Cover
Title page
Chapter 1. Introduction
1.1. Presentation of the problem
1.2. Some known results
1.3. Main results
1.4. Main ideas
Chapter 2. Tools of paradifferential operators
2.1. Paradifferential operators
2.2. Functional spaces
2.3. Symbolic calculus
2.4. Tame estimates in Sobolev space
2.5. Tame estimates in Chemin-Lerner spaces
2.6. Commutator estimates
Chapter 3. Parabolic evolution equation
Chapter 4. Elliptic estimates in a strip
4.1. Elliptic boundary problem
4.2. Flattening the boundary and paralinearization
4.3. Elliptic estimates in Sobolev space
4.4. Tame elliptic estimates
4.5. Elliptic estimates in Besov space
4.6. Interior ^{1, } estimate
Chapter 5. Dirichlet-Neumann operator
5.1. Definition and paralinearization
5.2. Sobolev estimate of the remainder
5.3. Tame estimate of the remainder
5.4. Hölder estimate of the remainder
Chapter 6. New formulation and paralinearization
6.1. New formulation
6.2. Paralinearization
Chapter 7. Estimate of the pressure
7.1. ² estimate of the pressure
7.2. Hölder estimate of the pressure
7.3. Sobolev estimate of the pressure
7.4. Estimate of
Chapter 8. Estimate of the velocity
8.1. Sobolev estimate of the velocity
8.2. The estimate of the irrotational part
8.3. The estimate of the rotational part
Chapter 9. Proof of break-down criterion
9.1. The ¹ energy estimate
9.2. Energy estimate of the trace of the velocity and the free surface
9.3. Energy estimate of the vorticity
9.4. Nonlinear estimates
9.5. Energy functional
9.6. Proof of Theorem 1.3
Chapter 10. Iteration scheme
10.1. Strategy
10.2. Iteration scheme
10.3. Existence of iteration scheme
Chapter 11. Uniform energy estimates
11.1. Set-up
11.2. Energy functional.

11.3. Estimate of the velocity
11.4. Estimate of the pressure
11.5. Estimates of the remainder of DN operator
11.6. Energy estimates
11.7. Nonlinear estimates
11.8. Completion of the uniform estimate
Chapter 12. Cauchy sequence and the limit system
12.1. Set-up
12.2. Elliptic estimates with a parameter
12.3. Energy estimates
12.4. The limit system
Chapter 13. From the limit system to the Euler equations
Chapter 14. Proof of Theorem 1.1
14.1. Construction of approximate smooth solution
14.2. Uniform estimates and existence
14.3. Uniqueness of the solution
Acknowledgement
Bibliography
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