scholarly journals AlgebraicL2Decay for Weak Solutions of the Nonlinear Heat Equations in Whole SpaceR3

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yuexing Yang
Keyword(s):  
Weak Solutions ◽  
Nonlinear Term ◽  
Heat Equations ◽  
Energy Methods ◽  
Time Decay ◽  
Analysis Technique ◽  
Nonlinear Heat Equations ◽  
Whole Space ◽  
Fourier Analysis Technique ◽  
Time Decay Rate

We obtained the algebraicL2time decay rate for weak solutions of the nonlinear heat equations with the nonlinear term∇u2uin whole spaceR3. The methods are based on energy methods and Fourier analysis technique.

scholarly journals Error Estimates for Solutions of the Semilinear Parabolic Equation in Whole Space

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xiaomei Hu
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Error Estimates ◽  
Three Dimensional ◽  
Semilinear Parabolic Equation ◽  
Energy Methods ◽  
Analysis Technique ◽  
Whole Space ◽  
Fourier Analysis Technique ◽  
Semilinear Parabolic

This paper is focused on the error estimates for solutions of the three-dimensional semilinear parabolic equation with initial datau0∈L2(ℝ3). Employing the energy methods and Fourier analysis technique, it is proved that the error between the solution of the semilinear parabolic equation and that of linear heat equation has the behavior asO((1+t)−3/8).

Blood Flow Velocimetry in a Microchannel During Coagulation Using PIV and wOFV

2020 ◽  
Author(s):  
E. Kucukal ◽  
Y. Man ◽  
U. A. Gurkan ◽  
B. E. Schmidt
Keyword(s):  
Blood Flow ◽  
Blood Cells ◽  
Thrombus Formation ◽  
Blockage Ratio ◽  
Time Decay ◽  
Clotting Time ◽  
White Light Source ◽  
Flow Velocimetry ◽  
Flow Evolution ◽  
Time Decay Rate

Abstract This article describes novel measurements of the velocity of whole blood flow in a microchannel during coagulation. The blood is imaged volumetrically using a simple optical setup involving a white light source and a microscope camera. The images are processed using PIV and wavelet-based optical flow velocimetry (wOFV), both of which use images of individual blood cells as flow tracers. Measurements of several clinically relevant parameters such as the clotting time, decay rate, and blockage ratio are computed. The high-resolution wOFV results yield highly detailed information regarding thrombus formation and corresponding flow evolution that is the first of its kind.

References and Remarks on the Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Weak Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Seminal Paper ◽  
Global Existence Of Solutions ◽  
Regularity Properties ◽  
Whole Space ◽  
Two Space Dimensions ◽  
Fundamental Paper

The purpose of this chapter is to give some historical landmarks to the reader. The concept of weak solutions certainly has its origin in mechanics; the article by C. Oseen [100] is referred to in the seminal paper by J. Leray. In that famous article, J. Leray proved the global existence of solutions of (NSν) in the sense of Definition 2.5, page 42, in the case when Ω = R3. The case when Ω is a bounded domain was studied by E. Hopf in. The study of the regularity properties of those weak solutions has been the purpose of a number of works. Among them, we recommend to the reader the fundamental paper of L. Caffarelli, R. Kohn and L. Nirenberg. In two space dimensions, J.-L. Lions and G. Prodi proved in [91] the uniqueness of weak solutions (this corresponds to Theorem 3.2, page 56, of this book). Theorem 3.3, page 58, of this book shows that regularity and uniqueness are two closely related issues. In the case of the whole space R3, theorems of that type have been proved by J. Leray in.

scholarly journals Blow-up of solutions of nonlinear heat equations

1988 ◽  
Vol 129 (2) ◽  
pp. 409-419 ◽  
Author(s):  
Luis A. Caffarrelli ◽  
Avner Friedman
Keyword(s):  
Blow Up ◽  
Heat Equations ◽  
Nonlinear Heat Equations
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