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Roland Donninger

Publications and source records attributed to Roland Donninger.

At least 37 records · Page 2Linked to original sources
arXiv2018

Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow

We prove the existence of a (spectrally) stable self-similar blow-up solution $f_0$ to the heat flow for corotational harmonic maps from $\mathbb R^3$ to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear stability of $f_0$. At the heart of our analysis lies a new existence result of a monotone self-similar solution $f_0$. Although solutions of this kind have already been constructed before, our approach reveals substantial quantitative properties of $f_0$, leading to the stability result. A key ingredient is the use of interval arithmetic: a rigorous computer-assisted method for estimating functions. It is easy to verify our results by robust numerics but the purpose of the present paper is to provide mathematically rigorous proofs.

math.AP↗
arXiv2018

Stable blowup for the cubic wave equation in higher dimensions

We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in a backward light-cone and approaches the ODE blowup profile.

math.AP↗
arXiv2017

On blowup of co-rotational wave maps in odd space dimensions

We consider co-rotational wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere for $d\geq 3$ odd. This is an energy-supercritical model which is known to exhibit finite-time blowup via self-similar solutions. Based on a method developed by the second author and Schörkhuber, we prove the asymptotic nonlinear stability of the "ground-state" self-similar solution.

math.AP↗
arXiv2017

Hyperboloidal evolution and global dynamics for the focusing cubic wave equation

The focusing cubic wave equation in three spatial dimensions has the explicit solution $\sqrt{2}/t$. We study the stability of the blowup described by this solution as $t \to 0$ without symmetry restrictions on the data. Via the conformal invariance of the equation we obtain a companion result for the stability of slow decay in the framework of a hyperboloidal initial value formulation. More precisely, we identify a codimension-1 Lipschitz manifold of initial data leading to solutions which converge to Lorentz boosts of $\sqrt{2}/t$ as $t\to\infty$. These global solutions thus exhibit a slow nondispersive decay, in contrast to small data evolutions.

math.AP↗
arXiv2016

Stable self-similar blowup in the supercritical heat flow of harmonic maps

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.

math.AP↗
arXiv2016

Mode stability of self-similar wave maps in higher dimensions

We consider co-rotational wave maps from Minkowski space in $d+1$ dimensions to the $d$-sphere. Recently, Bizoń and Biernat found explicit self-similar solutions for each dimension $d\geq 4$. We give a rigorous proof for the mode stability of these self-similar wave maps.

math.AP↗
arXiv2016

Stable blowup for the supercritical Yang-Mills heat flow

In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.

math.AP↗
arXiv2016

Linear stability of the Skyrmion

We give a rigorous proof for the linear stability of the Skyrmion. In addition, we provide new proofs for the existence of the Skyrmion and the GGMT bound.

math-ph↗
arXiv2015

Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

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.

math.AP↗
arXiv2015

Stable blowup for wave equations in odd space dimensions

We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions $d \geq 5$. We prove that for every $p > \frac{d+3}{d-1}$ there exists an open set of radial initial data in $H^{\frac{d+1}{2}} \times H^{\frac{d-1}{2}}$ such that the corresponding solution exists in a backward lightcone and approaches the ODE blowup profile. The result covers the entire range of energy supercritical nonlinearities and extends our previous work for the three-dimensional radial wave equation to higher space dimensions.

math.AP↗
arXiv2015

On blowup in supercritical wave equations

We study the blowup behavior for the focusing energy-supercritical semilinear wave equation in 3 space dimensions without symmetry assumptions on the data. We prove the stability of the ODE blowup profile.

math.AP↗
arXiv2015

On the stability of self-similar solutions to nonlinear wave equations

We consider an explicit self-similar solution to an energy-supercritical Yang-Mills equation and prove its mode stability. Based on earlier work by one of the authors, we obtain a fully rigorous proof of the nonlinear stability of the self-similar blowup profile. This is a large-data result for a supercritical wave equation. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations.

math.AP↗
arXiv2014

A Proof for the Mode Stability of a Self-similar Wave Map

We study the fundamental self-similar solution to the SU(2) sigma model, found by Shatah and Turok-Spergel. We give a rigorous proof for its mode stability. Based on earlier results by the second author, the present paper constitutes the last building block for a completely rigorous proof of the nonlinear stability of the Shatah-Turok-Spergel wave map.

math.AP↗
arXiv2014

A vector field method on the distorted Fourier side and decay for wave equations with potentials

We study the Cauchy problem for the one-dimensional wave equation with an inverse square potential. We derive dispersive estimates, energy estimates, and estimates involving the scaling vector field, where the latter are obtained by employing a vector field method on the "distorted" Fourier side. In addition, we prove local energy decay estimates. 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.

math.AP↗
arXiv2014

Asymptotics and analytic modes for the wave equation in similarity coordinates

We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues with analytic eigenfunctions. Our results show that, for sufficiently regular data, the long time behaviour of the solution is governed by the analytic eigenfunctions. The same techniques are applied to the linear stability problem for the fundamental self--similar solution $χ_T$ of the wave equation with a focusing power nonlinearity. Analogous to the free wave equation, we show that the long time behaviour (in similarity coordinates) of linear perturbations around $χ_T$ is governed by analytic mode solutions. In particular, this yields a rigorous proof for the linear stability of $χ_T$ with the sharp decay rate for the perturbations.

math-ph↗