Recent Developments in the Navier-Stokes Problem

By P. Lemari'e-Rieusset. 2002

1,270 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/70c07566a39c5769e94e1b999462e1a4ee31aa14

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Citations in the corpus

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions (2012, background): “…under which condition this result can be localized in space: if u0 is a quite general initial condition for which a generalized suitable weak solution u in the sense of [19] is defined and u0|Br ∈ Lq(Br) for some q > 3, say, can we conclude that u is regular in B r 2 × [0, t1) for some time t1 > 0?”
On the uniqueness of strong solution to the incompressible Navier–Stokes equations with damping (2011, positive): “Since then, there has been a lot of literatures devoted to investigating the issue of uniqueness and regularity (see [4,6,7,11–13,15,17–20] and references therein).”
Generalized surface quasi‐geostrophic equations with singular velocities (2011, background): “These tools are now standard and can be found in several books, say [20], [66] and [91].”
Log Improvement of the Prodi-Serrin Criteria for Navier-Stokes Equations (2007, methodology): “We make use of the following result due to Kato [8] (see also the book of Lemarié-Rieusset [10]).”
Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor (2010, neutral): “, [6], [7], [9], [19], [21], [22], [24], [25], [32], [35], [36], [37] and references therein).”
Global cauchy problem of 2D generalized MHD equations (2014, positive): “By the wellknown L∞-estimates and Lq(0, T ; W 1,p)-estimates of the heat equations [1,6,8], we have ‖b‖L∞(0,T ;L∞) ≤ C, (2.”
A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation (2006, background): “We should point out that (1.4) is proved by Lemari´e-Rieusset[ 19 ] in the case α = 1, and by Danchin [14] when p is any even integer.”
Existence and Regularity of Weak Solutions to the Quasi-Geostrophic Equations in the Spaces Lp or $$\dot{H}^{-1/2}$$ (2007, background): “This kind of estimate is in the spirit of Lemarie-Rieusset’s work, [ 13 ] or [14], on the weak infinite-energy solutions for the Navier-Stokes equations in R 3 .”
An alternative approach to regularity for the Navier–Stokes equations in critical spaces (2009, background): “Therefore u satisfies (see [38], Theorem 11.”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, methodology): “Therefore, it follows from Bernstein’s inequality for Fourier multipliers [65] that kRDkC0 .”
Regularity Criteria for the Three-dimensional Navier-Stokes Equations (2008, background): “Over the years there has been an intensive work by many authors attacking this problem (see, e.g., [11], [12], [14], [25], [28], [30], [31], [32], [40], [43], [44], [45] and references therein).”
The generalized incompressible Navier-Stokes equations in Besov spaces (2004, background): “This result was later extended to Ḃ 3 p−1 p,∞ with p > 3, but the uniqueness of such mild solutions is unknown ([8]).”
On the regularity of a class of generalized quasi-geostrophic equations (2010, background): “Bernstein’s inequality is fundamental in the analysis involving Besov spaces (see [16]).”
Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces (2018, neutral): “First, recall that Ṁ(2,3)(R(3)) is a homogeneous Banach space of degree −1 and then we have Ṁ(2,3)(R(3)) ⊂ Ḃ−1,∞ ∞ (R (3)) (see the Chapter 4 of the book [10]).”