Global regularity for some classes of large solutions to the Navier-Stokes equations

By Jean-Yves Chemin, Isabelle Gallagher, Marius Paicu. 2008

152 citations recorded by the source corpus.

In previous works by the first two authors, classes of initial data to the three-dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in one of those studies is that it varies slowly in one direction, though in some sense it is "well-prepared" (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize that setting to an "ill prepared" situation (the norm blows up as the small parameter goes to zero). As in those works, the proof uses the special structure of the nonlinear term of the equation.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/75c1bee7e7504267f0b06e31d89d93449e47a306

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In Applied Mathematics

In Control and Systems Engineering

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Citations in the corpus

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations (2013, background): “For more directly relevant works which use norms of this type, see [31, 52, 23, 47, 24, 68].”
Landau Damping: Paraproducts and Gevrey Regularity (2013, background): “Norms such as Gλ(t),σ ;s are common when dealing with analytic or Gevrey regularity, for example, see the works [10,21,22,25,27,45,50,67].”
Global existence and decay of solutions to Prandtl system with small analytic data. (2019, unknown): “It was used in the context of anisotropic Navier-Stokes system [5] ( see also [22, 23]), which implies the global well-posedness of three dimensional Navier-Stokes system with a class of “ill”
On the stability threshold for the 3D Couette flow in Sobolev regularity (2015, background): “arity (see e.g. the Cauchy-Kowalevskaya theorems of [39, 40]). The use of dissipation-like terms that appear in L2-based inﬁnite regularity estimates goes back to [25]; see also related ideas in e.g. [35, 32, 16, 38] and the references therein. The ghost energy of Alinhac [1] for quasilinear wave eq”
Global well‐posedness for compressible Navier‐Stokes equations with highly oscillating initial velocity (2009, background): “Let us emphasize that this result allows us to construct global solutions for highly oscillating initial data which may have a large norm in Ḣ n 2 −1 or Ln.”
A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical Lp Framework (2010, background): “In particular, as pointed out by M. Cannone in [4] (or more recently in [6]), owing to the fact that d/p − 1 is negative if p > d, velocity ﬁelds u 0 with a large modulus may have small norm in ˙ B − 1 p, 1 provided they have fast enough oscillations.”
The Incompressible Navier–Stokes Equations on Non-compact Manifolds (2014, background): “Moreover, there are global well-posedness results for some classes of large data in all the above spaces, which use the structure of the non-linear term (see, for example, [1,11,34]).”