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Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in R 3 with nontrivial swirl. Such solutions are not known to be globally defined, but it is shown in (, Partial regularity of suitable weak solutions of the Navier–Stokes equations. they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r… (More)
Via gauge theory, we give a new proof of partial regularity for harmonic maps in dimensions m ≥ 3 into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of " minimal " C 2 regularity. The proof we present moreover extends to a large class of elliptic systems of quadratic growth.
We prove that in dimensions three or four, for suitably chosen initial data, the short time smooth solution to the Landau-Lifshitz-Gilbert equation blows up at finite time.
We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the u5-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.
Seine Spezialgebiete sind partielle Differentialgleichungen und Variationsrechnung.
We analyze the possible concentration behavior of heat flows related to the Moser-Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser-Trudinger energy in a supercritical… (More)
We present new regularity criteria involving the integrability of the pressure for the Navier-Stokes equations in bounded domains with smooth boundaries. We prove that either if the pressure belongs to L γ,q x,t with 3/γ + 2/q ≤ 2 and 3/2 < γ ≤ ∞ or if the gradient of the pressure belongs to L γ,q x,t with 3/γ + 2/q ≤ 2 and 1 < γ ≤ ∞, then weak solutions… (More)