The Newton’s problem assuming non-constant density of the fluid

This paper investigates the Newton’s problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions u ( r ) satisfying the initial conditions u ( 0 ) = u ′ ( 0 ) = 0 using a fixed-point theorem. We show that the maximal domain of the solution is finite, [ 0 , r M ) , terminating at a critical slope u ′ ( r M ) = 1 3 .