# PDE Seminar Spring 2019

### Fridays, 4:10pm, Stevenson Center 1307

Date: **Friday, Feb 1, 2019.**

- Speaker:
**Mahdi Mohebbi, Department of Mechanical Engineering, SUNY Korea.** - Title: Resonance and the existence of time periodic solutions in hyperbolic-parabolic coupled systems.
- Abstract: We consider coupled systems of PDEs (with special interest to hyperbolic-parabolic ones). We begin by clarifying the notion of resonance from the physical point of view. This leads to a precise mathematical definition of resonance when our system can be represented by a dynamical system. The goal is to study the relation between resonance, the existence of periodic solutions and the asymptotic behavior of solutions to the corresponding homogeneous system. In the case of linear systems, where the dynamical system is described by a C_0-semigroup, a complete description can be given based on previously known results, which we will review quickly. Moving to nonlinear systems, a (simplest) case of a harmonic oscillator in an infinite channel filled with a Newtonian fluid will be considered. The existence of time periodic solutions corresponding to arbitrary periodic forces and the asymptotic behavior of solutions with no forcing term will be discussed.

Date: **Friday, Feb 8, 2019.**

- Speaker:
**Naian Liao, Department of Mathematics, Chongqing University, China.** - Title: Continuity of solutions to a logarithmic diffusion equation.
- Abstract: In this talk, I will first give a brief introduction on the regularity theory for elliptic and parabolic equations, including degenerate and singular ones. Then, I will report some recent progress on the local regularity of solutions to a logarithmically singular parabolic equation.

Date: **Friday, Feb 22, 2019.**

- Speaker:
**Qingtang Su, Department of Mathematics,University of Michigan.** - Title: Long time behavior of 2d water waves with point vortices.
- Abstract: We study the motion of the two dimensional inviscid incompressible, infinite depth water waves with point vortices in the fluid. We show that Taylor sign condition ∂P/∂n >= 0 can fail if the point vortices are sufficient close to the free boundary, so the water waves could be subject to the Taylor instability. Assuming the Taylor sign condition, we prove that the water wave system is locally wellposed in Sobolev spaces. Moreover, we show that if the water waves is symmetric with a certain symmetric vortex pair traveling downward initially, then the free interface remains smooth for a long time, and for initial data of size ε << 1, the lifespan is at least O(ε
^{-2}).

Date: **Friday, March 1, 2019.**

- Speaker:
**Irena Lasiecka, Department of Mathematics, University of Memphis.** - Title: Evolutions arising in flow-structure interactions.
- Abstract: Fluid-structure interactions and flow-structure interactions are ubiquitous in nature. Problems such as attenuation of turbulence or flutter in an oscillating structure are prime examples of relevant applications. Mathematically, the models are represented by nonlinear Partial Dierential Equations (Navier Stokes-Euler equations and nonlinear elasticity) which display a strong boundary-type coupling at the interface between the two media. Moreover, in most models, the dynamical character of the two PDEs evolving on their corresponding domains is different and the overall system may display a parabolic/hyperbolic or hyperbolic/hyperbolic coupling, separated by the interface. This provides for a rich mathematical structure opening the door to several unresolved problems in the area of nonlinear PDE’s, dynamical systems and related harmonic analysis and geometry. Of particular interest are models with mixed boundary conditions [such as Kutta Joukovsky boundary conditions] which lead to a plethora of open problems in elliptic theory with related Hilbert-Riesz transform theory. This talk aims at providing a brief overview of recent developments in the area along with a presentation of some recent advances addressing the issues of mixed boundary conditions arising in modeling of panels fluttering in a non-viscous environment.

Date: **Friday, March 22, 2019.**

- Speaker:
**Jürgen Saal, Heinrich-Heine-Universität Düsseldorf, Germany.** - Title: TBA.
- Abstract: TBA.

Date: **Friday, March 29, 2019.**

- Speaker:
**Huy Nguyen, Department of Mathematics, Brown University.** - Title: TBA.
- Abstract: TBA.

Date: **Friday, April 5, 2019.**

- Speaker:
**Mihaela Ignatova, Department of Mathematics, Temple University.** - Title: TBA.
- Abstract: TBA.

Date: **Friday, April 12, 2019.**

- Speaker:
**Jeremy LeCrone, Department of Mathematics, University of Richmond.** - Title: TBA.
- Abstract: TBA.

Date: **Friday, April 12, 2019, 5:10pm (note different time; two talks on April 12, see above).**

- Speaker:
**Yuanzhen Shao, Department of Mathematics, Georgia Southern University.** - Title: TBA.
- Abstract: TBA.

Date: **Friday, April 19, 2019.**

- Speaker:
**Christian Zillinger, Department of Mathematics, University of Southern California.** - Title: On the linear forced Euler and Navier-Stokes equations: damping and modified scattering.
- Abstract: We study the long-time asymptotic behavior of the linearized Euler and nonlinear Navier-Stokes equations close to Couette flow. As a main result we show that suitable forcing breaks asymptotic stability results at the level of the vorticity, but that solutions never the less exhibit convergence of the velocity field. Thus, here linear inviscid damping persists despite instability of the vorticity equations.

Date: **Friday, April 26, 2019.**

- Speaker:
**Gerard Misiolek, Department of Mathematics, University of Notre Dame.** - Title: TBA.
- Abstract: TBA.

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