Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces
Antonio Agresti, Mark Veraar
Abstract
Abstract In this paper we study the stochastic Navier–Stokes equations on the d -dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case $$\mathbb {B}^{d/q-1}_{q,p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mi>q</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> for $$q\in [2,2d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions $$d\geqslant 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Moreover, for $$d=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.