Chapter 10. The Cauchy Problem for the Navier-Stokes Equations with Spatially Almost Periodic Initial Data

2009 ◽  
pp. 213-222 ◽  
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Almost Periodic ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Periodic Initial Data

scholarly journals Properties at potential blow-up times for Navier-Stokes

2017 ◽  
Vol 35 (2) ◽  
pp. 127 ◽  
Author(s):  
Paulo R. Zingano ◽  
Jens Lorenz
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Incompressible Flows ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Solution Properties ◽  
Maximal Interval ◽  
Navier Stokes Equations ◽  
The Cauchy Problem

In this paper we consider the Cauchy problem for the 3D navier-Stokes equations for incompressible flows. The initial data are assume d to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solution can develop singularities in finite time. Assuming the maximal interval of existence to be finite, we give a unified discussion of various known solution properties as time approaches the blow-up time.

scholarly journals Quasilinear Evolution Equations in LμP-Spaces with Lower Regular Initial Data

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Qinghua Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Interpolation Spaces ◽  
Navier Stokes Equations ◽  
Sectorial Operators ◽  
The Cauchy Problem

We study the Cauchy problem of the quasilinear evolution equations in Lμp-spaces. Based on the theories of maximal Lp-regularity of sectorial operators, interpolation spaces, and time-weighted Lp-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in Lp,q-spaces with temporal weights.

scholarly journals A priori estimates in terms of the maximum norm for the solution of the Navier-Stokes equations with periodic initial data

2019 ◽  
Vol 36 (1-2) ◽  
pp. 39-50
Author(s):  
Santosh Pathak
Keyword(s):  
Initial Data ◽  
A Priori Estimates ◽  
Stokes Equations ◽  
A Priori ◽  
Maximum Norm ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Periodic Initial Data

In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations in Rn for n ≥ 3 with smooth periodic initial data and derive a priori estimtes of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a special case of a paper by H-O Kreiss and J. Lorenz which also generalizes the main result of their paper to higher dimension.

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