Suitable weak solution

On an open cylinder $Q\subset\mathbb R^3\times\mathbb R$ , a suitable weak solution is a weak Navier–Stokes pair with local energy bounds

u\in L^\infty_{t,\rm loc}L^2_{x,\rm loc}\cap L^2_{t,\rm loc}H^1_{x,\rm loc}, \qquad p\in L^{3/2}_{\rm loc}(Q),

that obeys the local energy inequality. Here the bounds hold on every smaller cylinder compactly contained in $Q$ . For this definition take $f\in L^2_{\rm loc}(Q)$ ; more general forces are allowed when the weak equation and the product $f\cdot u$ are well defined and the particular theorem permits them.

The notation uses mixed Lebesgue norms and local Sobolev spaces; the pressure has the specified local integrability power.

Local integrability of the energy flux

The three-dimensional Sobolev inequality and interpolation give $u\in L^{10/3}_{\rm loc}(Q)$ , hence $u\in L^3_{\rm loc}(Q)$ . Hölder's inequality makes $pu$ , $|u|^2u$ , and $f\cdot u$ locally integrable. Suitability is a local condition; a global initial-value statement may also impose the Leray–Hopf conditions.

References

Caffarelli, Kohn and Nirenberg, Partial regularity of suitable weak solutions (1982).