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1609.05070
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{Introduction}Behavior of complex fluids, i.e. fluids involving an internal structure (either induced by external forces - as e.g. structures in turbulent flows - or structures of suspended particles or macromolecules), is not well described by classical fluid mechanics. A more microscopic theory (i.e. a theory involving more details) is needed. The Boltzmann kinetic theory (in which the one particle distribution function $f(\rr,\vv)$ is playing the role of the state variable; $\rr$ is the position vector and $\vv$ momentum of one particle)appears to be a natural candidate for such theory. Of course, the Boltzmann theory addresses only the behavior of ideal gases but it can still serve as a useful starting point for developing an extended fluid mechanics. Indeed, Grad's reformulation of the Boltzmann kinetic equation (consisting of replacing $f(\rr,\vv)$ with an infinite number of fields $(c^{(0)}(\rr),...,c^{(\infty)}(\rr))$)has played a significant role in such research.Among the important questions that arise in the investigations of the passage \textit{Boltzmann theory} $\rightarrow$ \textit{extended fluid mechanics} we mention the following:How does the Boltzmann kinetic theory reduce to mesoscopic fluid mechanics theories involving less microscopic details?What are the properties of solutions of the Boltzmann kinetic equation that do and that do not pass to the mesoscopic fluid mechanics theories and eventually to equilibrium thermodynamics? What are the new properties of solutions that emerge in reduced theories?To answer this type of questions, we proceed in the following two steps.First, in Section \ref{BE}, we recognize in the Boltzmann kinetic equation a structure of physical significance and then we keep it in Grad's reformulation and in reductions. By a "structure of physical significance" we mean a structure of the time evolution equations that guarantees that their solutions agree with results of certain basic experimental observations. Our focus on structures is a mathematical representation of our focus on physical understanding and experimental validation of reduced theories.The structure that we require to be passed from the Boltzmann equation to all its reformulations and eventually to all its reductions is the following (see more details in Section \ref{BE}): (i) the vector field of the Boltzmann equation is a sum of time reversible part (the free flow term) and the time irreversible part (the Boltzmann collision term), (ii) the time reversible part represents Hamiltonian dynamics (i.e. it has the form $LE_f$, where $E_f=\frac{\partial E}{\partial f(\rr,\vv)}$ is the gradient of the energy $E(f)$, $f(\rr,\vv)$ is the one particle distribution function, and $L$ is a Poisson bivector transforming a co-vector into a vector), (iii) the time reversible part possesses also the Godunov structure, (iv) the entropy remains unchanged during the time reversible time evolution, (v)the time irreversible part represents a generalized gradient dynamics in which the energy $E(f)$ remains unchanged and the entropy growths. The structure (ii) is physically significant because it expresses the compatibility of the Boltzmann kinetic theory with mechanics (the classical mechanics of particles inherited in the reduced description that uses $f(\rr,\vv)$ as state variable). The structures (iv) and (v) express the compatibility of the Boltzmann theory with thermodynamics (i.e. an agreement between theoretical predictions and experimentally observed approach to equilibrium at which the classical equilibrium thermodynamics is found to describe well the observed behavior).Having identifies the mathematical structure of importance, we then make, in the second step, Grad's reformulation separately for every structure.Reduced equations (i.e. governing equations of extended fluid mechanics) are constructed as particular realizations of the Grad form of the structures. In fact, we are constructing in this way both reduced equations (i.e. equations governing the time evolution of $(c^{(0)}(\rr),...,c^{(\infty)}(\rr))$ ) and reducing equations (i.e. equations governing the time evolution of $(c^{(N+1)}(\rr),...,c^{(\infty)}(\rr))$).This two-step approach to reductions distinguishes the investigation presented in this paper from the investigations reported in\cite{MR}, \cite{Jou},\cite{Rugg}, \cite{Struch}. After developing our approach we discuss briefly its relation to other approaches in Section \ref{Compoth}.
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