A new approach to regularity and singularity questions for a class of non-linear evolutionary PDEs (eg, 3-D Navier-Stokes eqns)

Start: 03/09/2010 - 4:15pm
End  : 03/09/2010 - 5:15pm

Applied Math Seminar

Saleh Tanveer (The Ohio State University)


(Joint work with Ovidiu Costin, G. Luo.)

We consider a new approach to a class of evolutionary PDEs where question of global existence or lack of it is tied to the asymptotics of solution to a non-linear integral equation in a dual variable whose solution has been shown to exist a priori. This integral equation approach is inspired by Borel summation of a formally divergent series for small time, but has general applicability and is not limited to analytic initial data.

In this approach, there is no blow-up in the variable p, which is dual to 1/t or some power 1/t^n; solutions are known to be smooth in p and exist globally for p in R+. Exponential growth in p, for different choice of n, signifies finite time singularity. On the other hand, sub-exponential growth implies global existence.

Further, unlike PDE problems where global existence is uncertain, a discretized Galerkin approximation to the associated integral equation has controlled errors. Further, known integral solution for p in [0, p_0], numerically or otherwise, gives sharper analytic bounds on the exponents in p and hence better estimate on the existence time for the associated PDE.

We will also discuss particular results for 3-D Navier-Stokes and discuss ways in which this method may be relevant to numerical studies of finite time blow-up problems.

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