受控漂移的非线性Fokker-Planck方程解的存在性与爆破
Existence and Blow-Up for Non-linear Fokker-Planck with Controlled Drift
中文总结 AI 辅助
该论文将临界参数下非线性Fokker-Planck方程的全局存在性结果推广到多维非自治漂移情形，通过有序解差异分析证明有限增量可达任意解。
中文摘要 AI 辅助
我们将[Bianchini和Leccese, J. Math. Anal. Appl. (2024)]中临界参数的全局存在性结果推广到多维问题\begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = \Delta u \end{equation*}，其中$b:(0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d$为非自治场，且满足\begin{equation*} b\in L^\infty_{\mathrm{loc}} \bigl((0,\infty),L^{p,\infty}(\mathbb{R}^d)\bigr), \qquad p>d\ge 2. \end{equation*}对于临界情形，我们研究两个有序解之间的差异，其守恒的小质量吸收了漂移。有限个增量随后达到任意解。
英文摘要
We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = Δu \end{equation*} for $b:(0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d$ non-autonomous fields, in the case \begin{equation*} b\in L^\infty_{\mathrm{loc}} \bigl((0,\infty),L^{p,\infty}(\mathbb{R}^d)\bigr), \qquad p>d\ge 2. \end{equation*} For the critical case, we study the difference between two ordered solutions whose conserved small mass absorbs the drift. Finitely many increments then reach an arbitrary solution.
