# PDE Seminar 2013-2014

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

Date: **Friday, September 13, 2013**

- Speaker:
**Marcelo Disconzi, Vanderbilt University** - Title:
**Existence, regularity and convergence results for the compressible Euler equations** - Abstract: We study the problem of inviscid slightly compressible fluids in a bounded domain. We find a unique solution to the initial-boundary value problem and show that it is close to the analogous solution for an incompressible fluid. Furthermore we find that solutions to the compressible motion problem in Lagrangian coordinates depend differentiably on their initial data, an unexpected result for non-linear equations. This is joint work with David G. Ebin.

Date: **Friday, September 20, 3:10 pm**

- Speaker: Mariana Smit Vega Garcia, Purdue University
- Title:
**Optimal regularity in the thin obstacle problem with Lipschitz variable coefficients** - Abstract: We will describe the lower-dimensional obstacle problem for a uniformly elliptic, divergence form operator $L = div(A(x)\nabla)$ with Lipschitz continuous coefficients and discuss the optimal regularity of the solution. Our main result states that, similarly to what happens when $L = \Delta$, the variational solution has the optimal interior regularity. We achieve this by proving some new monotonicity formulas for an appropriate generalization of Almgren’s frequency functional. This is joint work with Nicola Garofalo.

Date: **Friday, September 27, 2013**

- Speaker: Cameron Browne, Vanderbilt University
- Title:
**Global analysis of an age-structured multi-strain virus model** - Abstract: We consider a general model of a within-host viral infection with multiple virus strains and explicit age-since-infection structure for infected cells. Existence and uniqueness of solutions, along with asymptotic smoothness of the nonlinear semigroup generated by the family of solutions, are established. For each viral strain, a quantity called the reproduction number is defined. The main result is that the single-strain equilibrium corresponding to the virus strain with maximal reproduction number is a global attractor (provided that this maximal reproduction number is greater than unity and all reproduction numbers are distinct). In other words, the virus strain with maximal reproduction number competitively excludes all other strains. As an application of the model, HIV evolution is considered and simulations are conducted.

Date: **Friday, October 4, 2013**

- Speaker:
**Yuanzhen Shao, Vanderbilt University** - Title:
**Analyticity of solutions to the Yamabe flow on non-compact manifolds** - Abstract: The Yamabe flow can be considered as an alternative approach to the famous Yamabe problem. Nowadays there is increasing interest in studying the Yamabe flow on non-compact manifolds. We show by means of continuous maximal regularity theory and the implicit function theorem that in every conformal class containing at least one real analytic metric, solutions to the Yamabe flow immediately become analytic jointly in time and space. In comparison with the existing results, we do not ask for a uniform bound on the curvatures of the initial metric. We will also briefly discuss a generalization of our results on singular manifolds.

Date: **Friday, November 1, 2013**

- Speaker:
**Naian Liao, Vanderbilt University** - Title:
**Analyticity of local solutions to a logarithmic diffusion equation** - Abstract: In this talk, I will explain some recent progress on the local behavior of a logarithmic diffusion equation. First of all, in spite of the singularity of the equation local solutions are analytic in space variable and arbitrarily differentiable in time variable if some appropriate conditions are satisfied. Secondly, a Harnack inequality in the topology of $L^1$ will be reported.

Date: **Friday, November 8, 2013**

- Speaker:
**Gieri Simonett, Vanderbilt University** - Title:
**Two-phase flows with phase transitions** - Abstract: A thermodynamically consistent model for two-phase flows including phase transitions driven by temperature is introduced and analyzed.

Date: **Friday, November 15, 2013**

- Speaker:
**Magdalena Czubak, Binghamton University (SUNY)** - Title:
**Uniqueness questions for the Navier-Stokes equation in the hyperbolic setting** - Abstract: The smoothness and uniqueness of the Leray-Hopf solutions to the Navier-Stokes equation is well-known in 2D. Contrary to what is known in the Euclidean setting, in our previous work we showed that there is non-uniqueness in 2D for simply connected, complete manifolds with negative sectional curvature. The goal of this talk is to show how we can restore uniqueness. In the process, we develop the theory of weak solutions to the Navier-Stokes equations on the 2D hyperbolic space. This is joint work with Chi Hin Chan.

Date: **Friday, December 6, 2013**

- Speaker: Caner Koca, Vanderbilt University
- Title:
**The Monge-Ampere Equations and Yau’s Proof of the Calabi Conjecture** - Abstract: The resolution of Calabi’s Conjecture by S.-T. Yau in 1977 is considered to be one of the crowning achievements in mathematics in 20th century. Although the statement of the conjecture is very geometric, Yau’s proof involves solving a non-linear second order elliptic PDE known as the complex Monge-Ampere equation. In this expository talk, I will start with the basic definitions and facts from geometry to understand the statement of the conjecture, then I will show how to turn it into a PDE problem, and finally I will highlight the important steps in Yau’s proof.

Date: **Friday, March 21, 2014**

- Speaker: Yuanzhen Shao
- Title: Analyticity of the interface of a thermodynamically consistent two-phase Stefan problem
- Abstract: We study the regularity of solutions to a thermodynamically consistent two-phase Stefan problem with or without kinetic undercooling. It is shown that the free interface of the problem immediately becomes analytic jointly in time and space, provided the initial surface satisfies a mild regularity assumption. The proof is based on a combination of a family of parameter-dependent diffeomorphisms, $L_p$-maximal regularity theory, and the implicit function theorem.

Date: **Friday, April 4, 2014**

- Speaker: Marcelo Disconzi, Vanderbilt University
- Title:
**The relativistic Navier-Stokes and Einstein’s equations.** - Abstract: Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein’s equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These last two conditions are given an appropriate relativistic interpretation. The solutions enjoy the domain of dependence or finite propagation speed property. We shall also discuss a work in progress, in collaboration with Magdalena Czubak, which removes the irrotational hypothesis.

Date: **Friday, April 18, 2014**

- Speaker: Glenn Webb, Vanderbilt University
- Title:
**Nonlocal Phenomena in Partial Differential Equations.** - Abstract: Four examples of nonlocal phenomena in partial differential equations will be presented.(1) Partial differential equations with time delay (nonlocal in time – the future depends not only on the present time, but a history back through time); (2) Age structure population models (nonlocal in the boundary condition – offspring are born at age 0 at time t from mothers with age between a1 and a2); (3) cell-cell adhesion models (nonlocal in the transport term – cells have a sensing radius R, on the order of several cell diameters, that modulates their adhesion to other cells); (4) interference phenomena in quantum mechanics (nonlocal in the probability density P(x,t) – the detection of a quantum particle is determined only probabilistically).

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