Well-posedness and representation formulae for solutions to the transport equation, Laplace equation, heat equation, and wave equation. Fundamental solutions. Properties of harmonic functions. Green's function. Mean value formulae. Energy methods. Maximum principles. Method of characteristics for quasilinear equations. Burgers' equation. Shocks formation and entropy condition. Applications to fluid dynamics, elasticity problems and/or optimization problems.
Well-posedness and representation formulae for solutions to the transport equation, Laplace equation, heat equation, and wave equation. Fundamental solutions. Properties of harmonic functions. Green's function. Mean value formulae. Energy methods. Maximum principles. Method of characteristics for quasilinear equations. Burgers' equation. Shocks formation and entropy condition. Applications to fluid dynamics, elasticity problems and/or optimization problems.
