PDE Geometric Analysis seminar: Difference between revisions
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The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman. | The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman. | ||
===Hong Zhang=== | |||
On an elliptic equation arising from composite material | |||
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong. | |||
===Hiroyoshi Mitake=== | ===Hiroyoshi Mitake=== | ||
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Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem. | Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem. | ||
Revision as of 11:38, 5 February 2016
The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
Previous PDE/GA seminars
Tentative schedule for Fall 2016
Seminar Schedule Spring 2016
| date | speaker | title | host(s) |
|---|---|---|---|
| January 25 | Tianling Jin (HKUST and Caltech) | Holder gradient estimates for parabolic homogeneous p-Laplacian equations | Zlatos |
| February 1 | Russell Schwab (Michigan State University) | Neumann homogenization via integro-differential methods | Lin |
| February 8 | Jingrui Cheng (UW Madison) | Semi-geostrophic system with variable Coriolis parameter | Tran & Kim |
| February 15 | Paul Rabinowitz (UW Madison) | Tran & Kim | |
| February 22 | Hong Zhang (Brown) | On an elliptic equation arising from composite material | Kim |
| February 29 | Aaron Yip (Purdue university) | TBD | Tran |
| March 7 | Hiroyoshi Mitake (Hiroshima university) | Selection problem for fully nonlinear equations | Tran |
| March 15 | Nestor Guillen (UMass Amherst) | TBA | Lin |
| March 21 (Spring Break) | |||
| March 28 | Ryan Denlinger (Courant Institute) | The propagation of chaos for a rarefied gas of hard spheres in vacuum | Lee |
| April 4 | |||
| April 11 | |||
| April 18 | |||
| April 25 | Moon-Jin Kang (UT-Austin) | Kim | |
| May 2 |
Abstracts
Tianling Jin
Holder gradient estimates for parabolic homogeneous p-Laplacian equations
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u), where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.
Russell Schwab
Neumann homogenization via integro-differential methods
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.
Jingrui Cheng
Semi-geostrophic system with variable Coriolis parameter.
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.
Hong Zhang
On an elliptic equation arising from composite material
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.
Hiroyoshi Mitake
Selection problem for fully nonlinear equations
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.