Variational formulation of the Burgers' equation

Question

I am using quantum computing for solving several PDEs, and within this framework it is easier to rewrite PDEs as an optimization problem. Just as small example with the Poisson problem:

$$ -\nabla^2 u = f \quad \rightarrow \quad\min_{u} \int_\Omega \left(\frac{1}{2}|\nabla u|^2 - fu \right) d\Omega $$

Or the heat equation

$$ u_t -\nabla^2 u = f \quad \rightarrow \quad\min_{u} \int_\Omega \left(\frac{1}{2} (u^2)_t + \frac{1}{2}|\nabla u|^2 - fu \right) d\Omega \quad \forall t $$

The problem now is to find a functional $J(u)$ such that its minimization leads to the solution $u$ of the Burgers' equation (in 1D):

$$ u_t + u_x u - \nu u_{xx} = 0 $$

Note that I am neglecting the boundary conditions for brevity.

What I have tried so far:

• I tried to write the weak formulation of the Burgers' equation as $a(u;u,v) = \int_\Omega u_tv + u_x uv + \nu u_{x}v_x = 0$ for all $v=v(x)$, and set $J(u)=a(u;u,u)$. But, in this case, the convective term vanishes when computing the Fréchét derivative of $J(u)$, leading to the heat equation.
• I tried to have a look in the literature and I have found this method from He (https://doi.org/10.1016/S0960-0779(03)00265-0) which allows to write the variational formulation of any transport PDE, but I can't make it work. Even here, when I verify the Fréchét derivative, I get something different from the Burgers' equation.
• I tried to readapt the Navier-Stokes case (like in https://arxiv.org/pdf/1802.06606.pdf) by eliminating the additional terms, but I am not really able to do it, as most of the Navier-Stokes functional are the result of an integration over space and time, while I need something like

$$ \min J(u) = \min \int_\Omega j(u(x,t)) d\Omega \quad \forall t $$

PS: I am aware that saying "variational formulation" is an abuse of language, but due to my engineering background, I do not know exactly how this kind of formulation is called.

Answer 1

Let us follow He (2004). Setting $v_x = u$, we introduce the time-dependent cost functional $$ J(v) = \int_\Omega \left\lbrace f(t) \left(-\tfrac12 v_xv_t - \tfrac16 v_x^3\right) + F\right\rbrace d\Omega $$ where $f$ is an integrating factor to be determined. Making the first variation vanish, we get $$ \tfrac12 (f v_{x})_t + \tfrac12 (f v_{t})_x + \tfrac12 (f v_x^2)_x + \frac{\delta F}{\delta u} = 0 , $$ i.e. $$ \frac{f_t}{2f} v_{x} + v_{xt} + v_x v_{xx} + \frac{\delta F}{f\delta u} = 0 , $$ from which we deduce $f = \text{const}$ and ${\delta F}/{\delta u} = -\nu f u_{xx}$. Finally, according to this method, the choice $$ J(v) = \int_\Omega \left(-\tfrac12 v_xv_t - \tfrac16 v_x^3\right) + \tfrac12 \nu (u_x)^2\, d\Omega $$ should work (provided all the above steps are correct).

To recover the Burgers' equation case from the Navier-Stokes case, it suffices to carry out the minimisation of the proposed cost function in one space dimension while discarding the incompressibility constraint. This way, no Lagrangian multiplier $p$ appears, and we get the Burgers equation.

Note: Formulations of the heat equation as a minimisation problem are not as straightforward as suggested in OP, see related post. To deal with the odd derivative in $v_{xt} + v_x v_{xx} = \nu v_{xxx}$, one might also use Lagrange multipliers or other tricks.