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Series: PDE Seminar

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

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

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

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

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

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

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

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

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

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.