Analysis Seminar

In the talk we will focus on certain constructions of weak solutions to the Navier--Stokes inequality (NSI), \[ u \cdot \left( u_t - \nu \Delta + (u\cdot \nabla ) u+ \nabla p \right) \leq 0\] 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 NSI 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 (Comm. Math. Phys., 1985 & 1987) admit a finite-time blow-up on a Cantor set, which shows that the Caffarelli, Kohn & Nirenberg theory is sharp for such solutions. We will discuss the main ideas of his constructions, as well as present a stronger result: a construction of weak solutions to the NSI which follow a prescribed energy profile at the times leading to the blow-up on the Cantor set.