## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, April 19, 2011 - 11:00 , Location: Skiles 005 , Prof. Yuxi Zheng , Penn State University and Yeshiva University, , , Organizer: Ronghua Pan
We consider Riemann problems for the compressible Euler system in aerodynamics in two space dimensions. The solutionsinvolve shock waves, hyperbolic and elliptic regions. There are also regions which we call semi-hyperbolic. We have shownbefore the existence of such solutions, and now we show regularity of the boundaries of such regions.
Series: PDE Seminar
Tuesday, April 12, 2011 - 11:00 , Location: Skiles 255 , Prof. Divakar Viswanath , University of Michigan , , Organizer: Ronghua Pan
The incompressible Navier-Stokes equations provide an adequate physical model of a variety of physical phenomena. However, when the fluid speeds are not too low, the equations possess very complicated solutions making both mathematical theory and numerical work challenging. If time is discretized by treating the inertial term explicitly, each time step of the solver is a linear boundary value problem. We show how to solve this linear boundary value problem using Green's functions, assuming the channel and plane Couette geometries. The advantage of using Green's functions is that numerical derivatives are replaced by numerical integrals. However, the mere use of Green's functions does not result in a good solver. Numerical derivatives can come in through the nonlinear inertial term or the incompressibility constraint, even if the linear boundary value problem is tackled using Green's functions. In addition, the boundary value problem will be singularly perturbed at high Reynolds numbers. We show how to eliminate all numerical derivatives in the wall-normal direction and to cast the integrals into a form that is robust in the singularly perturbed limit. [This talk is based on joint work with Tobasco].
Series: PDE Seminar
Tuesday, March 15, 2011 - 11:00 , Location: Skiles 005 , , Department of Mathematics, University of Chicago , , Organizer: Ronghua Pan
I'll talk about a couple of commutator estimates and their role in the proofs of existence and uniqueness of solutions of active scalar equations with singular integral constitutive relations like the generalized SQG and Oldroyd B models.
Series: PDE Seminar
Tuesday, March 1, 2011 - 11:00 , Location: Skiles 005 , , Champlain College and McGill University , , Organizer: Ronghua Pan
In this talk, we consider the n-dimensional bipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations. In 1-D case, when the difference between the initial electron mass and the initial hole mass is non-zero (switch-on case), the stability of nonlinear diffusion wave has been open for a long time. In order to overcome this difficulty, we ingeniously construct some new correction functions to delete the gaps between the original solutions and the diffusion waves in L^2-space, so that we can deal with the one dimensional case for general perturbations, and prove the L^\infty-stability of diffusion waves in 1-D case. The optimal convergence rates are also obtained. Furthermore, based on the results of one-dimension, we establish some crucial energy estimates and apply a new but key inequality to prove the stability of planar diffusion waves in n-D case, which is the first result for the multi-dimensional bipolar hydrodynamic model of semiconductors, as we know. This is a joint work with Feimin Huang and Yong Wang.
Series: PDE Seminar
Tuesday, February 22, 2011 - 11:00 , Location: Skiles 005 , Prof. Nicola Gigli , University of Nice , , Organizer: Ronghua Pan
Aim of the talk is to make a survey on some recent results concerning analysis over spaces with Ricci curvature bounded from below. I will show that the heat flow in such setting can be equivalently built either as gradient flow of the natural Dirichlet energy in L^2 or as gradient flow if the relative entropy in the Wasserstein space. I will also show how such identification can lead to interesting analytic and geometric insights on the structures of the spaces themselves. From a collaboration with L.Ambrosio and G.Savare
Series: PDE Seminar
Tuesday, February 1, 2011 - 10:00 , Location: Skiles 005 , , Department of Mathematics, University of Houston , , Organizer: Ronghua Pan
Mathematical modeling, analysis and numerical simulation, combined with imagingand experimental validation, provide a powerful tool for studying various aspects ofcardiovascular treatment and diagnosis. At the same time, problems motivated bycardiovascular applications give rise to mathematical problems whose studyrequires the development of sophisticated mathematical techniques. This talk willaddress two examples where such a synergy led to novel mathematical results anddirections. The first example concerns a mathematical study of the benchmarkproblem of fluid‐structure interaction (FSI) in blood flow. The resulting problem is anonlinear moving‐boundary problem coupling the flow of a viscous, incompressiblefluid with the motion of a linearly viscoelastic membrane/shell. An existence resultfor an effective, reduced model will be presented.The second example concerns a novel dimension reduction/multi‐scale approach tomodeling of endovascular stents as 3D meshes of 1D curved rods. The resultingmodel is in the form of a nonlinear hyperbolic network, for which no generalexistence results are available. The modeling background and the challenges relatedto the analysis of the solutions will be presented. An application to the study of themechanical properties of the currently available coronary stents on the US marketwill be shown.This talk will be accessible to a wide scientific audience.Collaborators include: Josip Tambaca (University of Zagreb, Croatia), Ando Mikelic(University of Lyon 1, France), Dr. David Paniagua (Texas Heart Institute), and Dr.Stephen Little (Methodist Hospital in Houston).
