Difference between revisions of "PDE Geometric Analysis seminar"

From UW-Math Wiki
Jump to navigation Jump to search
Line 21: Line 21:
 
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]
 
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]
 
|[[#Guo Luo (Caltech) |
 
|[[#Guo Luo (Caltech) |
TBA. ]]
+
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]
 
|Kiselev
 
|Kiselev
 
|-
 
|-
Line 67: Line 67:
 
upper-half plane with a conformal metric.
 
upper-half plane with a conformal metric.
 
This is a joint work with Stephen Kleene.
 
This is a joint work with Stephen Kleene.
 +
 +
===Guo Luo (Caltech)===
 +
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''
 +
 +
Abstract:
 +
Whether the 3D incompressible Euler equations can develop a singularity in
 +
finite time from smooth initial data is one of the most challenging problems in
 +
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this
 +
long-standing open question from a numerical point of view, by presenting a class of
 +
potentially singular solutions to the Euler equations computed in axisymmetric
 +
geometries. The solutions satisfy a periodic boundary condition along the axial direction
 +
and no-flow boundary condition on the solid wall. The equations are discretized in space
 +
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially
 +
designed adaptive (moving) meshes that are dynamically adjusted to the evolving
 +
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the
 +
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and
 +
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a
 +
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and
 +
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity
 +
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup
 +
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and
 +
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also
 +
suggests that the blowing-up solution develops a self-similar structure near the point of
 +
the singularity, as the singularity time is approached.

Revision as of 07:43, 6 October 2013

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Previous PDE/GA seminars

Seminar Schedule Fall 2013

date speaker title host(s)
September 9 Greg Drugan (U. of Washington)
Construction of immersed self-shrinkers
Angenent
October 7 Guo Luo (Caltech)

Potentially Singular Solutions of the 3D Incompressible Euler Equations.

Kiselev
November 18 Roman Shterenberg (UAB)

TBA.

Kiselev


Seminar Schedule Spring 2014

date speaker title host(s)
March 3 Hongjie Dong (Brown)

TBA.

Kiselev
April 7 Zoran Grujic (University of Virginia)

TBA.

Kiselev

Abstracts

Greg Drugan (U. of Washington)

Construction of immersed self-shrinkers

Abstract: We describe a procedure for constructing immersed self-shrinking solutions to mean curvature flow. The self-shrinkers we construct have a rotational symmetry, and the construction involves a detailed study of geodesics in the upper-half plane with a conformal metric. This is a joint work with Stephen Kleene.

Guo Luo (Caltech)

Potentially Singular Solutions of the 3D Incompressible Euler Equations

Abstract: Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a \emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also suggests that the blowing-up solution develops a self-similar structure near the point of the singularity, as the singularity time is approached.