In this paper, we develop and analyze an implicit $a$ $posteriori$ error estimates for the local discontinuous Galerkin (LDG) method for nonlinear third-order Korteweg-de Vries (KdV) equations in one space dimension. First, we show that the LDG error on each element can be split into two parts. The first part is proportional to the $(p+1)$-degree right Radau polynomial and the second part converges with order $p$ $+$ $\frac{3}{2}$ in the $L^2$-norm, when piecewise polynomials of degree at most $p$ are used. These results allow us to construct $a$ $posteriori$ LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these $a$ $posteriori$ error estimates converge at a fixed time to the exact spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p$ $+$ $\frac{3}{2}$. Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed error estimator.