Global existence of weak solution to Navier-Stokes equations with large external force and general pressure

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

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

Communications in Contemporary Mathematics ◽

10.1142/s0219199712500319 ◽

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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The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations

Journal of Applied Mathematics ◽

10.1155/2013/321427 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6

Author(s):

Wen-Juan Wang ◽

Yan Jia

Keyword(s):

Weak Solution ◽

Asymptotic Stability ◽

Stokes Equations ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Perturbed System ◽

Regular Class ◽

Stability Issue ◽

The Stability

We study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solutionuof the Navier-Stokes equations lies in the regular class∇u∈Lp(0,∞;Bq,∞0(ℝ3)),(2α/p)+(3/q)=2α,2<q<∞,0<α<1, then every weak solutionv(x,t)of the perturbed system converges asymptotically tou(x,t)asvt-utL2→0,t→∞.

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GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE COMPRESSIBLE NAVIER–STOKES EQUATIONS FOR A 1D ISOTHERMAL VISCOUS GAS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202508003078 ◽

2008 ◽

Vol 18 (08) ◽

pp. 1383-1408 ◽

Cited By ~ 5

Author(s):

YUMING QIN ◽

YANLI ZHAO

Keyword(s):

Asymptotic Behavior ◽

Global Existence ◽

Stokes Equations ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Navier Stokes ◽

Asymptotic Behavior Of Solutions ◽

Navier Stokes Equations ◽

Viscous Gas ◽

Initial Boundary Value

In this paper, we prove the global existence and asymptotic behavior of solutions in Hi(i = 1, 2) to an initial boundary value problem of a 1D isentropic, isothermal and the compressible viscous gas with an non-autonomous external force in a bounded region.

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