Local Well-Posedness and Break-Down Criterion of the Incompressible Euler Equations with Free Boundary.

Format:

Book

Author/Creator:

Wang, Chao.

Contributor:

Zhang, Zhifei.

Zhao, Weiren.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.270

Language:

English

Subjects (All):

Fluid mechanics.

Physical Description:

1 online resource (132 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"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 3 2+. 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"-- Provided by publisher.

Contents:

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

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470465247

1470465248

OCLC:

1256821473