I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem for Boltzmann Equations ("R.J. DiPerna and P.L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math., 130")
Shortly summarized they show that classical solutions of the Boltzmann Equation converge weakly in $L^1$ to a renormalized solution and from this they deduce global existence of a solution to the Cauchy Problem.
What makes this so attractive? Well they also show that $f$ is a distributional solution if and only if $f$ is a renormalized solution, they also show that this is also equivalent that $f$ is mild solution.
Question: What is purpose for doing that? What can I deduce from the knowledge that if $f$ is renormalized solution than it has to be also a distributional solution?