# Past Partial Differential Equations Seminar

We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections, and existence of regular reflection solutions for potential flow equation. Then we will talk about recent results on uniqueness and stability of regular reflection solutions for potential flow equation in a natural class of self-similar solutions. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, and prove uniqueness by a version of method of continuity. A property of solutions important for the proof of uniqueness is convexity of the free boundary.

This talk is based on joint works with G.-Q. Chen and W. Xiang.

The Prandtl equation was derived in 1904 by Ludwig Prandtl in order to describe the behavior of fluids with small viscosity around a solid obstacle. Over the past decades, several results of ill-posedness in Sobolev spaces have been proved for this equation. As a consequence, it is natural to look for more sophisticated boundary layer models, that describe the coupling with the outer Euler flow at a higher order. Unfortunately, these models do not always display better mathematical properties, as I will explain in this talk. This is a joint work with Helge Dietert, David Gérard-Varet and Frédéric Marbach.

In recent years it has been discovered that also non-linear, degenerate equations like the $p$-Poisson equation $$ -\mathrm{div}(A(\nabla u))= - \mathrm{div} (|\nabla u|^{{p-2}}\nabla u)= -{\rm div} F$$ allow for optimal regularity. This equation has similarities to the one of power-law fluids. In particular, the non-linear mapping $F \mapsto A(\nabla u)$ satisfies surprisingly the linear, optimal estimate $\|A(\nabla u)\|_X \le c\, \|F\|_X$ for several choices of spaces $X$. In particular, this estimate holds for Lebesgue spaces $L^q$ (with $q \geq p'$), spaces of bounded mean oscillations and Holder spaces$C^{0,\alpha}$ (for some $\alpha>0$).

In this talk we show that we can extend this theory to Sobolev and Besov spaces of (almost) one derivative. Our result are restricted to the case of the plane, since we use complex analysis in our proof. Moreover, we are restricted to the super-linear case $p \geq 2$, since the result fails $p < 2$. Joint work with Anna Kh. Balci, Markus Weimar.

I will present a construction of large families of singularity-free stationary solutions of Einstein equations, for a large class of matter models including vacuum, with a negative cosmological constant. The solutions, which are of course real-valued Lorentzian metrics, are determined by a set of free data at conformal infinity, and the construction proceeds through elliptic equations for complex-valued tensor fields. One thus obtains infinite dimensional families of both strictly stationary spacetimes and black hole spacetimes.

We consider vector-valued maps which minimize an energy with two terms: an elastic term penalizing high gradients, and a potential term penalizing values far away from a fixed submanifold N. In the scaling limit where the second term is dominant, minimizers converge to maps with values into the manifold N. If the elastic term is the classical Dirichlet energy (i.e. the squared L^2-norm of the gradient), classical tools show that this convergence is uniform away from a singular set where the energy concentrates. Some physical models (as e.g. liquid crystal models) include however more general elastic energies (still coercive and quadratic in the gradient, but less symmetric), for which these classical tools do not apply. We will present a new strategy to obtain nevertheless this uniform convergence. This is a joint work with Andres Contreras.

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under the only assumption of $L^1$ weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary $L^1$ vorticity. Relations with previously known notions of solutions are shown.