I am a Computer Science student. For the fun I would like to have a better understanding about Navier-Stokes Existence and Smoothness problem. I am reading a paper from L. Caffarelli, R. Kohn and L. Nirenberg. The paper is titled: Partial Regularity of Suitable Weak Solutions of the Navier-Stokes Equations. Their are some inequalities which I do not understand. I would like to understand them so therefore I will describe them here.
Proposition 1. There are absolute constants $\epsilon_{1}$ and $C_{1} > 0$, and a constant $\epsilon_{2}(q)$ depending only on $q$, with the following property. Suppose $(u, p)$ is a suitable weak solution of the Navier-Stokes system on $Q_{1}$ with force $f \in L^{q}$, for some $q > \frac{5}{2}$, suppose further that \begin{align} \iint_{Q_{1}} \left(|u|^{3} + |u||p|\right) + \int^{0}_{-1} \left( \int_{|x| < 1} |p| dx \right)^{5/4} dt \leq \epsilon_{1} \end{align}
and \begin{align} \iint_{Q_{1}} |f|^{q} \leq \epsilon_{2} \end{align}
Then \begin{align} |u(x, t)| \leq C_{1} \end{align}
The proposition is later used in the paper as a hypothesis. My primary question is:
What does the following inequality means: \begin{align} \iint_{Q_{1}} \left(|u|^{3} + |u||p|\right) + \int^{0}_{-1} \left( \int_{|x| < 1} |p| dx \right)^{5/4} dt \leq \epsilon_{1} \end{align}
The primary question leads to two secondary questions: - How are such hypothesis established? - Is there literature available about proving dedicated about this field of Mathematics?
Could someone provide some help? Thank you very much.