We deform the contact form by the (normalized) CR Yamabe flow on a closed spherical CR 3-manifold. We show that if a contact form evolves with positive Tanaka-Webster curvature and vanishing torsion from initial data, then we obtain a new Li-Yau-Hamilton inequality for the CR Yamabe flow. By combining this parabolic subgradient estimate with a compactness theorem of a sequence of contact forms, it follows that the CR Yamabe flow exists for all time and converges smoothly to, up to the CR automorphism, a unique limit contact form of positive constant Webster scalar curvature on a closed CR. 3-manifold, which is CR equivalent to the standard CR 3-sphere with positive Tanaka-Webster curvature and vanishing torsion.
