On uniqueness and stability for supercritical nonlinear wave and Schrödinger equations

We show that smooth solutions to nonlinear wave and Schrödinger equations involving coercive nonlinearities of polynomial type with arbitrarily strong growth are unique among distribution solutions satisfying the energy inequality. The result also yields the stability of classical solutions in the energy norm and may be used to show convergence of the approximate solutions obtained by standard approximation schemes to the true solution in this nor

On a Serrin-Type Regularity Criterion for the Navier-Stokes Equations in Terms of the Pressure

Abstract.: We prove a Serrin-type regularity result for Leray-Hopf solutions to the Navier-Stokes equations, extending a recent result of Zhou [28

A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page

A ‘Super-Critical' Nonlinear Wave Equation in 2 Space Dimensions

These notes are an extended exposion of lectures given at the conference "Nonlinear Analysis”, Verbania, Sept. 25-29, 2010, where we reviewed the results from [11] on global well-posedness of the Cauchy problem for wave equations with exponential nonlinearities in 2 space dimensions for smooth, arbitrarily large radially symmetric dat

Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions

Extending the work of Ibrahim etal. (Commun Pure Appl Math 59(11): 1639-1658, 2006) on the Cauchy problem for wave equations with exponential nonlinearities in two space dimensions, we establish global well-posedness also in the super-critical regime of large energies for smooth, radially symmetric dat

Bedeutung von Nanomaterialien beim Recycling von Abfällen

Produkte, die Nanomaterialien enthalten, verbreiten sich zunehmend. In Herstellungsund Verarbeitungsprozessen von Nanomaterial-basierten Produkten gilt den gesundheitlichen Risiken für die Beschäftigten ein besonderes Augenmerk. Da bisher keine abschließenden Befunde vorliegen, wird im Produktionsbereich umfassend auf Präventionsmaßnahmen zum Gesundheits- und Arbeitsschutz gesetzt. Die Studie der Prognos AG zur Bedeutung von Nanomaterialien beim Recycling von Abfällen zeigt, dass viele dieser nanomaterialhaltigen Produkte nach ihrem Nutzungsende wieder stofflich verwertet, d.h. recycelt, werden. Da im Recyclingprozess eine Wiederfreisetzung nanomaterialhaltiger Stäube nicht ausgeschlossen werden kann, wird die Anwendung spezifischer Präventionsmaßnahmen zum Gesundheitsschutz auch für die Beschäftigten im Bereich Recycling empfohlen. Es besteht darüber hinaus Forschungsbedarf zu Möglichkeiten einer Wiederfreisetzung von Nanomaterialien im Zuge des Recyclings und zu ihrer technischen Minimierung ebenso wie zur weiteren Verbreitung von Nanomaterialien in Recyclingprodukten

Global pointwise decay estimates for defocusing radial nonlinear wave equations

We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping adjusted to the nonlinearity. As a result we obtain a pointwise bound on the solutions for arbitrarily large Cauchy data, provided the solutions exist globally. The decay rates are identical with those for small data and hence seem to be optimal. A generalization beyond the spherical symmetry is suggested.Comment: 9 pages, 1 figur

Quantization for an elliptic equation of order 2 m with critical exponential non-linearity

On a smoothly bounded domain $${\Omega\subset\mathbb{R}^{2m}}$$ we consider a sequence of positive solutions $${u_k\stackrel{w}{\rightharpoondown}0}$$ in H m (Ω) to the equation $${(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}}$$ subject to Dirichlet boundary conditions, where 0<λ k → 0. Assuming that $$0 < \Lambda:=\lim_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx < \infty,$$ we prove that Λ is an integer multiple of Λ1 :=(2m − 1)! vol(S 2m ), the total Q-curvature of the standard 2m-dimensional spher