# Poster Session and Reception

Thursday, June 23, 2016 - 3:15pm - 5:15pm

Lind 400

**On the Explicit Construction of Solvable Structures for ODEs from Non-solvable Symmetry Lie algebras: The Case of sl(2).**

Adrian Ruiz Servan (University of Cádiz)

A solvable structure for third order ordinary differential equations admitting the non-solvable symmetry algebra sl(2) is explicitly computed by means of the generators of the algebra. Once the solvable structure is determined, three functionally independent first integrals of the given equation can be derived by using quadratures alone. This leads to the complete integration of the equation by quadratures as in the case of solvable symmetry algebras.**Quasiperiodic Solutions to Elliptic Equations in R^d**

Dario Valdebenito (University of Minnesota, Twin Cities)

We establish sufficient conditions for the existence of solutions of certain semilinear elliptic equations, which are quasiperiodic in one variable, and decaying the remaining variables. These results apply for nonlinearities which are either small or have a small quadratic leading term. Such solutions are found using a center manifold reduction and results from the KAM theory. These sufficient conditions are satisfied by a large class of potentials and nonlinearities, and we focus specifically on the case where these functions are radially symmetric.**The Study of Global Stabilities of Leslie-type and Holling-Tanner Predator Prey Models**

Yi Zhu (University of Central Florida)

This poster shows that we study the global stabilities of diffusive predator-prey systems of Leslie-type and Holling-Tanner type in a bounded domain $\Omega\subset R^N$ with no-flux boundary condition. By using a new approach, we establish much improved global asymptotic stability of the unique positive equilibrium solution. We also show how to extend the result to more general type of systems with heterogeneous environment and/or wider class of kinetic terms.**On Thermodynamic Equations in Layered Tornado Model**

Mikhail Shvartsman (University of St. Thomas)

We explore the energy balance in a thunderstorm, in particular, how energy is redistributed on a local level inside a tornado-like flow. The notions of non equilibrium thermodynamics are used to describe the problem. We show that fluctuations on a macroscopic level play an especially important role in this model.**Power Laws for Axisymmetric Vortical Flows**

Pavel Belik (Augsburg College)

We explore the rates of decay in the tangential velocity and axial vorticity in various idealized axisymmetric vortical flows that include bathtub vortices, vortex chamber vortices, dust devils, supercell tornadoes, mesocyclones, and tropical cyclones. These all exhibit a common power law where the velocity decays like $r^{-b}$ for some 0 < $b$ < 1. We review the empirical evidence for such a law, provide results from a simple numerical simulation, and provide an analytical model extending Serrin's swirling vortex model to include the $r^{-b}$ decay.**Sobolev Stability Threshold for 2D Shear flows Near Couette**

Fei Wang (University of Southern California)

We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\ U(y) - y\_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon \ll \nu^{1/2}$, where $\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-close in $H^1$ to $(e^{t \nu \partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \gg \nu^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales as $\nu^{1/2}$ for 2D shear flows close to the Couette flow.**Center Manifold Computations via the Lyapunov-Perron Method**

Yu-Min Chung (College of William and Mary)Emily Schaal (College of William and Mary)

M.S. Jolly and R. Rosa in 2005 develop an iterative algorithm based on the Lyapunov-Perron method to compute the center manifold. The main problem is how to discretize the Lyapunov-Perron operator, which is an integral operator we re-define on what we show to be a proper functional space. We also find a simple formulation so that generic schemes, such as the Runge-Kutta methods, can be utilized. Using a test problem, we demonstrate the implementation of the algorithm.**Inertial Manifolds for the 3D Modified Leray-α (ML-α) Model with Periodic Boundary Conditions**

Anna Kostianko (University of Surrey)

The existence of an inertial manifold for the ML-α model (which is one of the truncated versions of the Navier-Stokes equations) with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averaging principle. This method was introduced by G. Sell and J. Mallet-Paret in order to treat reaction-diffusion equations on a 3D torus. The novelty of this work lies in the application of the spatial averaging principle to the system of equations, which cannot be done in general situation but appears to be possible in our case, because the spatial averaging of the derivative of the non-linearity of the ML-α model is identically zero. Furthermore, the adaptation of classical Perron method for constructing inertial manifolds is proposed for the particular case of zero spatial averaging.**On the Ghost Solutions of 2D Navier-Stokes Equations**

Bingsheng Zhang (Texas A & M University)

We study the geometric properties for the solutions in the global attractor of 2D NSE whose projection into the energy-enstropy plane is a single point.**On Gevrey Regularity of the supercritical SQG Equation**

Vincent Martinez (Tulane University)

Gevrey regularity for the supercritical SQG equation is established in the scale of critical Besov spaces, thereby extending the Gevrey norm energy-technique of Foias-Temam to a subanalytic, $L^p$ setting. The approach is based on an approach of Lemarie-Rieusset, who established a suitable product estimate in an analytic Gevrey class by establishing boundedness of an associated bilinear multiplier operator. It is shown that although the symbol of our operator satisfies decay estimates different from the Coffman-Meyer condition, one can nevertheless deduce boundedness by exploiting additional spectral localizations made available by working in the Besov space setting.**New Spiral Solutions for the Navier-Stokes Equations**

Julien Guillod (University of Minnesota, Twin Cities)

New explicit solutions to the incompressible Navier-Stokes equations in $\mathbb{R}^2\setminus\{0\}$ are determined, which generalize the scale-invariant solutions found by Hamel. These new solutions are invariant under a particular combination of the scaling and rotational symmetries. They are the only solutions invariant under this new symmetry in the same way as the Hamel solutions are the only scale-invariant solutions. While the Hamel solutions are parameterized by a discrete parameter $n$, the flux $\Phi$, and an angle $\theta_0$, the new solutions generalize the Hamel solutions by introducing an additional parameter $\kappa$ which produces a rotation. The new solutions decay like $x^{-1}$ as the Hamel solutions and exhibit spiral behavior. The new variety of asymptotes induced by the existence of these solutions further emphasizes the difficulties faced when trying to establish the asymptotic behavior of the Navier-Stokes equations in a two-dimensional exterior domain or in the whole plane. This is a joint work with Peter Wittwer from the University of Geneva.