scholarly journals Results on existence for generalized nD Navier-Stokes equations

2019 ◽  
Vol 17 (1) ◽  
pp. 1652-1679
Author(s):  
Khaled Zennir
Keyword(s):  
Global Existence ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Rocky Mountain ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
New Approach ◽  
Kirchhoff Type ◽  
New Class ◽  
Global Existence Of Solutions

Abstract In this paper we consider a class of nD Navier-Stokes equations of Kirchhoff type and prove the global existence of solutions by using a new approach introduced in [Jday R., Zennir Kh., Georgiev S.G., Existence and smoothness for new class of n-dimentional Navier-Stokes equations, Rocky Mountain J. Math., 2019, 49(5), 1595–1615].

THE POINTWISE ESTIMATES OF SOLUTIONS TO THE ISENTROPIC NAVIER–STOKES EQUATIONS IN EVEN SPACE-DIMENSIONS

2005 ◽  
Vol 02 (03) ◽  
pp. 673-695 ◽  
Author(s):  
WEIKE WANG ◽  
XIONGFENG YANG
Keyword(s):  
Existence Of Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Pointwise Estimates ◽  
Navier Stokes Equations ◽  
Estimates Of Solutions ◽  
Global Existence Of Solutions ◽  
Linearized System ◽  
The Green Function ◽  
Time Asymptotic Behavior

We study the dissipation of solutions for the isentropic Navier–Stokes equations in even space-dimensions. Based on the global existence of solutions and on the analysis of the Green function associated with the linearized system, the pointwise estimates of solution for the isentropic Navier–Stokes equations are established. It is observed here that in even dimensions the time-asymptotic behavior of the solutions follows from the weak Huygen's principle.

scholarly journals CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

2010 ◽  
Vol 20 (07) ◽  
pp. 1049-1087 ◽  
Author(s):  
BORIS HASPOT
Keyword(s):  
Shallow Water ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Water System ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Viscosity Coefficients ◽  
Global Existence Of Solutions ◽  
Dependent Viscosity

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

GLOBAL EXISTENCE TO THE NAVIER–STOKES EQUATIONS THROUGH A SELF-SIMILAR-LIKE UPPER SOLUTION

2012 ◽  
Vol 14 (05) ◽  
pp. 1250031
Author(s):  
GUY BERNARD
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Velocity ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Navier Stokes Equations ◽  
Upper Solution ◽  
Self Similar

A global existence result is presented for the Navier–Stokes equations filling out all of three-dimensional Euclidean space ℝ3. The initial velocity is required to have a bell-like form. The method of proof is based on symmetry transformations of the Navier–Stokes equations and a specific upper solution to the heat equation in ℝ3× [0, 1]. This upper solution has a self-similar-like form and models the diffusion process of the heat equation. By a symmetry transformation, the problem is transformed into an equivalent one having a very small initial velocity. Using the upper solution, the equivalent problem is then solved in the time interval [0, 1]. This local solution is then extended to the time interval [0, ∞) by an iterative process. At each step, the problem is extended further in time in an interval of time whose length is greater than one, thus producing the global solution. Each extension is transformed, by an appropriate change of variables, into the first local problem in the time interval [0, 1]. These transformations exploit the diffusive and self-similar-like nature of the upper solution.

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