New Trends in Nonlinear PDEs


9:00-9:45. Marshall Slemrod

Title: The problem with Hilbert's 6th problem.

Abstract: In his famous 1900 ICM address Hilbert proposed his famous list of problems for the 20th century. Among these was his 6th problem which was less clearly formulated than the others but dealt with a rigorous derivation of the macroscopic equations of continuum mechanics from the available microscopic theory of his time, i.e. statistical mechanics and specifically Boltzmann's kinetic theory of gases. The problem has drawn attention from analysts over the years and even Hilbert himself made a contribution. In this talk I will note how an exact summation of the Chapman-Enskog expansion for the Boltzmannuation due to Ilya Karlin ( ETH) and Alexander Gorban (Leicester) can be used to represent solutions of the Boltzmann equation and then show that these solutions CANNOT converge the classical balance laws of mass, momentum, and energy associated the Euler equation of compressible gas dynamics. Hence alas  Hilbert's program (at least with respect to gas dynamics) has a negative outcome.

Additional Links:

Beyond Navier Stokes

https://www.quantamagazine.org/20150721-famous-fluid-equations-are-incomplete/

9:50-10:35. Athanasios Tzavaras

King Abdullah University of Science and Technology (KAUST)

Title: Kinetic models for the description of sedimenting suspensions

Abstract: I review some works on modeling and the mathematical theory for dilute suspensions of rigid rods. Such problems appear in modeling  sedimentation of suspensions of particles. Similar in spirit models are also used for modeling swimming micro-organisms. Here, we focus on a  class of models introduced by  Doi and describing suspensions of rod--like molecules in a solvent fluid. They couple a microscopic Fokker-Planck type equation for the probability distribution of rod orientations to a macroscopic Stokes flow. One objective is to compare such models with traditional models used in macoscopic viscoelasticity as the well known Oldroyd model. In particular: For the problem of sedimenting rods under the influence of gravity we discuss the instability of the quiescent flow and  the derivation of effective equations describing the collective response. We derive two such effective theories: (i) One ammounts to a classical diffusive limit and produces a Keller-Segel type of model. (ii) A second approach involves the derivation of a moment closure theory and the approximation of moments via a quasi-dynamic approximation. This produces a model that belongs to the classof flux-limited Keller-Segel systems. The two theories are compared numerically with the kinetic equation. (joint work with Christiane Helzel, Univ. Duesseldorf).


11:10- 11:55. Emmanouil Milakis, UCY

Title: On the Parabolic Fractional Obstacle Problem

Abstract: Obstacle problems are characterised by the fact that the solution must satisfy unilateral constraint i.e. must remain, on its domain of definition or part of it, above a given function the so called obstacle. Parabolic obstacle problems, i.e. when the involved operators are of parabolic type, can be formulated in various ways such as a system of inequalities, variational inequalities or Hamilton- Jacobi equation. In the first part of the talk, I will briefly explain the formulation of elliptic and parabolic obstacle problems and will connect them with the corresponding extension problems for the fractional Laplacian and fractional Ηeat. In the second part of the talk, I will present some of our recent result on the so-called  non-dynamic parabolic Fractional Obstacle Problem. We will discuss how to obtain higher regularity as well as optimal regularity of the space derivatives of the solution. Furthermore, at free boundary points of positive parabolic density, I will describe how the Holder continuity of the time derivative is obtained. Based on joint works with I. Athanasopoulos and L. Caffarelli.

12:00- 12:45. Yiorgos-Socratis Smyrlis, UCY

Title: Optimal analyticity estimates for non-linear dissipative equations

Abstract: We investigate the spatial analyticity of solutions of a class of evolutionary pseudo-differential equations with Burgers' nonlinearity, which are periodic in space, and possess global attractors. We examine their analyticity by utilising a criterion involving the rate of growth of suitable norms of the n-th derivative of the solution, growth of the n-th spatial derivative is obtained by a spectral method. We prove that the solutions are analytic if the order of the pseudo-differential operator is higher than one. We also present numerical evidence suggesting that this is optimal, i.e., if the order is not larger that one, then the solution is not in general analytic. These ideas can be applied to a wide class of dissipative-dispersive pseudo-differential equations.



Date: May 19, 2017

Speakers and Abstracts