Quantitative Compactness Estimates for Hamilton–Jacobi Equations

We study quantitative compactness estimates in W1,1 for the map St , t > 0 loc that is associated with the given initial data u0 ∈ Lip(RN ) for the corresponding solution St u0 of a Hamilton–Jacobi equation u t + H ∇x u = 0 , t 0 , x ∈ R N , with a uniformly convex Hamiltonian H = H(p). We provide upper and lower estimates of order 1/εN on the Kolmogorov ε-entropy in W1,1 of the image through the map St of sets of bounded, compactly supported initial data. Estimates of this type are inspired by a question posed by Lax (Course on Hyperbolic Systems of Conservation Laws. XXVII Scuola Estiva di Fisica Matematica, Ravello, 2002) within the context of conservation laws, and could provide a measure of the order of “resolution” of a numerical method implemented for this equation.