# PDE Seminar Spring 2018

### Fridays, 4:10pm, Stevenson Center 1307

Date: **Friday, January 26, 2018.**

- Speaker:
**Naian Liao, Department of Mathematics, Chongqing University (China)** - Title: A Wiener-type estimate for singular diffusion equations.
- Abstract: In this talk, I will give an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions of singular parabolic equations of p-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity.

Date: **Thursday, February 8, 2018, 5:15pm, SC 1431 **(note different day, time, and location).

- Speaker:
**David G. Ebin, Department of Mathematics, Stony Brook University** - Title: Motions with Strong Constraining Force.
- Abstract: We will begin describing a general theory of motions with a strong constraining force. Let N be a Riemannian manifold and let M be a submanifold. Consider a geodesic x(t) in M. Let V be a function on N for which M is a strict minimum. Let
*v*(M) be the normal bundle of M in N and assume that D^{2}V gives a positive definite bilinear form on*v*(M). Let y(t) be a curve in M which would be a geodesic in N except that it is accelerated by -∇V. Using the exponential map of M we can construct a tubular neighborhood of M in N. Give it coordinates (u,v) where u describes position in M and v is the deviation from M in the normal direction (u and v can be vector valued. In fact if M and the fibers of of*v*(M) are infinite dimensional, u and v will take values in a Banach space). With these coordinates we write y(t) = (u(t),v(t)). Then the differential equation for y gives a a coupled pair of equations for u and v. The equation for u will be the geodesic equation for x with some error terms and the equation for v will be an oscillatory equation with an inhomegeneous term which is quadratic in u. We consider the case that y starts on M and dy/dt is tangent to M. Then v(0) and dv(0)/dt will be zero. If D^{2}V has all eigenvalues greater than k, we will find that v has magnitude like 1/k and the distance from x to u will be bounded by a term of order 1/k. Thus the curve y will be close to x so the motion is approximated by motion in the submanifold. The above construction has several physical applications.

We will first discuss the case of slightly compressible fluids in some domain Ω. In this case N will be the group of diffeomorphisms of Ω and M will be the submanifold of volume preserving diffeomorphisms. V will be constructed from the relation between pressure and density in the fluid and k will correspond to the derivative of pressure with respect to density so slight compressibility corresponds to large k. We will see how slightly compressible motion differs from incompressible motion. The difference will be related to propagation of sound in the fluid. Second case: if Ω is n-dimensional, we will discuss the space of volume preserving embeddings of Ω into R^{n}. This should be the configuration space for an incompressible fluid that starts in Ω, but whose boundary is free to move. The forcing function will then be formed from the surface tension of the fluid with k being the constant of surface tension. The submanifold will again be the group of volume-preserving diffeomorphisms. If k is large the boundary will move only slightly and in the limit of infinite k, the motion will be in the submanifold so the boundary will be fixed.

Date: **Friday, March 30, 2018.**

- Speaker:
**Tarek Elgindi, Department of Mathematics, UC San Diego** - Title: Singularity Formation in Incompressible Fluids.
- Abstract: We will discuss some recent results on finite-time and infinite-time singularity formation for strong solutions to the incompressible Euler equations and related fluid models. Our basic approach is to use scale-invariance to derive simpler lower dimensional equations where singularity formation can be established in a relatively straightforward fashion. Based on joint works with I. Jeong.

Date: **Friday, April 6, 2018.**

- Speaker:
**Thomas Hagen, Department of Mathematics, University of Memphis** - Title: Volume Scavenging of networked droplets.
- Abstract: The Florida Palm Beetle has a clever defense mechanism against predators: excreting oily liquids through tarsal channels and maintaining strong ground suction by controlling the resultant liquid bridges with the substrate. I will discuss the underlying grab-and-release mechanism in terms of capillarity-induced flows of liquids in networks of channels. These channel flows are driven by volume scavenging where fluid droplets of varying sizes leech off one another to increase in volume.The underlying model is given by a large gradient system of differential equations on a simple, connected graph. It generalizes an earlier Newtonian model by van Lengerich, Vogel and Steen. The presentation will report about previously unknown equilibrium solutions which are most relevant for the suction process, a (surprisingly) complete, rigorous classification of their stability in dependence on the relevant bifurcation parameters as well as related results on forward invariant sets and hierarchies of equilibria. The predicted long-term dynamics will be demonstrated with some animations.I will also briefly address the “infinite droplet limit” on specific graphs. In this limit the governing equations formally reduce to a single evolution partial differential equation which exhibits both forward and backward diffusion.This is joint work with Paul H. Steen (Cornell).

Date: **Friday, April 13, 2018.**

- Speaker:
**Pierre Magal, University of Bordeaux (France)** - Title: Some Epidemic models with age of infection.
- Abstract: We study an infection-age model of disease transmission, where both the infectiousness and the removal rate may depend on the infection age. In order to study persistence, the system is described using integrated semigroups. If the basic reproduction number R_0<1, then the disease-free equilibrium is globally asymptotically stable. For R_0>1, a Lyapunov functional is used to show that the unique endemic equilibrium is globally stable amongst solutions for which disease transmission occurs.

Date: **Friday, April 20, 2018.**

- Speaker:
**Dan Popovici, Department of Mathematics, University of Toulouse (France)** - Title: Non-Kähler Mirror Symmetry of the Iwasawa Manifold.
- Abstract: We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold X, a well-known non-Kähler compact complex manifold of dimension 3, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical Kähler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of X.

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