Stokes Equations In Critical Spaces Research Articles

Maximal $L^{1}$-regularity and free boundary problems for the incompressible Navier–Stokes equations in critical spaces

Maximal $L^{1}$-regularity and free boundary problems for the incompressible Navier–Stokes equations in critical spaces

Long-time asymptotics of axisymmetric Navier–Stokes equations in critical spaces

Long-time asymptotics of axisymmetric Navier–Stokes equations in critical spaces

Near-invariance under dynamic scaling for Navier–Stokes equations in critical spaces: a probabilistic approach to regularity problems

We make a detailed comparison between the Navier–Stokes equations and their dynamically scaled counterpart, the so-called Leray equations. The Navier–Stokes equations are invariant under static scaling transforms, but are not generally invariant under dynamic scaling transforms. We will study how closely they can be brought together using the critical dependent variables and discuss the implications on the regularity problems. Assuming that the Navier–Stokes equations written in the vector potential have a solution that blows up at t = 1, we derive the Leray equations by dynamic scaling. We observe: (1) the Leray equations have only one term extra on top of those of the Navier–Stokes equations; (2) we can recast the Navier–Stokes equations as a Wiener path integral and the Leray equations as another Ornstein–Uhlenbeck path integral. Using the Maruyama–Girsanov theorem, both equations take the identical form modulo the Maruyama–Girsanov density, which is valid up to by the Novikov condition; (3) the global solution of the Leray equations is given by a finite-dimensional projection of a functional of an Ornstein–Uhlenbeck process and a probability measure. If remains smooth beyond t = 1 under an absolute continuous change of the probability measure, we can rule out finite-time blowup by contradiction. There are two cases: (A) given by a finite number of Wiener integrals, and (B) otherwise. Ruling out blowup in (A) is straightforward. For (B), a condition based on a limit passage in the Picard iterations is identified for such a contradiction to come out. The whole argument equally holds in for any .

An alternative approach to regularity for the Navier–Stokes equations in critical spaces

In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier–Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space H˙12 do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Šverák using a different approach. We use the method of “concentration-compactness” + “rigidity theorem” using “critical elements” which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authorsʼ knowledge, this is the first instance in which this method has been applied to a parabolic equation.