Date:  September 19, 2017 
Time:  3:00pm 
Location:  BLOC 628 
Speaker:  James Kelliher, UC Riverside 
Title:  Nonlinear PDEs Seminar 
Abstract:  Title: Bounded Vorticity, unbounded velocity solutions to the 2D Euler equations.
Abstract: The pioneering work on bounded vorticity solutions to the 2D Euler equations was done by Yudovich in the early 1960s, working in a bounded domain. He proved existence and uniqueness of such solutions. Subsequently, existence was shown to hold in much weaker settings (vorticity lying in any Lebesgue space), but uniqueness has only ever been extended to an incrementally larger class of initial data.
Yudovich's theory extends easily to the full plane (indeed it is slightly less technical there) as long as the velocity is assumed to decay sufficiently rapidly at infinity that the BiotSavart law holds. In 1995, Ph. Serfati established the existence and uniqueness of bounded vorticity solutions having no decay at infinity. Such solutions very much violate the BiotSavart law, but Serfati discovered an identity that the solutions hold that can be used as a kind of substitute for that law.
The boundedness of the velocity was very important in Serfati's argument, yet there is room in his identity to accommodate some growth of the velocity at infinity. I will speak on ongoing joint work with Elaine Cozzi in which we exploit Serfati's identity to obtain existence and uniqueness classes allowing growth at infinity as large as possible (without assuming any special symmetry of the initial data). Roughly speaking, we show that existence can be achieved only for very slowly growing velocities, but that uniqueness holds for velocities growing slower than the square root of the distance from the origin. We also consider the issue of continuous dependence on initial data, which is already an interesting problem even in Yudovich's original setting. 

Date:  September 26, 2017 
Time:  3:00pm 
Location:  BLOC 628 
Speaker:  Giles Auchmuty, University of Houston 
Title:  Nonlinear PDEs Seminar 
Abstract:  Title: Energy Bounds for Planar Divcurl Boundary Value Problems
Abstract: This talk will describe the derivation of sharp L^2 norm estimates for the solutions of divcurl boundary value problems on bounded Lipschitz planar regions Nontrivial prescribed tangential, normal or mixed boundary condition problems are considered of the type that arise in stationary electromagnetic eld modeling. The solutions are found using special decompositions of the vector elds in terms of scalar potentials, stream functions and harmonic elds. Explicit spectral formulae for the solutions are derived that involve various eigenvalues and eigenfunctions of the Laplacian including the Steklov eigenvalues and eigenfunctions. The bounds depend on certain least eigenvalues and the given data. 

Date:  October 17, 2017 
Time:  3:00pm 
Location:  BLOC 628 
Speaker:  Xin Liu, Texas A&M University 
Title:  Nonlinear PDEs Seminar 
Abstract:  Title: Some gasvacuum interface problems of compressible NavierStokes equations in spherically symmetric motions
Abstract:
I will talk about the wellposedness of two problems concerning the evolution of a flow connected with vacuum. The flow, or gas, connects the vacuum area in a way that the sound speed across the gasvacuum interface has only Holder continuity. A typical example is the LaneEmden solution for gaseous stars, where the sound speed is only 1/2Holder continuity on the gasvacuum interface. As pointed out by T.P. Liu in 1996, the classical hyperbolic method fails due to such singularity. Only recently, Jang and Masmoudi, Coutand, Lindblad and Shkoller independently developed some weighted energy estimates to show the wellposedness of the inviscid isentropic flows. This work is to investigate how the viscosity will help resolve such singularity. In particular, the equilibrium and the wellposedness of a model based on the thermodynamic model listed in Chandrasekhar’s book (An introduction to the study of stellar structure) is studied. Also, we investigate the global wellposedness of the NavierStokes equations, which allows the density and velocity to be large, the gas to connect to vacuum in a general manner but the energy to be small.
This is based on my Ph.D. thesis as a student of Prof. Zhouping Xin in the Chinese University of Hong Kong, Hong Kong.
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Date:  October 31, 2017 
Time:  3:00pm 
Location:  BLOC 628 
Speaker:  Alex Mahalov, Arizona State University 
Title:  Nonlinear PDE's seminar 
Abstract:  Title:Stochastic ThreeDimensional NavierStokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics
Abstract:
We consider stochastic threedimensional NavierStokes equations + Waves on long time intervals. Regularity results are established by bootstrapping from global regularity of the averaged stochastic resonant equations and convergence theorems. The averaged covariance operator couples stochastic and wave effects. The energy injected in the system by the noise is large, the initial condition has large energy, and the regularization time horizon is long. Regularization is the consequence of precise mechanisms of relevant threedimensional nonlinear interactions. We establish multiscale stochastic averaging, convergence and regularity theorems in a general framework. We also present theoretical and computational results for threedimensional nonlinear dynamics.


Date:  November 7, 2017 
Time:  3:00pm 
Location:  BLOC 628 
Speaker:  Suncica Canic, University of Houston 
Title:  Nonlinear PDEs Seminar 
Abstract:  Title: A mathematical framework for proving existence of
weak solutions to a class of fluidstructure interaction problems
Abstract:
The focus of this talk will be on nonlinear movingboundary problems involving
incompressible, viscous fluids and elastic structures. The fluid and structure are
coupled via two sets of coupling conditions, which are imposed on a deformed
fluidstructure interface. The main difficulty in studying this class of problems
from the analysis and numerical points of view comes from the strong geometric
nonlinearity due to the nonlinear fluidstructure coupling. We have recently
developed a robust framework for proving existence of weak solutions to this
class of problems, allowing the treatment of various structures (Koiter shell, multi
layered composite structures, meshsupported structures), and various coupling
conditions (noslip and Navier slip). The existence proofs are constructive: they
are based on the timediscretization via Lie operator splitting, and on our
generalization of the famous LionsAubinSimon’s compactness lemma to
moving boundary problems. The constructive proof strategy can be used in the
design of a looselycoupled partitioned scheme, in which the fluid and structure
subproblems are solved separately, with the cleverly designed boundary
conditions to enforce the coupling in a way that approximates well the continuous
energy of the coupled problem. This provides stability and uniform energy
estimates, important for the convergence proof of the numerical scheme.
Applications of this strategy to the simulations of reallife problems will be shown.
They include the flow of blood in a multilayered coronary artery treated with
vascular devices called stents (with Dr. Paniagua (Texas Heart Institute) and Drs.
Little and Barker, Methodist Hospital, Houston), and optimal design of micro
swimmers and biorobots (with biomed. engineer Prof. Zorlutuna, Notre Dame).
Parts of the mathematical work are joint with B. Muha 

Date:  November 10, 2017 
Time:  1:50pm 
Location:  BLOC 628 
Speaker:  Luan T. Hoang, Texas Tech University 
Title:  Nonlinear PDEs (joint with Mathematical Physics Seminar) 
Abstract:  Title: Largetime asymptotic expansions for solutions of NavierStokes equations
Abstract: We study the longtime behavior of solutions to the threedimensional NavierStokes equations of viscous, incompressible fluids with periodic boundary conditions. The body forces decay in time either exponentially or algebraically. We establish the asymptotic expansions of FoiasSauttype for all LerayHopf weak solutions. If the force has an asymptotic expansion, as time tends to infinity, in terms of exponential functions or negativepower functions, then any weak solution admits an asymptotic expansion of the same type. Moreover, when the force's expansion holds in Gevrey spaces, which have much stronger norms than the Sobolev spaces, then so does the solution's expansion. This extends the previous results of Foias and Saut in Sobolev spaces for the case of potential forces.

