The basic mathematical concepts used in the book are presumed in this chapter. The necessary function spaces framework — Lebesgue spaces, Sobolev spaces, and spaces of continuous functions — are introduced here. In order to underline the physical background of the function spaces, the exact definitions introduced in Chapter 2 are followed by the energy estimates deduced in Chapter 3 directly for any smooth solution of the underlying equations (a priori estimates). Several other fundamental concepts are also treated at length: average continuity of weak solutions, renormalized solutions of the continuity equation, and instantaneous values of the state variables.
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