Liouville theorems for the Navier–Stokes equations and applications

2009/01/01 by Gabriel Koch, Gabriel S. Koch, Nikolaï Nadirashvili +5 · 300 citations
Mathematics · #Advanced Mathematical Physics Problems #Bounded function #Chen #Compressibility #Constant (computer programming) #Geometry #Mathematical analysis #Mathematics #Navier-Stokes equation solutions #Navier–Stokes equations #Nonlinear Partial Differential Equations #Physics #Pure mathematics #Rotational symmetry #Singularity #Space (punctuation)

paper · pdf · doi:10.1007/s11511-009-0039-6

published in Acta Mathematica 203(1), 83-105 (Mittag-Leffler Institute)

openalex publication_date 2009/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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].

Citations

• Sur le mouvement d'un liquide visqueux emplissant l'espace
• Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities

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