  In this paper, we consider the local existence and uniqueness result for the
inhomogeneous Prandtl equations in dimension two by energy method. First of
all, for the homogeneous case, the local-in-time well-posedness theory of
unsteady Prandtl equations was obtained by [Alexandre, Wang, Xu, Yang, J. Am.
Math. Soc., 28 (3), 745-784 (2015)] and [Masmoudi, Wong, Comm. Pure Appl.
Math., 68 (10), 1683-1741 (2015)] independently by energy method without any
transformation. However, for the inhomogeneous case, the appearance of density
will create some new difficulties for us to overcome the loss of tangential
derivative of horizontal velocity. Thus, our first result is to overcome the
loss of tangential derivative such that one can establish the local-in-time
well-posedness result for the inhomogeneous Prandtl equations by energy method.
Secondly, for the homogeneous case, the local-in-x well-posedness in higher
regular space for the steady Prandtl equations was obtained by [Guo, Iyer,
Comm. Math. Phys., 382 (3), 1403-447 (2021)] by energy method since they
firstly found the good quantity(called `quotient'). With the help of this
quotient, our second result is to establish the local-in-x well-posedness in
higher regular Sobolev space for the steady inhomogeneous Prandtl equations.
