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European Mathematical Society Publishing House
2024-03-29 16:39:51
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=11&iss=1&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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Zuerich, Switzerland
© European Mathematical Society
11
2009
1
Non-generic blow-up solutions for the critical focusing NLS in 1-D
Joachim
Krieger
Bâtiment des Mathématiques, LAUSANNE, SWITZERLAND
Wilhelm
Schlag
University of Chicago, CHICAGO, UNITED STATES
Nonlinear Schrödinger equations, L2-critical NLS, pseudo-conformal blow-up
We consider the L2-critical focussing nonlinear Schrödinger equation in 1+1-D. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a co-dimension one stable blow up manifold in the measurable category.
Partial differential equations
General
1
125
10.4171/JEMS/143
http://www.ems-ph.org/doi/10.4171/JEMS/143
On the Ladyzhenskaya�Smagorinsky turbulence model of the Navier�Stokes equations in smooth domains. The regularity problem
Hugo
Beirão da Veiga
Università di Pisa, PISA, ITALY
We establish regularity results up to the boundary for solutions to generalized Stokes and Navier-Stokes systems of equations in the stationary and in the evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat here the case $p\geq 2$. Actually, we are interested in proving regularity results in $L^q(\Omega)$ spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous scheme, introduced in references \cite{bvlali} and \cite{bvcubo} for the flat-boundary case, to the case of curvilinear boundaries.
Partial differential equations
Fluid mechanics
General
127
167
10.4171/JEMS/144
http://www.ems-ph.org/doi/10.4171/JEMS/144
Single-point blow-up for a semilinear parabolic system
Philippe
Souplet
Université de Paris XIII/CNRS, VILLETANEUSE, FRANCE
Semilinear parabolic system, reaction-diffusion, power nonlinearities, blow-up set, single point blow-up
We consider positive solutions of a semilinear parabolic system coupled by power nonlinearities, in a ball or in the whole space. Relatively little is known on the blow-up set for parabolic systems and, up to now, no result was available for this basic system except for the very special case of equal powers. Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A.~Friedman and Y.~Giga (1987). We also obtain lower pointwise estimates for the final blow-up profiles.
Partial differential equations
General
169
188
10.4171/JEMS/145
http://www.ems-ph.org/doi/10.4171/JEMS/145
On the topology of positively curved Bazaikin spaces
Luis
Florit
Estrada Dona Castorina 110, RIO DE JANEIRO RJ, BRAZIL
Wolfgang
Ziller
University of Pennsylvania, PHILADELPHIA, UNITED STATES
Positive curvature, Bazaikin spaces
In this note we study the topology of the positively curved Bazaikin spaces. We show that the only Bazaikin space that is homotopically equivalent to a homogeneous space is the Berger space. Moreover, we compute their Pontryagin classes and linking form to conclude that there is no pair of positively curved Bazaikin spaces which are homeomorphic, at least if the order of the sixth cohomology group with integer coefficients is less than 108.
Differential geometry
General
189
205
10.4171/JEMS/146
http://www.ems-ph.org/doi/10.4171/JEMS/146
Liouville theorems for self-similar solutions of heat flows
Jiayu
Li
Chinese Academy of Sciences, BEIJING, CHINA
Meng
Wang
Zhejiang University, HANGZHOU, CHINA
Harmonic sphere, self-similar solution, quasi-harmonic sphere, heat flow
Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from $({\bf R}^m, e^{-|x|^2/2(m-2)}ds_0^2)$ to $N$ ($m\geq 3$) with finite energy ([LnW]). Here $ds_0^2$ is the Euclidean metric in ${\bf R}^m$. It arises from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.
Partial differential equations
Global analysis, analysis on manifolds
General
207
221
10.4171/JEMS/147
http://www.ems-ph.org/doi/10.4171/JEMS/147