PDE Geometric Analysis seminar: Difference between revisions
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|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)] | |[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)] | ||
|[[#Guo Luo (Caltech) | | |[[#Guo Luo (Caltech) | | ||
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]] | |||
|Kiselev | |Kiselev | ||
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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 13:43, 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) | Kiselev |
Seminar Schedule Spring 2014
| date | speaker | title | host(s) |
|---|---|---|---|
| March 3 | Hongjie Dong (Brown) | Kiselev | |
| April 7 | Zoran Grujic (University of Virginia) | 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.