In this paper, we study the Jacobian equation from unbalanced transport maps. For a given volume form \(\mu _0\textrm{d}x\) on a bounded open domain \(\Omega \subset \mathbb {R}^{n}\) and \(\mu _0\in C^{k,\alpha }(\bar{\Omega };\mathbb {R})\) and \(W^{m,p}(\bar{\Omega };\mathbb {R})\) , respectively, we prove the corresponding existence of \(u_{t}\in C^{k+1,\alpha }(\bar{\Omega };\mathbb {R}^{n})\) and \(W^{m+1,p}(\bar{\Omega };\mathbb {R}^{n})\) of the equation, where \(u_{1}\) transform \(\mu _{0}\textrm{d}x\) into the Lebesgue measure \(\textrm{d}x\) .
