Non-uniqueness of weak solutions to hyperviscous Navier–Stokes equations: on sharpness of J.-L. Lions exponent
Tianwen Luo, Edriss S. Titi
Abstract
Abstract Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity $$(-\Delta )^{\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>θ</mml:mi></mml:msup></mml:math> , whenever the exponent $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>θ</mml:mi></mml:math> is less than Lions’ exponent 5/4, i.e., when $$\theta < 5/4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>θ</mml:mi><mml:mo><</mml:mo><mml:mn>5</mml:mn><mml:mo>/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> .