scholarly journals Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption

2021 ◽  
Author(s):  
Wei‐Xi Li ◽  
Nader Masmoudi ◽  
Tong Yang
Keyword(s):  
Function Space ◽  
Well Posedness ◽  
Prandtl Equations ◽  
Structural Assumption

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

Well-posedness of Prandtl equations with non-compatible data

Nonlinearity ◽  
2013 ◽  
Vol 26 (12) ◽  
pp. 3077-3100 ◽  
Author(s):  
M Cannone ◽  
M C Lombardo ◽  
M Sammartino
Keyword(s):  
Well Posedness ◽  
Prandtl Equations

scholarly journals A well-posedness theory for the Prandtl equations in three space variables

2017 ◽  
Vol 308 ◽  
pp. 1074-1126 ◽  
Author(s):  
Cheng-Jie Liu ◽  
Ya-Guang Wang ◽  
Tong Yang
Keyword(s):  
Well Posedness ◽  
Prandtl Equations ◽  
Three Space

scholarly journals A well-posedness result for a system of cross-diffusion equations

2021 ◽  
Author(s):  
Christian Seis ◽  
Dominik Winkler
Keyword(s):  
Fixed Point ◽  
Function Space ◽  
Weak Solutions ◽  
Diffusion System ◽  
Diffusion Equations ◽  
Well Posedness ◽  
Cross Diffusion ◽  
Bounded Functions ◽  
Hopping Model ◽  
Point Argument

AbstractThis work’s major intention is the investigation of the well-posedness of certain cross-diffusion equations in the class of bounded functions. More precisely, we show existence, uniqueness and stability of bounded weak solutions under a smallness assumption on the intial data. As an application, we provide a new well-posedness theory for a diffusion-dominant cross-diffusion system that originates from a hopping model with size exclusions. Our approach is based on a fixed point argument in a function space that is induced by suitable Carleson-type measures.

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

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