- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 09:58:14
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=6&iss=6&update_since=2024-03-29
Zeitschrift für Analysis und ihre Anwendungen
Z. Anal. Anwend.
ZAA
0232-2064
1661-4534
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
10.4171/ZAA
http://www.ems-ph.org/doi/10.4171/ZAA
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
6
1987
6
Stability Properties of Space Periodic Standing Waves
B.
Scarpellini
Universität Basel, BASEL, SWITZERLAND
Equilibrium solutions of parabolic systems of the form $u = \Delta u + F(\alpha, u)$ are considered, where $D$ designates as $2 \times 2$ diagonal matrix, $\alpha$ a bifurcation parameter, $u = (u_1, u_2)$ a state vector and $F$ a polynomial nonlinearity. A trivial solution branch $u(\delta) \in \mathbb R^2, \delta \in I = (– \epsilon, \epsilon)$, is supposed to be given, i.e. $F(\alpha_0 + \delta, u(\delta)) = 0$ for some $\alpha_0$ and every $\delta \in I$. Then a period $L$ is fixed and under suitable assumptions space-$L$-periodic bifurcating standing waves are constructed. It is shown that these bifurcating branches become generically unstable as $L \uparrow \infty$. Under the condition of $d_{uu}F(\alpha_0, u(0)) = 0$ however, they will remain stable against $nL$-periodic perturbations $(1 < n \in \mathbb N)$, provided that the trivial solution-branch $u(\delta)$ behaves alike for small $\delta < 0$. The so-called Landau-Ginzburg equations arising in Landau’s theory of phase transitions constitute a special example in physics.
General
485
503
10.4171/ZAA/267
http://www.ems-ph.org/doi/10.4171/ZAA/267
Second Microlocalization and Propagation Theorems for the Wave Front Sets
Patrick
Esser
Université de Liège, LIÈGE 1, BELGIUM
There is showed how to define the second analytic wave front set in order to obtain propagation properties for differentiable, Gevrey, or analytic singularities of distributions without second analytic support.
General
505
515
10.4171/ZAA/268
http://www.ems-ph.org/doi/10.4171/ZAA/268
Die Bevorzugung hexagonaler Strukturen bei symmetrischen Oberflächenformen magnetischer Flüssigkeiten
Karin
Quasthoff
Universität Leipzig, LEIPZIG, GERMANY
A magnetic ideal fluid under the influence of a vertical homogeneous magnetic field is considered. Over rhombic lattices with arbitrary angle equilibrium configurations of the free surface are determined. Especially, for some lattice length $l_c$ the bifurcation equations are solved and over regular hexagonal lattices with lattice length $I > l_c$ solutions of the bifurcation equations are given. The regular hexagonal structures, in an interval depending on physical parameters, have smallest energy.
General
517
528
10.4171/ZAA/269
http://www.ems-ph.org/doi/10.4171/ZAA/269
A Class of Nonlinear Generalized Riemann-Hilbert-Poincaré Problems for Holomorphic Functions
Lothar
von Wolfersdorf
Technische Universität, FREIBERG, GERMANY
By means of the theory of pseudo-monotone operators the existence of a solution of a class of nonlinear generalized Riemann-Hilbert-Poincaré problems for a holomorphic function in the unit disk is proved.
General
529
538
10.4171/ZAA/270
http://www.ems-ph.org/doi/10.4171/ZAA/270
Stabilitätskriterien für Näherungsverfahren bei singulären Integralgleichungen in $L^p$
Siegfried
Prössdorf
Angewandte Analysis und Stochastik, BERLIN, GERMANY
A.
Rathsfeld
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Approximation methods for one-dimensional singular integral equations with piecewise continuous coefficients on simple closed Liapunov curves are considered. For spline Calerkin methods and for spline and trigonometric collocation, necessary and sufficient conditions for their stability in $L^p$ are given. The central idea is a general stability criterion for sequences of approximate operators with circulant structure.
General
539
558
10.4171/ZAA/271
http://www.ems-ph.org/doi/10.4171/ZAA/271
The Convergence of Rothe’s Method for Parabolic Differential Equations
Rainer
Schumann
, LEIPZIG, GERMANY
The convergence of Rothe’s method for quasilinear parabolic differential equations is proved under typical assumptions which guarantee existence. The investigations are based on appropriate approximation schemes for evolution tripels and on the observation that Rothe’s method satisfies the requirements of stability and consistency.
General
559
574
10.4171/ZAA/272
http://www.ems-ph.org/doi/10.4171/ZAA/272
Dissipativität und globale Stabilität des komplexen Lorenz-Systems
G.A.
Leonov
St. Petersburg State University, ST. PETERSBURG, RUSSIAN FEDERATION
Volker
Reitmann
Technische Universität Dresden, DRESDEN, GERMANY
Using Liapunov functions it is shown that the complex Lorenz-System is dissipative and has a global asymptotic for some parameters.
General
575
582
10.4171/ZAA/273
http://www.ems-ph.org/doi/10.4171/ZAA/273