Georgia Institute of Technology College of Sciences

The no-hair theorem states that the only asymptotically flat stationary solutions of the Einstein vacuum equations are members of the Kerr family, parametrized by mass and angular momentum. Hawking’s rigidity theorem shows that under the hypothesis of analyticity, any such solution is axially symmetric. For axially symmetric solutions with a single connected event horizon, Robinson (1975) proved that the solution must be Kerr. However, the case of multiple black holes has remained largely open.

We study the sharp asymptotics for a class of quasilinear wave equations satisfying a weak null condition but not the classical null condition in three spatial dimensions. We prove that the asymptotics are very different from those for the equations satisfying the classical null condition. In particular, at leading order, the solution displays a continuous superposition of decay rates.

We will talk about recent work establishing a quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in (n+1) dimensions with n ≥ 4 even, which are determined by small scattering data at future infinity and past infinity. We will also explain why the case of even spatial dimension n poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature.

The compressible Euler equations readily form shocks, but in 1D the inclusion of viscosity prevents such singularities. In this talk, we will quantitatively examine the interaction between generic shock formation and viscous effects as the viscosity tends to zero. We thereby obtain sharp rates for the vanishing-viscosity limit in Hölder norms, and uncover universal viscous structure near shock formation. The results hold for large classes of viscous hyperbolic conservation laws, including compressible Navier–Stokes with physical rather than artificial viscosity.

In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John's counterexample $\Box u=u_tu_{tt}$ and the 3D compressible Euler equations without vorticity for $t\geq 0$.

We will talk about our results on the elasticity and stability of the 
collision of two kinks with low speed v>0 for the nonlinear wave 
equation of dimension 1+1 known as the phi^6 model. We will show that 
the collision of the two solitons is "almost" elastic and that, after 
the collision, the size of the energy norm of the remainder and the size 
of the defect of the speed of each soliton can be, for any k>0, of the 
order of any monomial v^{k} if v is small enough.

References:

We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition.

In this talk we will discuss an optimal control problem for stochastic differential delay equations. We will only consider the case with delays in the state. We will show how to rewrite the problem in a suitable infinite-dimensional Hilbert space.

I discuss some recent results, obtained jointly with David Wallauch, on the stability of self-similar wave maps under minimal regularity assumptions on the perturbation. More precisely, we prove the asymptotic stability of an explicitly known self-similar wave map in corotational symmetry. The key tool are Strichartz estimates for the linearized equation in similarity coordinates.