3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity

Examples of Singular Limits in Hydrodynamics

“…When there is no boundary (Ω = T 3 for instance) and when ν = η = 1 (the Navier-Stokes case) or ν = η = 0 (the Euler case), the problem was studied by several authors ( [81], [33], [8], [9], [10], [63], [68], [143]...) by using the group method of [154] and [81]. This method was first introduced to treat the compressible incompressible limit (see subsections 3.2.1 and 3.3.5).…”

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

“…In particular, by exploiting the special structure of the governing equations, Cao and Titi [4] proved the global well-posedness of the three-dimensional viscous primitive equations that model large-scale ocean and atmosphere dynamics. By taking advantage of the limiting property of some rapidly oscillating operators and using nonlinear averaging, Babin, Mahalov, and Nicolaenko [1] prove existence on infinite time intervals of regular solutions to the three-dimensional Navier-Stokes equations for some initial data characterized by uniformly large vorticity.…”

Dispersive Effect of the Coriolis Force and the Local Well-Posedness for the Navier-Stokes Equations in the Rotational Framework

“…Here W A R represents the speed of rotation around the vertical unit vector e 3 ¼ ð0; 0; 1Þ, which is called the Coriolis parameter. The purpose of this paper is to show the local existence and the uniqueness of a mild solution to (NSC) [4] obtained the global existence and regularity of solutions to (NSC) for large jWj with the periodic initial velocity. On the other hand, Giga, Inui, Mahalov and Saal [14] established the uniform global solvability of (NSC) for small initial velocity in FM for all l > 0, where u l 0 ðxÞ :¼ lu 0 ðlxÞ.…”