Lehigh online applied mathematics seminar; topics include but not limited to: differential equations, dynamical systems, fluid dynamics, neuronal networks, numerical simulations, scientific computing, and machine learning. The seminar will generally be scheduled at 10:00am-11:00am, Tuesdays and/or 11:00am-12:00pm, Fridays each weak.
All talks will be delivered remotely; please follow this link to all Zoom sessions. If a password is required, please contact the lead seminar organizer Prof. Linghai Zhang for more information.
| The list will be continuously updated and maintained throughout the semester. |
| April 4 (T) - | Michael Dabkowski (University of Michigan-Dearborn) |
| 11:00am-12:00pm | Title: Global Asymptotic Stability for the Diffusive Carr-Penrose Equation. |
| Abstract: The Carr-Penrose equations are a linear perturbation of the Lifshitz-Slyozov-Wagner Equations. The LSW Equations were introduced to understand the long time behavior and selection principle of certain particle coarsening problems. The Carr-Penrose Equations exhibit similar structure to the LSW equations and also have a one-parameter family of self similar solutions. We show that the diffusive Carr-Penrose Equation, with initial data of compact support, has a solution which at large times approximates physically relevant self-similar solution. | |
| April 11 (T) - | Baoxiang Wang (Jimei University, Xiamen) |
| 10:00am-11:00am | Title: Global solutions for the Navier-Stokes equation in supercritical spaces. |
| Abstract: We consider the Navier-Stokes equation (NS) in a class of super-critical spaces. We obtain that NS has a unique global solution for the initial data in a class of modulation spaces, which is supported in the first octant in the frequency spaces. We see that NS has a unique solution if the initial data is supported in the first octant in frequency spaces. | |
| April 11 (T) - | Katherine Zhiyuan Zhang (New York University) |
| 11:30am-12:30pm | Title: Outflow Problems in Plasmas and Fluids. |
| Abstract: We are concerned about problems in plasma and fluids with flow velocity being outward at the physical boundary. We consider the Vlasov-Poisson equation on a half line with outflow (completely absorbing) boundary conditions, and present a result on the nonlinear stability of a family of stationary solutions. This is a modeling of a plasma boundary layer (sheath). If time permits, I will also present a result about an outflow problem on the compressible Navier-Stokes equation. This is joint work with M. Suzuki and M. Takayama.. | |
| April 18 (T) - | Hongjun Gao (Southeast University, Nanjing) |
| 10:00am-11:00am | Title: Well-posedness and wave-breaking for the stochastic rotation-two-component Camassa-Holm system. |
| Abstract: We study the global well-posedness and wave-breaking phenomenon for the stochastic rotation-two-component Camassa-Holm (R2CH) system. First, we find a Hamiltonian structure of the R2CH system and use the stochastic Hamiltonian to derive the stochastic R2CH system. Then, we establish the local well-posedness of the stochastic R2CH system by the dispersion-dissipation approximation system and the regularization method. We also show a precise blow-up criterion for the stochastic R2CH system. Moreover, we prove the global existence of the stochastic R2CH system occurs with high probability. At last, we consider transport noise case and establish the local well-posedness and another blow-up criterion. | |
| April 28 (F) - | Hongjie Dong (Brown University) |
| 11:00am-12:00pm | Title: Recent results about the insulated conductivity problem. |
| Abstract: In the first part of the talk, I will present recent results about the insulated conductivity problem with closely spaced inclusions in a bounded domain in R^n. The gradient of solutions may blow up as the distance between inclusions approaches to 0. We obtained an optimal gradient estimate of solutions in terms of the distance, which settled down a major open problem in this area. In the second part, I will discuss a work regarding optimal estimates for higher derivatives of solutions of the conductivity problem with closely located circular inclusions in 2D, when the relative conductivities of inclusions have diff erent signs. This improves a recent result of Yong-Gwan Ji and Hyeonbae Kang. If time permits, I will also talk about a recent result about the insulated conductivity problem when the current-electric field relation is a power law.Based on joint work with Yanyan Li (Rutgers University), Zhuolun Yang (ICERM, Brown University), and Hanye Zhu (Brown University). | |
| May 2 (T) - | Sameer Iyer (University of California, Davis) |
| 11:00am--12:00pm | Title: Reversal in the Stationary Prandtl Equations. |
| Abstract: We investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which u > 0 and u < 0. The classical point of view of regarding the Prandtl equations as an evolution x completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi. |