Series: PDE Seminar
Tuesday, January 25, 2011 - 14:00 , Location: Skiles 006 , Dr. Suleyman Ulusoy , University of Maryland , Organizer: Ronghua Pan

Note the unusual time and room

We investigate the long-time behavior of weak solutions to the  thin-film type equation $$v_t =(xv - vv_{xxx})_x\ ,$$ which arises in the Hele-Shaw problem. We estimate the rate of  convergence of solutions  to the Smyth-Hill equilibrium solution, which has the form  $\frac{1}{24}(C^2-x^2)^2_+$,  in the norm $$|| f ||_{m,1}^2 = \int_{\R}(1+ |x|^{2m})|f(x)|^2\dd x + \int_{\R}|f_x(x)|^2\dd x\ .$$ We obtain exponential convergence in the $|\!|\!| \cdot |\!|\!|_{m,1}$ norm for all $m$ with $1\leq m< 2$, thus obtaining  rates of convergence in norms measuring both smoothness and  localization. The localization is the main novelty, and in fact, we  show that there is a close connection between the localization bounds and the smoothness  bounds: Convergence of second moments implies convergence in the $H^1$ Sobolev norm.   We then use methods of optimal mass  transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak  solutions for which all of the estimates on which our convergence  analysis depends may be rigorously derived. Though  our main results  on convergence can be stated without reference to optimal mass  transportation, essential use of this theory is made throughout our analysis.This is a joint work with Eric A. Carlen.
Series: PDE Seminar
Tuesday, November 30, 2010 - 15:00 , Location: Skiles 255 , Prof. Mikhail Perepelitsa , University of Houston , , Organizer: Ronghua Pan
In this talk we will discuss the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. We will follow the approach of R.DiPerna (1983) and reduce the problem to the study of a measure-valued solution of the Euler equations, obtained as a limit of a sequence of the vanishing viscosity solutions. For a fixed pair (x,t), the (Young) measure representing the solution encodes the oscillations of the vanishing viscosity solutions near (x,t). The Tartar-Murat commutator relation with respect to two pairs of weak entropy-entropy flux kernels is used to show that the solution takes only Dirac mass values and thus it is a weak solution of the Euler equations in the usual sense. In DiPerna's paper and the follow-up papers by other authors this approach was implemented for the system of the Euler equations with the artificial viscosity. The extension of this technique to the system of the Navier-Stokes equations is complicated because of the lack of uniform (with respect to the vanishing viscosity), pointwise estimates for the solutions. We will discuss how to obtain the Tartar-Murat commutator relation and to work out the reduction argument using only the standard energy estimates. This is a joint work with Gui-Qiang Chen (Oxford University and Northwestern University).
Series: PDE Seminar
Tuesday, November 9, 2010 - 15:05 , Location: Skiles 255 , Prof. Luis Silvestre , University of Chicago , , Organizer: Ronghua Pan
We prove a new Holder estimate for drift-(fractional)diffusion equations similar to the one recently obtained by Caffarelli and Vasseur, but for bounded drifts that are not necessarily divergence free. We use this estimate to study the regularity of solutions to either the Hamilton-Jacobi equation or conservation laws with critical fractional diffusion.
Series: PDE Seminar
Tuesday, November 2, 2010 - 15:00 , Location: Skiles 255 , , Ohio State University , , Organizer: Ronghua Pan
We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the combined effect of death rate, sinking or buoyant coe±cient, water column depth and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. This is based upon a joint work with Sze-Bi Hsu, National Tsing-Hua University.