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Person# Roland Donninger

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The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields – as they occur in classical physics –

In mathematical physics, Minkowski space (or Minkowski spacetime) (mɪŋˈkɔːfski,_-ˈkɒf-) combines inertial space and time manifolds (x,y) with a non-inertial reference frame of space and time (x',t')

In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A c

Roland Donninger, Joachim Krieger, Willie Wai Yeung Wong

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid. More precisely, we prove in a certain symmetry class the existence, in the neighborhood of the catenoid initial data, of a co-dimension 1 Lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid.

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For the critical focusing wave equation $\Box u = u^5$ on $\mathbb{R}^{3+1}$ in the radial case, we construct a family of blowup solutions which are obtained from the stationary solutions $W(r)$ by means of a dynamical rescaling $\lambda(t)\frac{1}{2}W(\lambda(t)r) +$ correction with $\lambda(t) \rightarrow\infty$ as $t\rightarrow 0$. The novelty here lies with the scaling law $\lambda(t)$ which eternally oscillates between various pure-power laws.

Roland Donninger, Joachim Krieger

We study the Cauchy problem for the one-dimensional wave equation \[ \partial_t^2 u(t,x)-\partial_x^2 u(t,x)+V(x)u(t,x)=0. \] The potential $V$ is assumed to be smooth with asymptotic behavior \[ V(x)\sim -\tfrac14 |x|^{-2}\mbox{ as } |x|\to \infty. \] We derive dispersive estimates, energy estimates, and estimates involving the scaling vector field $t\partial_t+x\partial_x$, where the latter are obtained by employing a vector field method on the ``distorted'' Fourier side. Our results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the co-dimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space, see Donninger, Krieger, Szeftel, and Wong, "Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space"

2016