论文标题
关于非线性kolmogorov-fokker-Planck类型方程的规律性和存在薄弱的解决方案，具有粗糙的系数
On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients
论文作者
论文摘要
我们考虑表单\ begin {qore} \ label {abeqn}（\ partial_t+x \ cdot \ cdot \ nabla_y）u = \ nabla_x u = \ nabla_x \ cdot（a（a \ nabla_x u = nabla_x u，x，y，x，y，y，y，y，y y y y， \ end {qore}函数$ a = a = a（ξ，x，y，t）：\ r^m \ times \ r^m \ times \ times \ r^m \ r^m \ times \ r \ to \ r^m $相对于$ $ξ$是连续的，并且相对于$ x，y $和$ x，y $和$ t $。 $ a = a（ξ，x，y，t）$被允许是非线性，但线性增长。我们建立了较高的可集成率和弱点，弱的harnack和harnack不平等的局部界限，并具有定量估计值。此外，我们在某些有限的$ x $，$ y $和$ t $依赖性域中确定了针对Dirichlet问题的弱解决方案的存在和独特性。
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(ξ,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $ξ$, and measurable with respect to $X,Y$ and $t$. $A=A(ξ,X,Y,t)$ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and H{ö}lder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded $X$, $Y$ and $t$ dependent domains.
