A Short Survey of the Well-posedness of the Two-dimensional Burgers' Equation

Abstract

In this paper, we establish the existence and uniqueness of solutions to the two-dimensional Burgers equation using the framework of infinite-dimensional dynamical systems. The two-dimensional Burgers equation, which models the interplay between nonlinear advection and viscous dissipation, is given by: $$u_t+u\cdot \nabla u=\nu \Delta u+f,$$ where $u=(u_1,u_2)$ is the velocity field, $\nu >0$ is the viscosity coefficient, and $f$ represents an external force. We primarily employed Galerkin method to transform the partial differential equation into an ordinary differential equation. In addition, by employing Sobolev spaces, energy estimates, and compactness arguments, we rigorously prove the existence of global solutions and their uniqueness under appropriate initial and boundary conditions.