5,181 research outputs found
A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation
We prove that every weak solution to the 3D Navier-Stokes equation that
belongs to the class and \n u belongs to localy
away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized
energy equality. In particular every such solution is suitable.Comment: 10 page
Partial Regularity of solutions to the Four-dimensional Navier-Stokes equations at the first blow-up time
The solutions of incompressible Navier-Stokes equations in four spatial
dimensions are considered. We prove that the two-dimensional Hausdorff measure
of the set of singular points at the first blow-up time is equal to zero.Comment: 19 pages, a comment regarding five or higher dimensional case is
added in Remark 1.3. accepted by Comm. Math. Phy
Nonlinear softening as a predictive precursor to climate tipping
Approaching a dangerous bifurcation, from which a dynamical system such as
the Earth's climate will jump (tip) to a different state, the current stable
state lies within a shrinking basin of attraction. Persistence of the state
becomes increasingly precarious in the presence of noisy disturbances. We
consider an underlying potential, as defined theoretically for a saddle-node
fold and (via averaging) for a Hopf bifurcation. Close to a stable state, this
potential has a parabolic form; but approaching a jump it becomes increasingly
dominated by softening nonlinearities. If we have already detected a decrease
in the linear decay rate, nonlinear information allows us to estimate the
propensity for early tipping due to noise. We argue that one needs to extract
information about the nonlinear features (a "softening") of the underlying
potential from the time series to judge the probability and timing of tipping.
This analysis is the logical next step if one has detected a decrease of the
linear decay rate. If there is no discernable trend in the linear analysis,
nonlinear softening is even more important in showing the proximity to tipping.
After extensive normal form calibration studies, we check two geological time
series from paleo-climate tipping events for softening of the underlying well.
For the ending of the last ice age, where we find no convincing linear
precursor, we identify a statistically significant nonlinear softening towards
increasing temperature. The analysis has thus successfully detected a warning
of the imminent tipping event.Comment: 22 pages, 11 figures, changed title back, corrected smaller mistakes,
updated reference
Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations
We present new interior regularity criteria for suitable weak solutions of
the 3-D Navier-Stokes equations: a suitable weak solution is regular near an
interior point if either the scaled -norm of the velocity
with , , or the -norm of the
vorticity with , , or the
-norm of the gradient of the vorticity with , , , is sufficiently small near
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