PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Kris Jenssen.

Title:Global solutions to the homogeneous and inhomogeneous Navier-Stokes equations
Seminar:CCMA PDEs and Numerical Methods Seminar Series
Speaker:Tepper L. Gill, Howard University
Abstract:
In this talk, I discuss the construction of the largest separable Hilbert space SD2[R3], for which the Leray-Hopf (weak) solutions on R3 are strong solutions in SD2 [R3 ] . In addition, we provide a new proof of global-in-time strong solutions, which leads to essentially the same proof for both bounded and unbounded domains and for a homogeneous or inhomogeneous incompressible fluid. In particular, on SD2[R3] we are able to obtain strong a priori estimates for the nonlinear term. When the body forces are zero, we prove that there exists a positive constant u_+ , such that, for all divergence-free vector fields in a dense set D contained in the closed ball B of radius 1/2 (1-\epsilon )u_+ , 0 < \epsilon < 1, the Navier-Stokes initial value problem has unique global strong solutions in C1[(0,!), B]. When body forces are present, we obtain the same result for divergence-free vector fields in a dense set D contained in the annulus bounded by constants u_- and 1/2 u_+ . In either case (with or without body forces), we obtain uniqueness for the Leray-Hopf (weak) solutions on R3 . Moreover, with mild conditions on the decay properties of the initial data, we obtain point-wise and time-decay of the solutions. However, our methods do not allow us to resolve the singularity question. If time permits, I will discuss the relationship of SD2[R3] to other spaces used to study global solutions on R3 .

Room Reservation Information

Room Number:MB216
Date:11 / 07 / 2011
Time:03:35pm - 04:25pm