On a non-solenoidal approximation to the incompressible Navier-Stokes equations

We establish an asymptotic profile that sharply describes the behavior as $t\to\infty$ for solutions to a non-solenoidal approximation of the incompressible Navier-Stokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical Navier-Stokes, e.g., in $L^3\_{\rm loc} (R^+ \times R^3)$, provided $\epsilon\to0$, where $\epsilon>0$ is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of Navier-Stokes for large times: indeed, its solutions can decay much slower as $t\to+\infty$ than the corresponding solutions of Navier-Stokes.Comment: Submitted to the Journal of the London Mathematical Society (under revision

Characterization of solutions to dissipative systems with sharp algebraic decay

Post refereeing version. To appear on SIAM J. Math. Anal.International audienceWe characterize the set of functions $u_0\in L^2(R^n)$ such that the solution of the problem $u_t=\mathcal{L}u$ in $R^n\times(0,\infty)$ starting from $u_0$ satisfy upper and lower bounds of the form $c(1+t)^{-\gamma}\le \|u(t)\|_2\le c'(1+t)^{-\gamma}$.Here $\mathcal{L}$ is in a large class of linear pseudo-differential operator with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discussed: in particular our characterization provides necessary and sufficient conditions on $u_0$ for a solution of the Navier--Stokes system to satisfy sharp upper-lower decay estimates as above.In doing so, we will revisit and improve the theory of \emph{decay characters} by C. Bjorland, C. Niche, and M.E. Schonbek, by getting advantage of the insight provided by the Littlewood--Paley analysis and the use of Besov spaces