#### Publication Date

Summer 7-2-2019

#### Abstract

In this doctoral dissertation, we consider the Cauchy problem for the 3D incompressible Navier-Stokes equations. Here, we are interested in a smooth periodic solution of the problem which happens to be a special case of a paper by Otto Kreiss and Jens Lorenz. More precisely, we will look into a special case of their paper by two approaches. In the first approach, we will try to follow the similar techniques as in the original paper for smooth periodic solution. Because of the involvement of the Fourier expansion in the process, we encounter with some intriguing factors in the periodic case which are absolutely not a part in the original paper. While in the second approach, we decompose our solution space using the Helmholtz-Weyl decomposition and introduce a new tool \say{the Leray projector} to eliminate the pressure term from the Navier-Stokes equations and go from there. This approach is completely different than the technique of dealing with the pressure term in the paper by Otto Kreiss and Jens Lorenz.

#### Degree Name

Mathematics

#### Level of Degree

Doctoral

#### Department Name

Mathematics & Statistics

#### First Committee Member (Chair)

Jens Lorenz

#### Second Committee Member

Cristina Pereyra

#### Third Committee Member

Stephen Lau

#### Fourth Committee Member

Laura De Carli

#### Language

English

#### Keywords

Navier-Stokes Equations, Maximum Norm, Periodic Initial Data

#### Document Type

Dissertation

#### Recommended Citation

Pathak, Santosh. "L^{\infty}-ESTIMATES OF THE SOLUTION OF THE NAVIER-STOKES EQUATIONS FOR PERIODIC INITIAL DATA." (2019). https://digitalrepository.unm.edu/math_etds/138