Michael Payne

Publication Date

9-5-2013

Abstract

In this paper we consider the Cauchy problem for the 3D \\NS equations for incompressible flows, and their solutions. We will discuss the results of a paper by Otto Kreiss and Jens Lorenz on the role of the pressure term in the \\NS equations, and its relationship to the fluid field $u(x,t)$. The focus here is to concentrate on solutions to the equation where the fluid field $u$ lies in the space $\\Ci(\\R^3)\\cap\\Li(\\R^3)$, and not necessarily in $L^2(\\R^3)$. If $u(x,0)=f(x)$, where $f\\in\\Ci(\\R^3)\\cap\\Li(\\R^3)$ we will consider the solutions for all $t$ in time interval $0\\leq t < T(f)$. In the original paper, estimates for the \\emph{derivatives} of the pressure were proved, but the definition of the pressure proved unsatisfactory due to the possibility of the divergence of the pressure term. The main object of this paper is to use the theory of singular integrals and the space of functions of \\BMO to properly address the pressure. In doing so, we will provide estimates on pressure term itself. This will allow us to strengthen the results of the original paper, and rigorously extend all results from the original paper to \\emph{any} function $u\\in\\Ci(\\R^3)\\cap\\Li(\\R^3)$.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Jens Lorenz

Second Committee Member

Maria Cristina Pereyra

Third Committee Member

Daniel Appelö

Fourth Committee Member

Francesco Sorrentino

Language

English

Keywords

Navier Stokes Equations

Document Type

Dissertation

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