Event Detail

Event Type: 
Ph.D.Defense
Date/Time: 
Monday, December 7, 2020 - 10:00 to 11:00
Location: 
Zoom - If you are interested in attending this presentation, please send an email to Nikki Sullivan - nikki.sullivan@oregonstate.edu - to request Zoom log in details.
Abstract: 

In physics, the Navier–Stokes equations (NSEs) are a set of partial differential equations which describe the motion of viscous fluid substances and can be viewed as Newton's second law of motion for fluids. In this talk, we explore the mathematical theory of the 3D homogeneous incompressible NSEs. In particular, we focus on how local perturbation of the initial condition influence the evolution of the solution to the NSEs, which is non-local in nature. The class the suitable weak solutions, whose concept was introduced by V. Scheffer in 1977, is our main interest in this work. We establish an upper bound for the growth rate of the local L^2 norm of the difference of two weak solutions u ,v that arise from the initial conditions u_0,v_0 from some suitable function spaces. The technical difficulty that lies in the lack of regularity of weak solutions makes the analysis challenging, and there are currently very few known results in this direction. Nevertheless, we establish an estimate for the local growth rate of the difference of their local kinetic energy – given the condition that the global (in space and time) kinetic energy for both of the solutions are bounded – that is approximately O(t^(1/4) ) when t>0 is small. The main merit of this estimate is the exploitation of the spatial localization technique and Sobolev embedding theorem. Moreover, our results confirm that the right-hand-side continuity of the local L^2norm at t=0 also holds in the class of suitable weak solutions.