Symmetry problems : the Navier-Stokes problem / Alexander G. Ramm.

Format:

Book

Author/Creator:

Ramm, A. G. (Alexander G.), author.

Series:

Synthesis digital library of engineering and computer science.

Synthesis lectures on mathemathics and statistics ; #23.

Synthesis lectures on mathemathics and statistics, 1938-1743 ; #23

Language:

English

Subjects (All):

Helmholtz equation.

Navier-Stokes equations.

Scattering (Mathematics).

Physical Description:

1 online resource (87 pages).

Place of Publication:

[San Rafael, California] : Morgan & Claypool, [2019]

Summary:

This book gives a necessary and sufficient condition in terms of the scattering amplitude for a scatterer to be spherically symmetric. By a scatterer we mean a potential or an obstacle. It also gives necessary and sufficient conditions for a domain to be a ball if an overdetermined boundary problem for the Helmholtz equation in this domain is solvable. This includes a proof of Schiffer's conjecture, the solution to the Pompeiu problem, and other symmetry problems for partial differential equations. It goes on to study some other symmetry problems related to the potential theory. Among these is the problem of "invisible obstacles." In Chapter 5, it provides a solution to the Navier-Stokes problem in R3. The author proves that this problem has a unique global solution if the data are smooth and decaying sufficiently fast. A new a priori estimate of the solution to the Navier-Stokes problem is also included. Finally, it delivers a solution to inverse problem of the potential theory without the standard assumptions about star-shapeness of the homogeneous bodies.

Contents:

1. Introduction

2. Necessary and sufficient conditions for a scatterer to be spherically symmetric

2.1. Scattering by potentials

2.2. Scattering by obstacles

3. Symmetry problems for the Helmholtz equation

3.1. A general symmetry problem

3.2. Old symmetry problem

3.3. Necessary and sufficient conditions for S to be a sphere

3.4. The Pompeiu problem

4. Other symmetry problems

4.1. Volume potential

4.2. Surface potential

4.3. Invisible obstacles

5. Solution to the Navier-Stokes problem

5.1. A new approach

5.2. Construction of G

5.3. Solution to integral equation for v satisfies NS equations

5.4. Uniqueness of the solution to the integral equation

5.5. Existence of the solution to integral equation

5.6. Energy of the solution

5.7. Auxiliary estimates

5.8. Proof of the uniqueness of the solution

5.9. Proof of the existence of the solution

5.10. Convolution and positiveness of distributions

6. Inverse problem of potential theory

6.1. Statement of the problem

6.2. Proofs.

Notes:

Title from PDF title page (viewed on April 2, 2019).

Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 65-69).

Cited in:

Compendex

INSPEC

Google scholar

Google book search

ISBN:

1-68173-506-7

OCLC:

1091193877