# PDE Seminar Spring 2018

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

Date: Friday, January 26, 2018.

• Speaker: Naian Liao, Department of Mathematics, Chongqing University
• 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 D2V 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 D2V 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 Rn. 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: TBA.
• Abstract: TBA.

Date: Friday, April 6, 2018.

• Speaker: Thomas Hagen, Department of Mathematics, University of Memphis
• Title: TBA.
• Abstract: TBA.