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On the binding of polarons in a mean-field quantum crystal

Mathieu Lewin, Nicolas Rougerie (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a multi-polaron model obtained by coupling the many-body Schrödinger equation for N interacting electrons with the energy functional of a mean-field crystal with a localized defect, obtaining a highly non linear many-body problem. The physical picture is that the electrons constitute a charge defect in an otherwise perfect periodic crystal. A remarkable feature of such a system is the possibility to form a bound state of electrons via their interaction with the polarizable background....

On the blow up criterion for the 2-D compressible Navier-Stokes equations

Lingyu Jiang, Yidong Wang (2010)

Czechoslovak Mathematical Journal

Motivated by [10], we prove that the upper bound of the density function ρ controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.

On the blow-up phenomenon for the mass-critical focusing Hartree equation in ℝ⁴

Changxing Miao, Guixiang Xu, Lifeng Zhao (2010)

Colloquium Mathematicae

We characterize the dynamics of the finite time blow-up solutions with minimal mass for the focusing mass-critical Hartree equation with H¹(ℝ⁴) data and L²(ℝ⁴) data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we analyze the mass concentration phenomenon of such blow-up solutions.

On the boundary convergence of solutions to the Hermite-Schrödinger equation

Peter Sjögren, J. L. Torrea (2010)

Colloquium Mathematicae

In the half-space ℝ d × ℝ ₊ , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on ℝ d . We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.

On the boundedness of the mapping f → | f | in Besov spaces

Patrick Oswald (1992)

Commentationes Mathematicae Universitatis Carolinae

For 1 ≤ p ≤ ∞ , precise conditions on the parameters are given under which the particular superposition operator T : f → | f | is a bounded map in the Besov space B p , q s ( R 1 ) . The proofs rely on linear spline approximation theory.

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