Construction of characteristic functions for the system of Navier--Stokes--Voigt equations and the BBM equation

A. P. Oskolkov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 10, Tome 69 (1977), pp. 136-148

Résumé

For the system of Navier–Stokes–Voigt equations \begin{equation} \frac{\partial\vec v}{\partial t}-\nu\Delta\vec v-x\frac{\partial\Delta\vec v}{\partial t}+v_k\frac{\partial\vec v}{\partial x_k}+\operatorname{grad}p=0,\quad \operatorname{div}\vec v=0 \tag{1} \end{equation} and the BBM equation \begin{equation} \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}-\frac{\partial^3v}{\partial t\partial x^2}=0 \tag{2} \end{equation} characteristic functions $\mathscr F(\vec \theta;t)$ of the measure $\mu_t(\omega)=\mu(V^{-1}_t(\omega))$, describing the evolution in time of the probability measure $\mu(\omega)$ defined on the set of initial conditions for the first initial boundary-value problem for system (1) or Eq. (2) are constructed and investigated. It is shown that the characteristic functions $\mathscr F(\overset{\to}\theta;t)$ constructed satisfy partial differential equations with an infinite number of independent variables $(t;\theta_1,\theta_2,\dots)$ [the statistical equations of E. Hopf for the system (1) or Eq. (2)].