# 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: Turbulence in Active Fluids Caused by Self-propulsion.
- Abstract: Self-sustained turbulent motion in microbiological suspensions present an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. This type so-called active or living fluids display turbulent behavior at low Reynolds regimes, a phenomenon that cannot be captured by classical fluid models. In a recent paper of Wensink et. al. a generalized Version of the Navier-Stokes equations is proposed to describe this so-called ‘active turbulence’.In my talk I will present an analytical approach to the generalized Navier-Stokes equations. The main purpose is to confirm the spontaneous formation of meso-scale vortices observed in experiments and simulations of self-propelled particles.

To capture the full asymptotics, we present an approach in three different frameworks:

– in L^2 on the whole euclidean space,

– in FM, i.e., in spaces of Fourier-transformed Radon measures, and

– in a setting with periodic boundary conditions.

Besides the active turbulence also furhter significant differences to the classical Navier-Stokes equations are observed, such as the existence of a global regular solution for arbitrary data in L^2.

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

- Speaker:
**Huy Nguyen, Department of Mathematics, Brown University.** - Title: A paradifferential approach for well-posedness and global

solutions for the Muskat problem. - Abstract: We study the Muskat problem in its full generality: one fluid or two fluids, with or without viscosity jump, with or without bottoms, and in arbitrary space dimension d of the interface. The Muskat problem is scaling invariant in the Sobolev space H
^{sc}(R^{d}) where s_{c}=1+d/2. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces H^{sc}(R^{d}), s>s_{c}. Moreover, in the infinite-depth case, if the initial interface is small in H^{sc}(R^{d}), s>s_{c}, we prove that the obtained solutions are global in time. The starting point of these results is a reformulation written solely in terms of the Drichlet-Neumann operator. The key elements of proofs are new paralinearization results for the Drichlet-Neumann operator in rough domains. This is joint work with B. Pausader (Brown U).

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

- Speaker:
**Mihaela Ignatova, Department of Mathematics, Temple University.** - Title: SQG in bounded domains.
- Abstract: I will describe results regarding the surface quasi-geostrophic equation (SQG) in bounded domains. The results concern global interior Lipschitz bounds for large data for the critical SQG in bounded domains. In order to obtain these, we establish nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. As an application global existence of weak solutions of SQG were obtained.

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

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

- Title: The Fractional Porous Medium Equation on Manifolds with Conic Singularities.
- Abstract: Due to the need to model long range diffusive interaction, during the last decade there has been a growing interest in considering diffusion equations involving non-local operators, e.g. the fractional powers of differential operators. In this talk, I will report some recent work with Nikolaos Roidos on the fractional porous medium equation on manifolds with cone-like singularities. I will show that most of the properties of the usual (local) porous medium equation, like existence, uniqueness of weak solution, comparison principle, conservation of mass, are inherited by the non-local version.

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: The L
^{2}exponential map in 2D and 3D hydrodynamics. - Abstract: In the 1960’s V. Arnold showed how solutions of the incompressible Euler equations can be viewed as geodesics on the group of diffeomorphisms of the fluid domain equipped with a metric given by fluid’s kinetic energy. The study of the exponential map of this metric is of particular interest and I will describe a few recent results its properties as well as some necessary background.

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