On a fiber bundle over a disk with the cantor set as a fiber

We construct a Pontryagin fiber bundle ξ = (N, p, S ¹), the total space N of which cannot be imbedded into any two-dimensional oriented manifold but can be imbedded into an arbitrary nonoriented two-dimensional manifold.     

H 4-boundedness of pullback attractor for a 2D non-Newtonian fluid flow

We prove the H4-boundedness of the pullback attractor for a two- dimensional non-autonomous non-Newtonian fluid in bounded domains.     

Remarks on a Bernstein Type Theorem for Entire Willmore Graphs in R 3

In this note we prove that every two-dimensional entire Willmore graph in R ³ with square integrable mean curvature is a plane.     

Preservation of approximative properties of Chebyshev sets and suns in a plane

Approximative properties of intersections of suns and Chebyshev sets in a two-dimensional Banach space X ₂ with subsets of ?² are studied. Subsets of ?² whose intersections with suns from X ₂ are suns are characterized. A similar problem for Chebyshev sets in X ₂ is studied.     

On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC ^∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC ^∞ boundary such that Ω ? ?².     

A non-autonomous two-phase flow model with oscillating external force and its global attractor

In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. Assuming that the external force is singularly oscillating and depends on a small parameter ?, we prove the existence of the uniform global attractor A^?. Furthermore, using the method similar to that of Chepyzhov and Vishik (2007) [22] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of A^? as ? goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier-Stokes system studied in Chepyzhov and Vishik (2007) [22].     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.     

The critical velocity for vortex existence in a two-dimensional rotating Bose-Einstein condensate

We investigate a model corresponding to the experiments for a two-dimensional rotating Bose-Einstein condensate. It consists in minimizing a Gross-Pitaevskii functional defined in R² under the unit mass constraint. We estimate the critical rotational speed Ω₁ for vortex existence in the bulk of the condensate and we give some fundamental energy estimates for velocities close to Ω₁.     

Pullback attractors of non-autonomous micropolar fluid flows

Pullback attractors of the two-dimensional non-autonomous micropolar fluid motion model in a bounded domain are investigated. It is shown that a compact pullback attractor in H¹³(Ω) exists when its external driven function is translation bounded with respect to L₂³(Ω).     

On the Brauer group of a two-dimensional local field

The two-dimensional local field K = F _q((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have x ^p ? x = ut ^?p =: h, then the condition that (y, f | h]_K = 0 for any y ε K is equivalent to the condition f ε Im(Nm(L*)).     

Vortices in the three-dimensional thin-film Ginzburg�CLandau model of superconductivity

Building on the results of Chapman et?al. (Z Angew Math Phys 47:410?C431, 1996) on the behavior of minimizers in the Ginzburg?CLandau thin-film model, we show that the vortices in the three-dimensional superconducting thin films are located in the cylinders whose cross sections coincide with the disks that contain the vortices in the two-dimensional model. To arrive at this conclusion, we prove that the three-dimensional minimizers converge to the two-dimensional counterparts in H ¹ and in C ^?? . We also give examples of regimes in which the vortex structure of the two-dimensional minimizers is well understood. Our results, in particular, provide insight into the behavior of the three-dimensional vortices in these regimes.     

Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge

For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the L¹ well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the L¹ stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the L¹ norm, and prove that the functional decreases in the flow direction. Then the L¹ stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained.     

On the dynamical coherence of structurally stable 3-diffeomorphisms

We prove that each structurally stable diffeomorphism f on a closed 3-manifold M ³ with a two-dimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.     

Diffraction of sound pulses by a rigid cylinder in an inhomogeneous medium

We consider the problem of scattering of two-dimensional sound pulses by a rigid circular cylinder embedded in a cylindrically stratified inhomogeneous medium. The line source is parallel to the axis of the cylinder. It is assumed that the velocity of soundc is given byc ^?1=pr ^q, wherep andq are real constants andp>0. The method of dual integral transformation developed by Friedlander is used. The solution in terms of pulse propagation modes gives the diffracted pulse and the method of steepest descents yields the geometrical acoustic field.     

Large time behavior and L−L estimate of solutions of 2-dimensional nonlinear damped wave equations

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L^p−L^q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L^p−L^q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|^αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2.     

On irreversibility of von Neumann additive cellular automata on grids

The von Neumann cellular automaton appears in many different settings in Operations Research varying from applications in Formal Languages to Biology. One of the major questions related to it is to find a general condition for irreversibility of a class of two-dimensional cellular automata on square grids (σ⁺-automata). This question is partially answered here with the proposal of a sufficient condition for the irreversibility of σ⁺-automata.     

Weak type (1,1) bounds for a Marcinkiewicz integral

In this paper, the two-dimensional Marcinkewicz integral introduced by Stein μ(f)(x)=(∫_0~x|∫_(|x-y|≤1) _(|x-y|)~(Ω(x-y))f(y)dy|~2t~(-3)dt)~2is shown to be of weak type (1,1) and weighted weak type (1,1) with respect to power weight |x|~" if- 1< α< 0, where Ω is homogeneous of degree 0. has mean value 0 and belongs to Llog~+L(S~1).     

Averaging of the planetary 3D geostrophic equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional planetary geostrophic model of the ocean with a singularly oscillating external force depending on a small parameter ?. We prove the existence of the uniform global attractor A^?. Furthermore, using the method of [11] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of A^? as ? goes to zero.     

Fluid forces on a transversely oscillating cylinder

The numerical solution of the unsteady two-dimensional Navier-Stokes equations is used to investigate the fluid forces experienced by a translating and transversely oscillating cylinder. Calculations are first performed in an oscillatory frequency range outside the synchronization when oscillatory-to-translational velocity ratio is 1.5 and at a fixed Reynolds number R = 10³. The object of this study is to examine the effect of increase of forced oscillation frequency on the fluid forces. The results of this study are in good agreement with previous experimental predictions.