How to solve Nonlinear transport equation (Hamilton-Jacobi)

Question

I tried to solve the next nonlinear PDE with fenics

$$\frac{\partial u}{\partial t}+\nu(\vec{x},t) \|\nabla u\| = 0$$

How to solve this equation with fenics if $\nu$ is known?

first
$$\frac{\partial u_{k}}{\partial t}\approx\frac{u_{k}-u_{k-1}}{\Delta t}$$
therefore
$$u_{k}\left(2u_{k-1}-u_{k}\right)+\left(\Delta t\right)^{2}\nu_{k}^{2}\nabla u_{k}\cdot\nabla u_{k}=u_{k-1}^{2}$$

but I can't to make the bilinear form of this equation

Comments

This isn't a FEniCS questions - it's a finite element method question.

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Try asking at scicomp.stackexchange.com.