\{ \Vert \mathcal{L}u\Vert _{L^{p}(S_{T})}+\Vert u\Vert_{L^{p}(S_{T})}\big\}
\end{align*} with $S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) $ for any $T\in(-\infty,+\infty]$. Also, the well-posedness in $W_{X}^{2,p}(Ω_{T})$, with $Ω_{T}=\mathbb{R}^{N}\times(0,T) $ and $T\in\mathbb{R}$, of the Cauchy problem% $$ \begin{cases} \mathcal{L}u=f & \text{in $Ω_{T}$} \\ u(\cdot,0) =g & \text{in $\mathbb{R}^{N}$} \end{cases} $$ is proved, for $f\in L^{p}(Ω_{T}), g\in W_{X}^{2,p}(\mathbb{R}^{N})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and $VMO$ in space Biagi, Stefano Bramanti, Marco Analysis of PDEs 35K65, 35K70, 35B45, 35A08, 42B20, 42B25 We consider Kolmogorov-Fokker-Planck operators of the form $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-\partial_{t}u, $$ with $\left( x,t\right) \in\mathbb{R}^{N+1},N\geq q\geq1$. We assume that $a_{ij}\in L^{\infty}\left( \mathbb{R}^{N+1}\right) $, the matrix $\left\{ a_{ij}\right\} $ is symmetric and uniformly positive on $\mathbb{R}^{q}$, and the drift \[ Y=\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}-\partial_{t} \] has a structure which makes the model operator with constant $a_{ij}$ hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and $2$-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients $a_{ij}$ are $VMO$ w.r.t. the space variable, and only bounded measurable in $t$. We prove, for every $p\in\left( 1,\infty\right) $, global Sobolev estimates of the kind: \begin{align*} \Vert u\Vert _{W_{X}^{2,p}(S_{T})} \equiv & \sum_{i,j=1}^{q}\Vert u_{x_{i}x_{j}}\Vert_{L^{p}(S_{T})} +\Vert Yu\Vert _{L^{p}(S_{T})} +\sum_{i=1}^{q}\Vert u_{x_{i}}\Vert _{L^{p}(S_{T})} +\Vert u\Vert _{L^{p}(S_{T})} \\ & \leq c\big\{ \Vert \mathcal{L}u\Vert _{L^{p}(S_{T})}+\Vert u\Vert_{L^{p}(S_{T})}\big\} \end{align*} with $S_{T}=\mathbb{R}^{N}\times\left( -\infty,T\right) $ for any $T\in(-\infty,+\infty]$. Also, the well-posedness in $W_{X}^{2,p}(Ω_{T})$, with $Ω_{T}=\mathbb{R}^{N}\times(0,T) $ and $T\in\mathbb{R}$, of the Cauchy problem% $$ \begin{cases} \mathcal{L}u=f & \text{in $Ω_{T}$} \\ u(\cdot,0) =g & \text{in $\mathbb{R}^{N}$} \end{cases} $$ is proved, for $f\in L^{p}(Ω_{T}), g\in W_{X}^{2,p}(\mathbb{R}^{N})$. |
| title | Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and $VMO$ in space |
| topic | Analysis of PDEs 35K65, 35K70, 35B45, 35A08, 42B20, 42B25 |
| url | https://arxiv.org/abs/2405.09358 |