Blow-up of Solutions to the Generalized Inviscid Proudman-Johnson Equation

<p>The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence.</p> <p>In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will be measured via a direct approach which involves the derivation of representation formulae for solutions to the problem. For a real parameter lambda, several classes of initial data are considered. These include the class of smooth functions with either zero or nonzero mean, a family of piecewise constant functions, and a large class of initial data with a bounded derivative that is, at least, continuous almost everywhere and satisfies Holder-type estimates near particular locations in the domain. Amongst other results, our analysis will indicate that for appropriate values of the parameter, the curvature of the data in a neighborhood of these locations is responsible for an eventual breakdown of solutions, or their persistence for all time. Additionally, we will establish a nontrivial connection between the qualitative properties of L-infinity blow-up in u_x, and its Lp regularity. Finally, for smooth and non-smooth initial data, a special emphasis is made on the study of regularity of stagnation point-form solutions to the two and three dimensional incompressible Euler equations subject to periodic or Dirichlet boundary conditions.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2012

Author dc:creator

• Sarria, Alejandro

Contributors dc:contributor

• Ralph Saxton
• Dongming Wei
• Craig Jensen

• generalized Proudman-Johnson equation
• blow-up, Euler equations
• Analysis
• Other Applied Mathematics
• Partial Differential Equations

Repository record dc:identifier

https://scholarworks.uno.edu/td/1555

OAI identifier oai:identifier

oai:scholarworks.uno.edu:td-2565