Monge-Amp\`ere方程是一类非常重要的完全非线性偏微分方程, 在最优运输、仿射几何、气象学和流体力学中都有广泛的应用. 近半个世纪以来, 这类方程解的存在性与正则性问题一直是偏微分方程中重要的研究领域, 也得到了很大的发展. 本文主要围绕三类Monge-Amp\`ere方程解的正则性展开, 包括具有第二边值条件的椭圆型Monge-Amp\`ere方程, 和具有Dirichlet边界条件或第二边界条件的抛物型Monge-Amp\`ere方程. 假设$f$和$g$是连续的正函数, $\Omega$和$\Omega^*$是边界$C^{1,1}$光滑的有界凸区域. 对于椭圆方程 $$\left\{\begin{aligned} &\det D^2u=f/g(Du)\quad\text{ 在 }\,\Omega\text{ 内},\\ &Du(\Omega)=\Omega^*, \end{aligned} \right.$$ 我们建立了$Du$在$\overline{\Omega}$上的连续性估计, 这一估计依赖于$\log(f/g)$的连续性模, 并在$f$的最优增长性条件下得到了$Du$的log-Lipschitz连续性. 这个研究使用了扰动方法, 并将原来的内部估计推广到了全局估计. 对于第一初边值问题的抛物型Monge-Amp\`ere方程 \begin{equation*} \left\{\begin{aligned} u_t&=\det D^2u\quad\quad\ \text{ 在 }\,Q_T\text{ 中 },\\ u&=\phi\quad\quad\quad\quad\quad \text{ 在 }\;\partial_pQ_T\text{ 上 }, \end{aligned}\right.\end{equation*}其中$Q_T=\Omega\times (0,T]$. 当$\Omega$, $\phi$充分光滑且$\phi(\cdot,0)$一致凸、$\phi_t$严格正时, 我们建立了光滑解的全局$C^{2,\alpha}$先验估计，并由此证明了抛物凸解的存在性与唯一性. 对于$\gamma$-高斯曲率流, 其中$0<\gamma\le 1$, 我们也证明了类似的结论. 这个问题的核心困难在于方程凹性的缺失, 我们通过两次Legendre变换和特殊的边界估计技巧解决了这一问题. 对于第二边界条件的抛物型Monge-Amp\`ere方程\begin{equation*} \left\{\begin{aligned} &u_t=\det D^2u\quad \text{ 在 }\,\Omega\times(0,+\infty)\text{ 中 },\\ &Du(\Omega,t)=\Omega^*\quad \forall t>0,\\ &u(0)=\varphi, \end{aligned}\right.\end{equation*}我们利用先验估计方法证明了方程解的无穷时间存在性, 以及解的全局$C^{2,\alpha}$估计. 作为解的全局正则性的推论，我们证明了解在$t\to\infty$时渐近于方程的平移解.

This article focuses on the existence and regularity of solutions to three types of Monge-Amp\`ere equations, including the elliptic and parabolic Monge-Amp\`ere equations. Assume that $f$ and $g$ are continuous positive functions, and $\Omega$ and $\Omega^*$ are bounded convex domains with $C^{1,1}$ smooth boundaries. For the elliptic equation $$\left\{\begin{aligned} &\det D^2u=f/g(Du)\text{ in }\Omega,\\ &Du(\Omega)=\Omega^*, \end{aligned} \right.$$ we establish the continuity estimate of $Du$ on $\overline{\Omega}$ in terms of the continuity modulus of $\log(f/g)$, and obtain the log-Lipschitz continuity of $Du$ under optimal growth conditions for $f$. This study employs the perturbation method and extends the original interior estimate to a global one. We then study the first initial-boundary value problem of the parabolic Monge-Amp\`ere equation, \begin{equation*} \left\{\begin{aligned} u_t&=\det D^2u\quad\quad\ \text{ in } Q_T,\\ u&=\phi\quad\quad\quad\quad\quad \text{ on } \partial_pQ_T. \end{aligned}\right. \end{equation*} where $\Omega$ and $\phi$ are sufficiently smooth and $\phi(\cdot,0)$ is uniformly convex, with $\phi_t$ strictly positive. We prove the existence and uniqueness of convex solutions to this equation after a global $\tilde{C}^{2,\alpha}$ estimate. Similar results are also obtained for the $\gamma$-Gauss curvature flow, where $0<\gamma\le 1$. For the parabolic Monge-Amp\`ere equation with the second boundary condition, \begin{equation*} \left\{\begin{aligned} &u_t=\det D^2u\quad \text{ in } \Omega\times (0,+\infty),\\ &Du(\Omega,t)=\Omega^* \quad \forall t>0,\\ &u(0)=\varphi. \end{aligned}\right. \end{equation*} we establish the long time existence of solutions and a global $\tilde{C}^{2,\alpha}$ estimate for the solutions. Finally, we obtain the asymptotic behavior of the solutions as $t\to\infty$, converging to translating solutions of the equation.