Liouville theorems for the Navier–Stokes equations and applications

By G. Koch, Nikolai Nadirashvili, G A Seregin, Vladimír Sverák. 2007

340 citations recorded by the source corpus.

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in Rn × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0feee27b335e240170c83e913ba3a297ad8bc543

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Citations in the corpus

Lower Bound on the Blow-up Rate of the Axisymmetric Navier-Stokes Equations (2007, positive): “2) has been improved in two recent preprints [3] and [13].”
ON THE LIOUVILLE THEOREM FOR THE STATIONARY NAVIER-STOKES EQUATIONS IN A CRITICAL SPACE (2011, result): “[6] G. Koch, N. Nadirashvili, G. Seregin and V. Sverak, Liouville theorems for the Navier?”
A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations (2010, methodology): “The authors in [15] proved such kind of Liouville theorems in the three-dimensional axially-symmetric case without swirl, or under the condition r|v| being bounded.”
Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations (2015, background): “2 in [13], they proved that for any bounded weak solution u of the unsteady Navier-Stokes equations in R3 × (−∞, 0), if u is axially symmetric with no swirl, thenu = (0, 0, b3(t)) for some bounded measurable function b3 : (−∞, 0) → R.”
Analyticity of the Stokes semigroup in spaces of bounded functions (2011, methodology): “For example, a blow-up argument was used to conclude non-existence of type I blow-up for axisymmetric solutions [36], [49] and solutions having continuously varying vorticity directions [33].”
On the interior regularity criteria and the number of singular points to the Navier-Stokes equations (2014, methodology): “Recently, for the axisymmetric Navier-Stokes equations, important progress has been made by ChenStrain-Yau-Tsai [4, 5] and Koch-Nadirashvili-Seregin-Šverák [14], who showed that the solution does not develop type I singularities (i.”
Criticality of the Axially Symmetric Navier-Stokes Equations (2015, background): “See, for instance, [1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 14, 15, 18, 20, 21, 22, 23, 24, 27] and many others.”
Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces (2018, background): “For more references on the Liouville problem for the stationary Navier-Stokes equations see also the articles [1], [3] and [6] and the references therein.”
The L∞-Stokes semigroup in exterior domains (2012, background): “A blow-up argument is a typical indirect argument to obtain an a priori upper bound for solutions; see [23,26,39,40] (also [21]) for semilinear heat equations and [28,31] for the Navier–Stokes equations.”
On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$ (2016, methodology): “ows the problem is solved by GilbargandWeinberger in[3], while the special case ofthe axially symmetric 3D ﬂows without swirl is studied recently by Korobkov, M. Pileckas and R. Russo in [5](see also [4]). In this paper we present three theorems, which present suﬃcient conditions to guarantee the tr”