55 research outputs found
A note on the Prodi-Serrin conditions for the regularity of a weak solution to the Navier-Stokes equations
The paper is concerned with the regularity of weak solutions to the
Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin
condition in order to obtain regularity for t > 0. The most interesting aspect
of the result is that no compatibility condition is required to the initial
data v_0\in J^2(\OO) J2({\Omega})
Phase transitions in the fractional three-dimensional Navier-Stokes equations
The fractional Navier-Stokes equations on a periodic domain
differ from their conventional counterpart by the replacement of the
Laplacian term by , where is the Stokes operator and is the viscosity
parameter. Four critical values of the exponent have been identified where
functional properties of solutions of the fractional Navier-Stokes equations
change. These values are: ; ; and
. In particular, in the fractional setting we prove an analogue
of one of the Prodi-Serrin regularity criteria (), an equation
of local energy balance () and an infinite hierarchy of
weak solution time averages (). The existence of our analogue
of the Prodi-Serrin criterion for suggests that the convex
integration schemes that construct H\"older-continuous solutions with epochs of
regularity for are sharp with respect to the value of
Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and NavierâStokes equations
Whether the 3D incompressible Euler and NavierâStokes equations can develop a finite-time singularity from smooth initial data with finite energy has been one of the most long-standing open questions. We review some recent theoretical and computational studies which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. We also investigate the dynamic stability of the 3D NavierâStokes equations and the stabilizing effect of convection. A unique feature of our approach is the interplay between computation and analysis. Guided by our local non-blow-up theory, we have performed large-scale computations of the 3D Euler equations using a novel pseudo-spectral method on some of the most promising blow-up candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. Moreover, we observe that the support of maximum vorticity becomes severely flattened as the maximum vorticity increases and the direction of the vortex filaments near the support of maximum vorticity is very regular. Our numerical observations in turn provide valuable insight, which leads to further theoretical breakthrough. Finally, we present a new class of solutions for the 3D Euler and NavierâStokes equations, which exhibit very interesting dynamic growth properties. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions
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