  This paper presents a minimum flow approach applicable to a wide range of
doubly nonlinear diffusion problems. We introduce a minimum flow steepest
descent algorithm that seeks an optimal traffic flow by minimizing an internal
energy function, while incorporating a specific minimum flow constraint for the
transition work. This flexible framework surpasses traditional methods by
generalizing (among other things) the established $H^{-1}$-theory for linear
diffusion to the nonlinear setting. It offers distinct advantages in handling
diverse applications, leveraging both the intrinsic internal energy and the
inherent traffic flow concepts. The approach demonstrably tackles various
applications, including fluid flow in porous media and many other scenarios
like the Stefan/Hele-Shaw problem. Notably, it can handle diverse differential
operators, encompassing even nonlinear ones like the $p-$Laplacian and
Leray-Lions operators.
  Furthermore, it allows for the simultaneous handling convection/transport
phenomena and complex boundary conditions within both the energy and transition
work functions, contrasting favorably with other steepest descent algorithm
like the JKO scheme in Wasserstein spaces. The paper delves into details of
this comparison and offers additional theoretical insights involving $p'-$curve
and metric gradient flow in Sobolev dual spaces.
