2003/03/24 by Giuseppe Maria Coclite, Coclite, Giuseppe Maria, Nils Henrik Risebro +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.math/0303288
20 pages, 1 figure
arxiv created 2003/03/24 · openalex publication_date 2003/03/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to ``internal boundaries''. By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in L^∞, and that the limits are viscosity solutions.