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I have noticed that the Navier Stokes equations can be written as a semilinear symmetric first order system $$ u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u) $$ for a 9 by 1 vector $u$ containing the velocities and momentum fluxes. The question is: Has such a system been studied?

To be specific, consider the system for velocity $u_1,u_2,u_3$ and a matrix $\varphi$: $$ \rho u_t + {\rm div\,}\varphi=0, $$ $$ \varphi = \rho uu^T+\frac{1}{3}({\rm tr\,}\varphi-\rho|u|^2)I-\mu Du-\mu (Du)^T, $$ and determine pressure by $p=\frac{1}{3}({\rm tr\,}\varphi-\rho|u|^2)$). These are equivalent, for smooth solutions, to the Navier Stokes equations. To set up the system define functions $u_4,\ldots,u_9$ by writing the flux matrix as $$ \varphi = \mu\begin{bmatrix} 2u_4 & u_5 & u_6 \cr u_5 & 2u_7 & u_8 \cr u_6 & u_8 & 2u_9 \end{bmatrix}. $$ The system becomes $$ \begin{array}{ccccccccccccc} \rho u_{1,t}\!\! & \ & \ &\!\!\!\!\!\!\!\!\!+2\mu u_{4,1}\!&\!\!\!\!\!\!+\mu u_{5,2}\!\!\!\!&\!\!\!+\mu u_{6,3}\! & \ & \ & \ & \!\!\!= & 0 \cr \ & \rho u_{2,t}\!\! & \ & \ &\!\!\!\!\!\!+\mu u_{5,1}\!& \ &\!\!\!\!\!\!+2\mu u_{7,2}\!&\!\!\!\!\!\!+\mu u_{8,3}\!& \ & \!\!\!= & 0 \cr \ & \ & \rho u_{3,t}\!\! & \ & \ &\!\!\!\!\!\!+\mu u_{6,1}\! & \ &\!\!\!\!\!\!\!\!+\mu u_{8,2}\!\!\!\!&\!\!\!+2\mu u_{9,3}\! & \!\!\!= & 0 \cr 2\mu u_{1,1}\!\! & \ & \ & \ & \ & \ & \ & \ & \ & \!\!\!= & \!\!\!\rho u_1^2\!+\!p-2\mu u_4 \cr \mu u_{1,2}\!\! &\!\!\!\!\!\!+\mu u_{2,1}\!& \ & \ & \ & \ & \ & \ & \ & \!\!\!= & \rho u_2u_1-\mu u_5 \cr \mu u_{1,3}\!\! & \ &\!\!\!\!\!\!+\mu u_{3,1}\!\! & \ & \ & \ & \ & \ & \ & \!\!\!= & \rho u_3u_1-\mu u_6 \cr \ & 2\mu u_{2,2}\!\! & \ & \ & \ & \ & \ & \ & \ & \!\!\!= & \!\!\!\rho u_2^2\!+\!p-2\mu u_7 \cr \ & \mu u_{2,3}\!\!&\!\!\!\!\!\!+\mu u_{3,2}\!\! & \ & \ & \ & \ & \ & \ & \!\!\!= & \rho u_3u_2-\mu u_8 \cr \ & \ & 2\mu u_{3,3}\!\! & \ & \ & \ & \ & \ & \ & \!\!\!= & \!\!\!\rho u_3^2\!+\!p-2\mu u_9 \cr \end{array} $$ and $p = \frac{1}{3}\big(2\mu(u_4+u_7+u_9)-\rho(u_1^2+u_2^2+u_3^2)\big)$.

Edit: I would like to add a reason for my curiosity about this system. Successive approximations can be made which are positive symmetric in the sense of Friedrichs (write-up here ) and the admissible boundary conditions, in addition to the usual initial and boundary conditions, include some I haven't seen.(I'm asking about one of those on physics stack exchange.)

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  • $\begingroup$ It looks like you only have an equation for the first three components $u_{1,t}, u_{2,t}, u_{3,t}$ of $u_t$, but not for the remaining six components $u_{4,t}, \dots, u_{9,t}$, so this is not yet in the form of a first-order system. $\endgroup$ – Terry Tao Apr 19 '14 at 0:14
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    $\begingroup$ Also, Navier-Stokes is a parabolic equation rather than a hyperbolic one, so it is highly unlikely that a first-order system formulation is viable. (As a warmup, you might try to see if you can write the 1D heat equation $u_t + u_{xx}=0$ (the simplest example of a parabolic equation) as a first-order system.) $\endgroup$ – Terry Tao Apr 19 '14 at 0:18
  • $\begingroup$ @Terry Tao: I agree it is certainly not hyperbolic. $\endgroup$ – Bob Terrell Apr 19 '14 at 11:46

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