  We make the elementary observation that the differential equation associated
with Newton's second law $m\ddot\gamma(t)=-D V(\gamma(t))$ always has a
solution for given initial conditions provided that the potential energy $V$ is
semiconvex. That is, if $-D V$ satisfies a one-sided Lipschitz condition. We
will then build upon this idea to verify the existence of solutions for the
Jeans-Vlasov equation, the pressureless Euler equations in one spatial
dimension and the equations of elastodynamics under appropriate semiconvexity
assumptions.
