arXiv subjects

Kui Yan

Publications and source records attributed to Kui Yan.

3 recordsLinked to original sources

arXiv2026

A Liouville theorem for ancient solutions of the parabolic Monge-Amp\`ere equation with periodic data

This article is concerned with the parabolic Monge-Amp\`ere equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of the form $-\tau t+p(x)+v(x,t)$, where $\tau$ is a positive constant, $p(x)$ is a convex quadratic polynomial, and $v$ inherits both the spatial and temporal periodicity from $f$. This work extends previous contributions by Caffarelli-Li \cite{cl04} on periodic frameworks for the elliptic Monge-Amp\`ere equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for $f_2\equiv1$ in parabolic case.

math.AP↗

arXiv2026

Optimal Asymptotic Behavior at Infinity of Ancient Solution to the Parabolic Monge-Amp\`ere Equation with Slow Perturbation Term

In this paper, we obtain optimal asymptotic behavior of parabolically convex $C^{2,1}$ solution to the parabolic Monge-Amp\`ere equation $-u_t\det D_x^2u=f$, where $f$ converges to $1$ at infinity with a slow rate. This result extends the elliptic estimate in \cite{lb5} to the parabolic setting.

math.AP↗

arXiv2026

Liouville theorem and sharp solvability for solutions of the parabolic Monge-Amp\`ere equation with periodic data

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Amp\`ere equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.

math.AP↗