scholarly journals Global classical solution of the Cauchy problem to 1D compressible Navier–Stokes equations with large initial data

2014 ◽  
Vol 257 (2) ◽  
pp. 311-350 ◽  
Author(s):  
Quansen Jiu ◽  
Mingjie Li ◽  
Yulin Ye
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Classical Solution ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Global Classical Solution ◽  
Large Initial Data ◽  
Navier Stokes Equations ◽  
The Cauchy Problem

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 GLOBAL REGULARITY TO THE NAVIER-STOKES EQUATIONS FOR A CLASS OF LARGE INITIAL DATA

2018 ◽  
Vol 23 (2) ◽  
pp. 262-286
Author(s):  
Bin Han ◽  
Yukang Chen
Keyword(s):  
Initial Data ◽  
Initial Velocity ◽  
Global Regularity ◽  
Stokes Equations ◽  
Regularity Of Solutions ◽  
Navier Stokes ◽  
Large Initial Data ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Well Posed

In [5], Chemin, Gallagher and Paicu proved the global regularity of solutions to the classical Navier-Stokes equations with a class of large initial data on T2 × R. This data varies slowly in vertical variable and has a norm which blows up as the small parameter ( represented by ǫ in the paper) tends to zero. However, to the best of our knowledge, the result is still unclear for the whole spaces R3. In this paper, we consider the generalized Navier-Stokes equations on Rn(n ≥ 3): ∂tu + u · ∇u + Dsu + ∇P = 0, div u = 0. For some suitable number s, we prove that the Cauchy problem with initial data of the form u0ǫ(x) = (v0h(xǫ), ǫ−1v0n(xǫ))T , xǫ = (xh, ǫxn)T , is globally well-posed for all small ǫ > 0, provided that the initial velocity profile v0 is analytic in xn and certain norm of v0 is sufficiently small but independent of ǫ. In particular, our result is true for the n-dimensional classical Navier-Stokes equations with n ≥ 4 and the fractional Navier-Stokes equations with 1 ≤ s < 2 in 3D.

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