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 Some Properties of Blow up Solutions in the Cauchy Problem for 3D Navier–Stokes Equations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1523
Author(s):  
Vladimir I. Semenov
Keyword(s):  
Cauchy Problem ◽  
Fluid Flow ◽  
Kinetic Energy ◽  
Blow Up ◽  
Stokes Equations ◽  
Invariant Form ◽  
Navier Stokes ◽  
Solution Properties ◽  
Navier Stokes Equations ◽  
The Cauchy Problem

Up to now, it is unknown an existence of blow up solutions in the Cauchy problem for Navier–Stokes equations in space. The first important property of hypothetical blow up solutions was found by J. Leray in 1934. It is connected with norms in Lp(R3),p>3. However, there are important solutions in L2(R3) because the second power of this norm can be interpreted as a kinetic energy of the fluid flow. It gives a new possibility to study an influence of kinetic energy changing on solution properties. There are offered new tools in this way. In particular, inequalities with an invariant form are considered as elements of latent symmetry.

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.

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