  The purpose of this paper is twofold. First, we use a classical method to
establish Gaussian bounds of the fundamental matrix of a generalized parabolic
Lam\'{e} system with only bounded and measurable coefficients. Second, we
derive a maximal $L^1$ regularity result for the abstract Cauchy problem
associated with a composite operator. In a concrete example, we also obtain
maximal $L^1$ regularity for the Lam\'{e} system, from which it follows that
the Lipschitz seminorm of the solutions to the Lam\'{e} system is globally
$L^1$-in-time integrable. As an application, we use a Lagrangian approach to
prove a global-in-time well-posedness result for a viscous pressureless flow
provided that the initial velocity satisfies a scaling-invariant smallness
condition. The method established in this paper might be a powerful tool for
studying many issues arising from viscous fluids with truly variable densities.
