Analysis Seminar

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Weak solutions to the Navier--Stokes inequality with arbitrary energy profiles

Speaker: Wojciech Ożański, Warwick University

Location: Warren Weaver Hall 1302

Date: Thursday, February 28, 2019, 11 a.m.

Synopsis:

In the talk we will focus on certain constructions of weak
solutions to the Navier-Stokes inequality,
\[ u \cdot \left( u_t - \nu \Delta + (u\cdot \nabla ) u+ \nabla p \right)
\leq 0\]
on $\mathbb{R}^3$. Such vector fields satisfy both the strong energy
inequality and the local energy inequality (but not necessarily solve the
Navier-Stokes equations). Given $T>0$ and a nonincreasing energy profile
$e: [0,T] \to [0,\infty )$ we will construct a weak solution to the
Navier--Stokes inequality that is localised in space and whose energy
profile $\| u(t)\|_{L^2 (\mathbb{R}^3 )}$ stays arbitrarily close to $e(t)$
for all $t\in [0,T]$.
The relevance of such solutions is that, despite not satisfying the
Navier-Stokes equations, they do satisfy the partial regularity theory of
Caffarelli, Kohn \& Nirenberg (Comm. Pure Appl. Math., 1982). In fact,
Scheffer's constructions of weak solutions to the Navier-Stokes inequality
with blow-ups (Comm. Math. Phys., 1985 \& 1987) show that the Caffarelli,
Kohn \& Nirenberg's theory is sharp for such solutions. Namely, his
construction admits a finite-time blow-up on a Cantor set whose Hausdorff
dimension is greater than $\xi$, for any preassigned $\xi \in (0,1)$.
Moreover, we will show how our approach can be used to obtain a stronger
result than Scheffer's. Namely, we obtain weak solutions to the
Navier-Stokes inequality with both blow-up and a prescribed energy profile.