Radially symmetric solutions for Navier–Stokes–Smoluchowski system: Global existence in unbounded annular domain and center singularity

Models with a vanishing anisotropic viscosity in the vertical direction are of relevance for the study of turbulent flows in geophysics. This motivates us to study the two-dimensional Boussinesq system with horizontal viscosity in only one equation. In this paper, we focus on the global existence issue for possibly large initial data. We first examine the case where the Navier–Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transport–diffusion equation with diffusion in the horizontal direction only. For both systems, we construct global weak solutions à la Leray and strong unique solutions for more regular data. Our results rest on the fact that the diffusion acts perpendicularly to the buoyancy force.

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

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.3012 ◽

2013 ◽

Vol 37 (17) ◽

pp. 2716-2727 ◽

Cited By ~ 1

Author(s):

Anthony Suen

Keyword(s):

Weak Solution ◽

Global Existence ◽

External Force ◽

Stokes Equations ◽

Navier Stokes ◽

Navier Stokes Equations

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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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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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