Eric A. Carlen, Michael Loss
Optimal Smoothing and Decay Estimates for Viscously Damped Conservation Laws, with Application to the 2-D Navier Stokes Equation
The paper is published: Duke Math. J. 81, 1 (1995) 135-157

MSC:

35B40 Asymptotic behavior of solutions

35L65 Conservation laws

35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07,

Abstract: Optimal bounds on the $L^p(\IR^n)$ smoothing and decay
are established for certain viscously damped conservations laws, of which
the vorticity formulation of the Navier--Stokes equation on $\IR^2$ is a
basic example. From the smoothing bounds, we obtain pointwise bounds
that provide optimal control on the spatial decay of solutions. We
apply this in a study of the physically important case
in which the integral of
the initial data (i.e., the total vorticity in the example) vanishes.
We show in this case that as
the time $t$ increases, the $L^1(\IR^n)$ norm of the solution decays to
zero in two stages: for large initial data, there is a slow decay period
during which the $L^1(\IR^n)$ norm falls off with an inverse power of
the logarithm of $t$. Then, once the norm has fallen below a critical value,
it decays away to zero with $t^{-1/2}$.
Again, this is optimal, and all of the constants
in these estimates are explicitly computable in terms of the initial data.

Keywords: vorticity; $L\sb p-L\sb q$-estimate