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Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary
About this Title
Chao Wang, Zhifei Zhang, Weiren Zhao and Yunrui Zheng
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 270, Number 1318
ISBNs: 978-1-4704-4689-5 (print); 978-1-4704-6524-7 (online)
DOI: https://doi.org/10.1090/memo/1318
Published electronically: May 21, 2021
Table of Contents
Chapters
- 1. Introduction
- 2. Tools of paradifferential operators
- 3. Parabolic evolution equation
- 4. Elliptic estimates in a strip
- 5. Dirichlet-Neumann operator
- 6. New formulation and paralinearization
- 7. Estimate of the pressure
- 8. Estimate of the velocity
- 9. Proof of break-down criterion
- 10. Iteration scheme
- 11. Uniform energy estimates
- 12. Cauchy sequence and the limit system
- 13. From the limit system to the Euler equations
- 14. Proof of Theorem
Abstract
In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschitz function and the free surface belongs to $C^{\frac {3}{2}+\varepsilon }$. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.- T. Alazard, N. Burq, and C. Zuily, On the water-wave equations with surface tension, Duke Math. J. 158 (2011), no. 3, 413–499. MR 2805065, DOI 10.1215/00127094-1345653
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