The vector field method was introduced in the 1980s by Sergiu Klainerman to analyze the decay properties of the linear wave equation. Since its historical debut, the vector field method has been at the forefront of several breakthrough results including the global stability of Minkowski space, the dynamical formation of black holes, and shock formation in 3D compressible fluids.This work showcases how the vector field method can be used in a systematic way to derive a priori estimates for nonlinear evolution equations. For nonlinear dispersive equations, these estimates can be married to the decay properties enjoyed by the solutions to derive quantitative asymptotics. This is done in this work through the lens of three concrete problems: a nonlocal kinetic model, the wave maps equation, and the relativistic membrane equation. For the kinetic model, the vector field method is paired with dispersive decay properties of the spatial density to prove global wellposedness of small data. This can be interpreted physically as "stability" of the trivial solution. For the wave maps equation, a stability result is proven for a "non-trivial" ODE geodesic solution. For the relativistic membrane equation, the vector field method is used to prove stability of large simple-traveling-waves. For the wave map and membrane equations, we intimately use several structural properties known as null conditions that preclude singular behavior.
Read
