# derivation of variational forms of PDE directly from conservation form

A lot of texts derive the variational form of a PDE as follows. First, life begins with a conservation law for the field $$q$$:

$$\partial_t \int_\omega G(q)\;dx + \int_{\partial\omega} F(q, \nabla q, \ldots)\cdot\nu\;ds = \int_\omega f\;dx$$

for all control volumes $$\omega$$, where $$\nu$$ is the unit outward normal vector to $$\partial\omega$$, $$F$$ is the flux function, $$G$$ takes $$q$$ to the density of some extensive quantity, and $$f$$ is a source term. We then apply the divergence theorem (assuming things are nice enough) to arrive at a strong form

$$\partial_t G(q) + \nabla\cdot F(q, \nabla q, \ldots) = f.$$

Then we multiply everything by a test function $$v$$ and push the divergence over onto $$v$$ using the usual tricks. So the argument goes conservation law $$\rightarrow$$ strong form $$\rightarrow$$ weak or variational form. I find the indirectness kind of annoying. Can we go directly from a conservation law to a variational form instead?

You could imagine that, for any $$u$$,

$$\int_\omega u\;dx = \int_\Omega u\cdot \mathbf 1_\omega\;dx$$

where $$\mathbf 1_\omega$$ is the indicator function of the set $$\omega$$. Likewise, assuming that the boundary of $$\omega$$ is nice enough, for some reasonable class of vector fields $$F$$, we can also say that

$$\int_{\partial\omega} F\cdot\nu\;ds = -\int_\Omega F\cdot\nabla\mathbf 1_\omega\;dx$$

in a distributional sense. Substituting these two relations into our original conservation law, we can then take arbitrary scalar multiples and sums to arrive at a variational form

$$\int_\Omega\left\{\partial_tG(q)\cdot v - F(q, \nabla q, \ldots)\cdot\nabla v - f\cdot v\right\}\,dx = \ldots$$

in a distributional sense for any simple function $$v$$, i.e. $$v = \sum_i\alpha_i\mathbf{1}_{\omega_i}$$ for some finite collection of control volumes $$\{\omega_i\}$$. (I've left off some boundary terms for brevity.) We can then take a limit as $$v$$ approaches a function in, say, $$H^1(\Omega)$$ or whatever space of test functions is most appropriate.

It's been a few years since I had measure theory or distribution theory. Can this argument be made rigorous, without an intermediate assumption that the PDE has a strong solution? I'd be perfectly happy if it worked for some common problem or class of problems; for example, the generalized Poisson equation where the conductivity coefficient has a jump discontinuity across some smooth hypersurface. I don't know that doing so has much practical use, I just find it more satisfying. Virtually every book I have on Galerkin methods or PDE theory either goes the conservation / strong / variational route or just starts with variational forms and ignores their derivation entirely.

Alright guess I'll have to try and do it myself then. Assumptions: $$\Omega$$ is a nice enough compact domain that $$C^\infty(\Omega)$$ is dense in $$H^1(\Omega)$$, and $$u$$ is a solution of the conservation form of the diffusion equation -- for all smooth control volumes $$\omega$$,

$$-\int_{\partial\omega} k\nabla u\cdot\nu\;ds = \int_\omega f\; dx$$

where $$\nu$$ is the unit outward normal vector, $$k$$ is the (strictly positive) diffusion coefficient, and $$f$$ is in $$L^2(\Omega)$$. We'll take the boundary conditions to be $$u|_{\partial\Omega} = 0$$ for simplicity; adding an extra term takes care of the Neumann or Robin cases. Assume additionally that $$u$$ is in $$H_0^1(\Omega)$$. Let $$v$$ be an arbitrary smooth function that vanishes on $$\partial\Omega$$. Using the smooth coarea formula, we can write

$$\int_\Omega k\nabla u\cdot\nabla v\;dx = \int_\Omega k\nabla u\cdot \frac{\nabla v}{|\nabla v|}|\nabla v|\; dx = \int_{-\infty}^\infty\int_{\partial\{x: v(x) \ge t\}}k\nabla u\cdot\frac{\nabla v}{|\nabla v|}\;ds\;dt\ldots$$

but we also know that, if $$t$$ is a regular value of $$v$$, the unit outward normal to the hypersurface $$v^{-1}(t)$$ is equal to $$-\nabla v/|\nabla v|$$, hence

$$\ldots = -\int_{-\infty}^\infty\int_{\partial\{x: v(x) \ge t\}}k\nabla u\cdot\nu\;ds\;dt = \int_{-\infty}^\infty\int_{\{x: v(x) \ge t\}}f\;dx\;dt\ldots$$

The latter equality is because of our assumption that $$u$$ is a solution of the conservation form of the equations and that $$v$$ is smooth + Sard's theorem. Finally the latter integral is just a rearrangement of the integral

$$\ldots = \int_\Omega f\,v\;dx$$

We can then pass to the limit for $$v$$ an arbitrary element of $$H_0^1(\Omega)$$ to say that $$u$$ is a solution of the variational problem

$$\int_\Omega k\nabla u\cdot\nabla v\;dx = \int_\Omega fv\; dx$$

for all $$v$$, QED.

I thought this would directly involve some distribution theory in order to be able to make statements like the gradient of an indicator function being a surface measure. But the coarea formula lets you shove all that under the hood. The same idea (+ some extra terms for initial conditions) seems like it should work for more problems as long as they can be expressed as a divergence in spacetime.

Please let me know if there are any mistakes or holes in the argument.