  We provide some new integral estimates for solutions to Hamilton-Jacobi
equations and we discuss several consequences, ranging from $L^p$-rates of
convergence for the vanishing viscosity approximation and homogenization to
regularizing effects for the Cauchy problem in the whole Euclidean space and
Liouville-type theorems. Our approach is based on duality techniques \`a la
Evans and a careful study of advection-diffusion equations. The optimality of
the results is discussed by several examples.

Analysis of PDEs