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.

# Past Partial Differential Equations Seminar

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).

I will present a joint work with Leonid Kovalev and Jani Onninen. The proofs are based on topological arguments (degree theory) and the properties of planar quasiconformal mappings. These new ideas apply well to inner variational equations of conformally invariant energy integrals; in particular, to the Hopf-Laplace equation for the Dirichlet integral.

In this talk, we use Carleman type techniques to address uniqueness of self-shrinkers of mean curvature flow with given asymptotic behaviors.

A theory of Sobolev inequalities in arbitrary open sets in $R^n$ is offered. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set. This is a joint work with V. Maz'ya.

We will focus on vortex gradient fields of unit-length. The associated stream function solves the eikonal equation, more precisely it is the distance function to a point. We will prove a kinetic formulation characterizing such vector fields in any dimension.

We consider Euler equations on a fixed Lorentzian manifold. The fluid is initially supported on a compact domain and the boundary between the fluid and the vacuum is allowed to move. Imposing the so-called physical vacuum boundary condition, we will explain how to obtain a priori estimates for this problem. In particular, our functional framework allows us to track the regularity of the free boundary. This is joint work with S. Shkoller and J. Speck.