two space dimensions
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2022 ◽  
Vol 214 ◽  
pp. 112532
Author(s):  
Tohru Ozawa ◽  
Kenta Tomioka

2021 ◽  
pp. 2250003
Author(s):  
Chengfeng Sun ◽  
Qianqian Huang ◽  
Hui Liu

The stochastic two-dimensional Cahn–Hilliard–Navier–Stokes equations under non-Lipschitz conditions are considered. This model consists of the Navier–Stokes equations controlling the velocity and the Cahn–Hilliard model controlling the phase parameters. By iterative techniques, a priori estimates and weak convergence method, the existence and uniqueness of an energy weak solution to the equations under non-Lipschitz conditions have been obtained.


2021 ◽  
Vol 44 (7) ◽  
Author(s):  
A. Gong ◽  
S. Rode ◽  
G. Gompper ◽  
U. B. Kaupp ◽  
J. Elgeti ◽  
...  

Abstract  The eukaryotic flagellum propels sperm cells and simultaneously detects physical and chemical cues that modulate the waveform of the flagellar beat. Most previous studies have characterized the flagellar beat and swimming trajectories in two space dimensions (2D) at a water/glass interface. Here, using refined holographic imaging methods, we report high-quality recordings of three-dimensional (3D) flagellar bending waves. As predicted by theory, we observed that an asymmetric and planar flagellar beat results in a circular swimming path, whereas a symmetric and non-planar flagellar beat results in a twisted-ribbon swimming path. During swimming in 3D, human sperm flagella exhibit torsion waves characterized by maxima at the low curvature regions of the flagellar wave. We suggest that these torsion waves are common in nature and that they are an intrinsic property of beating axonemes. We discuss how 3D beat patterns result in twisted-ribbon swimming paths. This study provides new insight into the axoneme dynamics, the 3D flagellar beat, and the resulting swimming behavior. Graphic abstract


2021 ◽  
pp. 1-30
Author(s):  
Takashi Suzuki

We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.


2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Lijuan Nong ◽  
An Chen

The modified anomalous subdiffusion equation plays an important role in the modeling of the processes that become less anomalous as time evolves. In this paper, we consider the efficient difference scheme for solving such time-fractional equation in two space dimensions. By using the modified L1 method and the compact difference operator with fast discrete sine transform technique, we develop a fast Crank-Nicolson compact difference scheme which is proved to be stable with the accuracy of O τ min 1 + α , 1 + β + h 4 . Here, α and β are the fractional orders which both range from 0 to 1, and τ and h are, respectively, the temporal and spatial stepsizes. We also consider the method of adding correction terms to efficiently deal with the nonsmooth problems. Numerical examples are provided to verify the effectiveness of the proposed scheme.


Author(s):  
M. C. van der Weele ◽  
A. S. Fokas

AbstractStarting from the 3-wave interaction equations in 2+1 dimensions (i.e., two space dimensions and one time dimension), we complexify the independent variables, thus doubling the number of real variables, and hence we work in 4+2 dimensions: $$x_1$$ x 1 , $$x_2$$ x 2 , $$y_1$$ y 1 , $$y_2$$ y 2 and $$t_1$$ t 1 , $$t_2$$ t 2 . In this paper we solve the initial value problem of the 3-wave interaction equations in 4+2 dimensions.


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