Necessary conditions of potential blow up for Navier-Stokes equations
Assuming that $T$ is a potential blow up time, we show that $H^\frac 12$-norm of the velocity field goes to $\infty$ as time $t$ approaches $T$
math.AP↗
arXiv subjects
Gregory Seregin
Publications and source records attributed to Gregory Seregin.
On Divergence-free Drifts
We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t - Δ+b\cdot\nabla$ and $-Δ+b\cdot\nabla$ with a divergence-free drift $b$. We prove the Liouville theorem and Harnack inequality when $b\in L_\infty(BMO^{-1})$ resp. $b\in BMO^{-1}$ and provide a counterexample to such results demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BMO skew-symmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the non-scale-invariant norm $\|b\|_{L_1}$. In three dimensions, on the other hand, bounded solutions with $L_1$ drifts may be discontinuous.
math.AP↗
Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker-Planck equations
We provide a proof of global regularity of solutions of coupled Navier-Stokes equations and Fokker-Planck equations, in two spatial dimensions, in the absence of boundaries. The proof yields a priori estimates for the growth of spatial gradients.
math.AP↗
Holder Continuity of Solutions of 2D Navier-Stokes Equations with Singular Forcing
We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.
math.AP↗
Navier-Stokes equations: almost $L_{3,\infty}$-case
A sufficient condition of regularity for solutions to the Navier-Stokes equations is proved. It generalizes the so-called $L_{3,\infty}$-case.
math.AP↗
On Smoothness of $L_{3,\infty}$-Solutions to the Navier-Stokes Equations up to Boundary
We show that $L_{3,\infty}$-solutions to the three-dimensional Navier-Stokes equations near a flat part of the boundary are smooth.
math.AP↗