I have a function $$ g\left(x,v,t\right) = u\left(x,t\right)\cdot v + \theta\left(x,t\right)\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(x,t), v\in \mathbb R^N,N=2,3$. I also have a matrix valued function $X=X(v)\in\mathbb R^{N\times N}$, satisfying $$ \int_{\mathbb R^N} X \xi \, \text{d}v = 0 \qquad\text{ if }\quad\xi = 1,v_1,v_2,v_3,\left\lvert v\right\rvert^2 $$ and I write for shorthand $$ A := v\otimes v - \frac{1}{3} \left\lvert v\right\rvert^2 I $$ Where $I$ is the $n\times n$ identity matrix. I'm trying to verify the following integral identity: $$ \int_{\mathbb R^N} X v\cdot\nabla_xg \, \text{d}v = \nu \left[ \nabla_x u + \left(\nabla_xu\right)^T - \frac{2}{3} \,\nabla\cdot u I \right] $$ with proportionality constant $\nu = \int_{\mathbb R^N} X : A \ \text{d}v$.
For context this is the line on page 74 that starts "With (6.16) one easily finds that" in the following book chapter on the Boltzmann equation and its hydrodynamic limits. You might (rightly) guess that $\nu$ is the viscosity of the limiting solution which will solve the Navier-Stokes-Fourier system.
I'm honestly at a loss at how to arrive at the right hand side. I can't get very far other than the first obvious step,
$$ \nabla_x g = \left(\nabla_x u\right)^T v + \nabla_x \theta\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$
My immediate issues:
- how do I express the integral of the first term in a useful way?
- I suspect the second integral should disappear, but this isn't obvious if I only assume the $L^2$ orthogonality conditions $\int_{\mathbb R^N} X \xi \, \text{d}v = 0 \;\text{ if }\;\xi = 1,v,\left\lvert v\right\rvert^2$ because of the extra $v$ in $v\cdot \nabla_x$.
I would be very happy with just a first step even, or a source for how to deal with these kinds of integrals (seems a bit too specific but I can hope!)
(edit): Just in case it is useful, $X = \tilde A M_{1,0,1}$ where $M_{1,0,1}(v) = \dfrac{1}{\left(2\pi\right)^{3/2}}\,\exp\left(\dfrac{\left\lvert v\right\rvert^2}{2}\right)$ is the 'Maxwellian velocity distribution', and $\tilde A$ is the unique solution $L^2(M\,\text{d}v)$-orthogonal to $\operatorname{span}\left(1,v_1,v_2,v_3,\left\lvert v\right\rvert^2\right)$ to the equation $\mathcal{L} \tilde A = A$, where $\mathcal{L}$ is the linearization of the quadratic hard-sphere Boltzmann collision operator $Q$.
Just in case this is useful, I also type up the only useful looking computation I have: \begin{align} \int X_{ij} v\cdot \nabla_x g \, \text{d}v &= \int X_{ij}\left[ v^lv^k \partial_{x^k}u^l + \partial{x^k} \theta v^k \dfrac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right)\right]\text{d}v \\ &= \int X_{ij}\left[\left(v\otimes v : \nabla_x u\right) + \nabla_x \theta \cdot \left(\dfrac{v\left\lvert v\right\rvert^2 - 5v}{2}\right)\right] \text{d}v \end{align}