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 ‘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

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

Quantization for a fourth order equation with critical exponential growth

For concentrating solutions $$0 < u_k \rightharpoonup 0$$ weakly in H 2(Ω) to the equation $$\Delta^{2} u_{k}= \lambda_{k} u_{k} e^{2u_{k}^{2}}$$ on a domain $$\Omega \subset \mathbb{R}^{4}$$ with Navier boundary conditions the concentration energy $$\Lambda = \lim_{k \rightarrow \infty} \int_{\Omega} |\Delta u_k|^{2} dx$$ is shown to be strictly quantized in multiples of the number $$\Lambda_1 = 16 \pi^{2}$

Partial regularity for biharmonic maps, revisited

Extending our previous results with Tristan Rivière for harmonic maps, we show how partial regularity for stationary biharmonic maps into arbitrary targets can be naturally obtained via gauge theory in any dimensions m ≥

An Analytic Framework for the Supercritical Lane-Emden Equation and its Gradient Flow

The natural setting for the Lane-Emden equation −Δu=|u|p−2u on a domain $\Omega \subset \mathbb {R}^n$, n≥3, for supercritical exponents p>2*=2n/(n−2) is identified as the space of functions $u\in H^1_0\cap L^p(\Omega)$ with finite scale-invariant Morrey norms. We show that this Morrey regularity is propagated by the heat flow associated with this equation, and we study the blow-up profile