# Past Partial Differential Equations Seminar

Leinster and Willerton have introduced the concept of the magnitude of a metric space, as a special case as that of an enriched category. It is a numerical invariant which is designed to capture the important geometric information about the space, but concrete examples of ts values on compact sets in euclidean space have hitherto been lacking. We discuss progress in some conjectures of Leinster and Willerton.

We study the adaptive finite element approximation of the Dirichlet problem $-\Delta u = f$ with zero boundary values using newest vertex bisection. Our approach is based on the minimization of the corresponding Dirichlet energy. We show that the maximums strategy attains every energy level with a number of degrees of freedom, which is proportional to the optimal number. As a consequence we achieve instance optimality of the error. This is a joint work with Christian Kreuzer (Bochum) and Rob Stevenson (Amsterdam).

We establish sharp Sobolev inequalities of order four on Euclidean $d$-balls for $d$ greater than or equal to four. When $d=4$, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean $2$-balls. Our method relies on the use of scattering theory on hyperbolic $d$-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$-balls. This is joint work with Alice Chang.

This will be an overview of Prof Stroffolini's research and precursor to the eight-hour mini-course Prof Stroffolini will be giving later in October.

The general problem of shock formation in three space dimensions was solved by Christodoulou in 2007. In his work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but in addition has singular initial data because of the behavior of the solution at the blowup surface. In my talk I will present the solution to this problem in the case of spherical symmetry. This is joint work with Demetrios Christodoulou.

We consider the system of nonlinear differential equations

\begin{equation}

(1) \qquad

\begin{cases}

\dot u_n(t) + \lambda^{2n} u_n(t)

- \lambda^{\beta n} u_{n-1}(t)^2 + \lambda^{\beta(n+1)} u_n(t) u_{n+1}(t) = 0,\\

u_n(0) = a_n, n \in \mathbb{N}, \quad \lambda > 1, \beta > 0.

\end{cases}

\end{equation}

In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A. Cheskidov proved that the system (1) has a unique "strong" solution if $\beta \le 2$, whereas the "strong" solution does not exist if $\beta > 3$. (Note, that the 3D-Navier-Stokes equations correspond to the value $\beta = 5/2$.) We show that for sufficiently "good" initial data the system (1) has a unique Leray-Hopf solution for all $\beta > 0$.

We look at the construction of radial metrics with an isolated singularity for the constant fractional curvature equation. This is a semilinear, non-local equation involving the fractional Laplacian, and appears naturally in conformal geometry.

In the talk we present a survey of recent results (see [4]-[6]) on the existence theorems for the steady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D cases for bounded and exterior domains (the so called *Leray problem*, inspired by the classical paper [8]). One of the main tools is the Morse-Sard Theorem for the Sobolev functions $f\in W^2_1(\mathbb R^2)$ [1] (see also [2]-[3] for the multidimensional case). This theorem guaranties that almost all level lines of such functions are $C^1$-curves besides the function $f$ itself could be not $C^1$-regular.

Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equations for axially symmetric 3D solutions in the absence of swirl (see [1]).

**References**

- Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of Sobolev and BV functions, Rev. Mat. Iberoam.,
**29**, No. 1, 1-23 (2013). - Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on $\mathbb R^n$, Journal fur die reine und angewandte Mathematik (Crelles Journal) (Online first 2013).
- Korobkov M. V., Kristensen J., On the Morse-Sard Theorem for the sharp case of Sobolev mappings, Indiana Univ. Math. J.,
**63**, No. 6, 1703-1724 (2014). - Korobkov M. V., Pileckas K., Russo R., The existence theorem for steady Navier-Stokes equations in the axially symmetric case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5),
**14**, No. 1, 233-262 (2015). - Korobkov M. V., Pileckas K., Russo R., Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains, Ann. of Math.,
**181**, No. 2, 769-807 (2015). - Korobkov M. V., Pileckas K., Russo R., The existence theorem for the steady Navier-Stokes problem in exterior axially symmetric 3D domains, 2014, 75 pp., http://arXiv.org/abs/1403.6921.
- Korobkov M. V., Pileckas K., Russo R., The Liouville Theorem for the Steady-State Navier-Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl, J. Math. Fluid Mech. (Online first 2015).
- Leray J., Étude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl.,
**9**, No. 12, 1- 82 (1933).