A Short Survey of the Well-posedness of the Two-dimensional Burgers' Equation
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| Format: | Preprint |
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2025
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| _version_ | 1866917947098791936 |
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| author | Zhang, Xiang Xie, Shuhan Sun, Yule |
| author_facet | Zhang, Xiang Xie, Shuhan Sun, Yule |
| contents | 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: $$ u_{t} + u \cdot \nabla u = νΔu + f, $$ where $ u = (u_1, u_2) $ is the velocity field, $ ν> 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04467 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Short Survey of the Well-posedness of the Two-dimensional Burgers' Equation Zhang, Xiang Xie, Shuhan Sun, Yule Analysis of PDEs 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: $$ u_{t} + u \cdot \nabla u = νΔu + f, $$ where $ u = (u_1, u_2) $ is the velocity field, $ ν> 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. |
| title | A Short Survey of the Well-posedness of the Two-dimensional Burgers' Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.04467 |