arXiv subjects

Witold Sadowski

Publications and source records attributed to Witold Sadowski.

2 recordsLinked to original sources

arXiv2021

Robustness of Regularity for the $3$D Convective Brinkman-Forchheimer Equations

We prove a robustness of regularity result for the $3$D convective Brinkman-Forchheimer equations $$ \partial_tu -μΔu + (u \cdot \nabla)u + \nabla p + αu + β\abs{u}^{r - 1}u = f, $$ for the range of the absorption exponent $r \in [1, 3]$ (for $r > 3$ there exist global-in-time regular solutions), i.e. we show that strong solutions of these equations remain strong under small enough changes of the initial condition and forcing function. We provide a smallness condition which is similar to the robustness conditions given for the $3$D incompressible Navier-Stokes equations by Chernyshenko et al. (2007) and Dashti & Robinson (2008).

math.AP↗

arXiv2006

Decay of weak solutions and the singular set of the three-dimensional Navier-Stokes equations

We consider the behaviour of weak solutions of the unforced three-dimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ($\|u\|^2=\int|u|^2$) but infinite enstrophy ($\|Du\|^2=\int|{\rm curl} u|^2$). We show that this has to be reflected in the solution for small times, so that in particular $\|Du(t)\|\to+\infty$ as $t\to0$. We also give some limitations on this `backwards blowup', and give an elementary proof that the upper box-counting dimension of the set of singular times can be no larger than one half. Although similar in flavour, this final result neither implies nor is implied by Scheffer's result that the 1/2-dimensional Hausdorff measure of the singular set is zero.

math.AP↗