
Date Time 
Location  Speaker 
Title – click for abstract 

09/19 3:00pm 
BLOC 628 
James Kelliher UC Riverside 
Nonlinear PDEs Seminar
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. 

09/26 3:00pm 
BLOC 628 
Giles Auchmuty University of Houston 
Nonlinear PDEs Seminar
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. 

10/17 3:00pm 
BLOC 628 
Xin Liu Texas A&M University 
Nonlinear PDEs Seminar
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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10/24 3:00pm 
BLOC 628 
Siran Li Rice University 
Nonlinear PDEs Seminar
Title: Singular limits of NavierStokes and Euler equations with geometric effects.
Abstract: The NavierStokes and Euler equations are the PDEs describing the conservation of mass and momentum of fluids. One of the important topics of mathematical hydrodynamics is the singular limit problems. In this talk we discuss three studies on singular limits, with linkages to differential geometry.
1. For the incompressible Euler equations on $\mathbb{R}^2$ or $\mathbb{T}^2$, we establish a direct analogue with the GaussCodazzi equations in differential geometry. Thus, an inherent direct application of the results on isometric embeddings can lead to the existence of ''wild'' Euler solutions with low regularity, which are continuous in time. This may provide further insight to the works by De Lellis, Szekelyhidi and coauthors.
2. For the incompressible NavierStokes equations with Navier boundary conditions on a regular surface embedded in $\mathbb{R}^3$, we prove that the strong solution in $H^{r+1}$ converges in $H^r$ to that of the Euler equations, when $r \geq 5/2$.
3. For a ''nozzle'' with $n$dimensional base manifold $M$ and $d$dimensional varying crosssections $\epsilon F$, we show that, as $\epsilon \rightarrow 0$, the suitable weak solutions to the compressible NavierStokes equations on the nozzle converge to weak solutions of some limiting equations on $M$, which contain the $d$dimensional measure of the crosssections as a variable. This extends the work by BellaFeireislLewickaNotovny.
(Joint with Profs. Amit Acharya, GuiQiang Chen, Zhongmin Qian, Marshall Slemrod and Dehua Wang) 

10/31 3:00pm 
BLOC 628 
Alex Mahalov Arizona State University 
Nonlinear PDE's seminar
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.


11/07 3:00pm 
BLOC 628 
Suncica Canic University of Houston 
Nonlinear PDEs Seminar
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 