Title: A Short Survey of the Well-posedness of the Two-dimensional Burgers' Equation Show Chinese title

Title: 二维Burgers方程的适定性简要综述

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. Show Chinese abstract

Abstract: 在本文中，我们利用无限维动力系统框架，建立了二维Burgers方程解的存在性和唯一性。 二维Burgers方程，用于描述非线性对流与粘性耗散之间的相互作用，其表达式为：$$ u_{t} + u \cdot \nabla u = \nu \Delta u + f, $$，其中$ u = (u_1, u_2) $是速度场，$ \nu > 0 $是粘性系数，$ f $表示外力。 我们主要采用了Galerkin方法将偏微分方程转化为常微分方程。 此外，通过采用Sobolev空间、能量估计和紧致性论证，我们严格证明了在适当初始和边界条件下全局解的存在性及其唯一性。